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A New Index for Quantifying Stability Adaptation Effort in Power Electronics-Dominated Power Systems Based on Frequency-Domain Impedance Identification

Luis A. Garcia-Reyes, Macarena Martin-Almenta, Edgar Nuño-Martínez, Javier Renedo, Antonio Cordón-Rodríguez, Oriol Gomis-Bellmunt, Eduardo Prieto-Araujo

arXiv:2610.03158v1eess.SYcs.SY

TL;DR

Black-box converter models make stability assessment and mitigation difficult, while existing eigenvalue and impedance methods leave gaps in identifying which converter should adapt. The paper combines Nyquist stability screening with modal sensitivity from fitted state-space models to quantify adaptation effort; in the tested GFM/GFL system, the GFL required less effort to restore stability.

  • Problem

    Black-box models limit conventional stability assessment, and existing eigenvalue and impedance methods do not jointly identify which converