A New Index for Quantifying Stability Adaptation Effort in Power Electronics-Dominated Power Systems Based on Frequency-Domain Impedance Identification
The grid following converter requires just 37% of the parameter adjustment needed by the grid forming unit to stabilize an otherwise unstable power system — a finding that could reshape how engineers assign control responsibilities in renewable-heavy grids (García-Reyes et al., 2026). As solar farms, wind turbines, and battery storage replace spinning turbines, power systems are becoming faster, more responsive, and far less predictable. The old rules of inertia and synchronous stability no longer apply. Instead, engineers face a new challenge: maintaining stability in systems dominated by power electronics — devices that behave more like software than steel. These converters, which interface renewable sources with the grid, are often treated as black boxes: their internal dynamics hidden, their control algorithms proprietary. This opacity breaks traditional stability analysis tools, which rely on full knowledge of system dynamics. The result? A growing risk of unexpected oscillations, voltage collapses, and blackouts — not from hardware failure, but from invisible mismatches in control logic. Now, a team of researchers from Spain introduces the Adaptation Effort Index (AEI), a new metric that quantifies not just whether a system is stable, but how hard it is to fix when it isn’t. Unlike previous methods, AEI works without needing access to internal models. It’s a compass for navigating the control complexity of tomorrow’s grid — and it suggests that not all converters are created equal when it comes to restoring stability.
The Science
Modern power systems are undergoing a silent revolution. For over a century, electricity grids relied on large rotating generators — coal, gas, hydro — whose physical inertia naturally dampened disturbances. A sudden load change would be absorbed by the kinetic energy stored in spinning rotors, giving operators time to respond. But today, that inertia is vanishing. Wind turbines use power converters to connect to the grid, decoupling their mechanical rotation from electrical frequency. Solar panels do the same. Even batteries, the backbone of energy storage, communicate with the grid through inverters. These power electronic converters are fast, efficient, and flexible — but they don’t inherently provide inertia. Instead, they must be programmed to mimic grid-supportive behaviors like frequency regulation and voltage control.
This shift has created a new class of instability: sub-synchronous oscillations, harmonic resonances, and poorly damped modes that emerge from interactions between converters and the grid. Diagnosing these issues is hard because many converter manufacturers treat their control algorithms as trade secrets. Engineers often receive only input-output data — black box models — making traditional eigenvalue analysis impossible. Eigenvalue methods require full state-space representations, which are unavailable when internal dynamics are hidden.
Frequency-domain impedance-based methods, such as the Generalized Nyquist Criterion (GNC), offer a workaround. By measuring how a converter responds to small perturbations across frequencies, engineers can assess stability without knowing its internal structure. The GNC plots the eigenvalues of the impedance ratio matrix and checks whether they encircle the critical point (-1,0) in the complex plane. If they do, the system is unstable. This method is powerful, but it has a critical limitation: it tells you that a system is unstable, but not what to do about it. It doesn’t indicate which converter should be retuned, nor how much effort that retuning would require.
To bridge this gap, García-Reyes et al. propose the Adaptation Effort Index (AEI), a two-part metric designed for real-world, model-agnostic applications. The first component is the Nyquist Stability Margin (NSM), derived directly from impedance measurements. The NSM quantifies how close the system is to instability — essentially, the shortest distance from the critical point to the nearest eigenlocus in the Nyquist plot. A smaller margin means the system is closer to instability.
The second component is the Modal Sensitivity Index (MSI), which captures how sensitive the system’s dominant unstable modes are to changes in converter parameters. To compute this, the researchers fit a state-space model to the measured impedance data using vector fitting — a technique that approximates frequency-domain responses with rational functions. From this fitted model, they calculate the sensitivity of eigenvalues to variations in key control parameters, such as current controller gains or filter time constants. The MSI is normalized to reflect the relative impact of parameter changes across different converters.
The AEI combines these two components multiplicatively:
A higher AEI indicates that a converter has a large influence on stability and requires only small parameter adjustments to restore it — in other words, it’s an efficient “knob” for stabilization. A lower AEI means either the system is far from instability (large NSM) or the converter’s parameters have little effect on dominant modes (small MSI), making it a poor candidate for retuning.
The method was tested on a simulated two-converter system: one grid forming (GFM) and one grid following (GFL). GFM converters emulate traditional generators, regulating voltage and frequency autonomously. GFL units, in contrast, assume a stable grid and simply inject power according to setpoints. Both types are essential in modern grids, but they behave very differently under stress.
What They Found
When the researchers induced instability by adjusting line impedance and load conditions, both converters could stabilize the system when retuned — but the effort required differed dramatically. The AEI revealed that the grid following converter had an index value of 4.8, while the grid forming unit scored only 1.8. This means the GFL converter was over twice as effective per unit of parameter change.
More concretely, to suppress a dominant oscillatory mode at 42 Hz, the GFL unit required a 12% reduction in its current controller proportional gain. The GFM unit, however, needed a 32% adjustment in its virtual inertia constant — a much more aggressive change that could compromise other performance objectives, such as frequency response or transient stability.
Adaptation Effort Index (AEI) for GFL vs GFM Converters
| Label | Value |
|---|---|
| Grid Following (GFL) | 4.8 |
| Grid Forming (GFM) | 1.8 |
**
Adaptation Effort Index (AEI) for GFL vs GFM Converters
| Label | Value |
|---|---|
| Grid Following (GFL) | 4.8 |
| Grid Forming (GFM) | 1.8 |
: Adaptation Effort Index (AEI) for GFL vs GFM Converters**
| Converter Type | AEI Value |
|---|---|
| Grid Following (GFL) | 4.8 |
| Grid Forming (GFM) | 1.8 |
Source: García-Reyes et al. (2026)
The researchers also examined how AEI varied across different operating points. They found that the GFL unit consistently outperformed the GFM converter across a range of loading conditions, with AEI values ranging from 3.9 to 5.1 for GFL versus 1.5 to 2.0 for GFM. This robustness suggests that the advantage isn’t an artifact of a specific scenario but a structural property of the control architectures.
Another key insight came from decomposing the AEI into its components. The NSM was nearly identical for both converters — meaning the system was equally close to instability regardless of which unit was considered. But the MSI differed sharply: the GFL converter’s parameters had a much stronger influence on the unstable mode. This implies that the GFL unit, despite being traditionally seen as a “follower,” actually exerts greater control authority over certain dynamic instabilities.
Modal Sensitivity Index (MSI) and Nyquist Stability Margin (NSM) by Converter Type
| Label | Value |
|---|---|
| NSM | 0.42 |
| MSI | 11.4 |
**
Modal Sensitivity Index (MSI) and Nyquist Stability Margin (NSM) by Converter Type
| Label | Value |
|---|---|
| NSM | 0.42 |
| MSI | 11.4 |
: Modal Sensitivity Index (MSI) and Nyquist Stability Margin (NSM) by Converter Type**
| Component | GFL | GFM |
|---|---|---|
| NSM | 0.42 | 0.41 |
| MSI | 11.4 | 4.4 |
Source: García-Reyes et al. (2026)
The paper includes a Nyquist plot (
) showing the eigenloci of the impedance ratio before and after stabilization. In the unstable case, one eigenlocus encircles the (-1,0) point, confirming instability via the GNC. After retuning the GFL converter, the locus shifts away, restoring stability. The same correction applied to the GFM unit required larger parameter shifts to achieve a similar shift in the locus.
: Generalized Nyquist plot showing eigenloci of the impedance ratio matrix. Left: unstable system with encirclement of (-1,0). Right: stabilized system after GFL retuning. Source: García-Reyes et al. (2026).*
A third figure (
) illustrates the fitted state-space model’s accuracy by comparing the measured impedance of the GFL converter with the vector-fitted approximation across a frequency range of 1–100 Hz. The close match validates the use of black-box identification for modal analysis.
: Measured vs. fitted impedance magnitude for the grid following converter. The vector-fitted model (dashed) closely matches the measured data (solid), enabling reliable modal sensitivity calculations. Source: García-Reyes et al. (2026).*
Why This Changes Things
The implications of the AEI extend far beyond a single simulation. As power systems worldwide accelerate their transition to renewables, the question of who stabilizes the grid is becoming urgent. In traditional grids, that responsibility fell naturally to large generators. Today, it must be explicitly assigned — and often, it’s unclear which devices are best suited for the task.
Regulators in Europe, the U.S., and Australia are already requiring renewable plants to provide “grid-forming” capabilities. The logic is intuitive: if inverters are replacing generators, they should behave like them. But this study suggests a counterintuitive possibility — that sometimes, the best stabilizer isn’t the one designed to emulate a generator.
The grid following converter, often dismissed as a passive actor, may actually be a more efficient stabilizer for certain types of instabilities. This doesn’t mean GFM is obsolete — far from it. Grid-forming converters are essential for black-start capability and voltage regulation in weak grids. But for damping specific oscillatory modes, especially those arising from impedance interactions, the simpler control structure of GFL units may offer finer, more responsive tuning.
This has practical consequences. Retuning a converter isn’t free. It can affect efficiency, thermal stress, and compliance with grid codes. A 32% change in virtual inertia might push a GFM unit outside its certified operating envelope, requiring retesting and recertification. A 12% adjustment in a current controller gain, by contrast, may be within normal tuning ranges.
Moreover, the AEI is model-agnostic. That’s a game-changer. Utilities and system operators don’t need to negotiate access to proprietary control algorithms. They can measure impedance at the point of connection, compute the AEI, and decide which device to reconfigure — all without violating intellectual property. This makes the metric not just technically useful, but institutionally feasible.
Consider a real-world scenario: a wind farm in Texas begins to oscillate after a transmission line outage. The grid operator detects the instability via phasor measurement units (PMUs) but doesn’t know whether to ask the wind inverters, a nearby battery, or a synchronous condenser to respond. Today, the decision is often based on rules of thumb or contractual obligations. With AEI, it could be based on physics: measure the impedance, compute the index, and direct the most effective responder.
The metric also opens the door to automated stability management. Imagine a control layer that continuously monitors AEI values across a network and triggers adaptive tuning in real time — a self-healing grid that doesn’t just detect problems but optimally assigns fixes.
What’s Next
The AEI is promising, but it’s not a panacea. The study tested only two converters in a simplified system. Real grids involve dozens or hundreds of interacting devices, with nonlinearities, time delays, and varying communication latencies. Extending AEI to large-scale, multi-converter systems will require efficient computational methods and robust data acquisition.
One challenge is impedance measurement itself. While possible in simulation, measuring accurate impedance in the field requires specialized equipment and careful signal injection. Ongoing work in harmonic current injection and wideband probing may help, but standardization is still lacking.
Another open question is how AEI behaves under transient conditions. The current formulation is based on small-signal analysis — it assumes linearized dynamics around an operating point. Large disturbances, such as faults or sudden load drops, may trigger nonlinear effects that the index doesn’t capture.
There’s also the question of parameter selection. The AEI depends on which control parameters are considered “tunable.” In practice, manufacturers may restrict access to certain gains or limits. Future work could incorporate constraints into the index, creating a “practical AEI” that reflects real-world configurability.
Finally, the metric assumes that stability can be restored by adjusting a single converter. In reality, coordination may be necessary. A natural next step is to extend AEI to a multi-agent framework, where the index guides not just which converter to tune, but how they should cooperate.
Still, the core insight stands: not all converters are equally effective at restoring stability, and we now have a way to measure that difference. As grids become more complex, we’ll need more such tools — not just to keep the lights on, but to do so efficiently, fairly, and transparently. The Adaptation Effort Index won’t solve all our grid stability problems. But it gives us something just as important: a way to ask better questions.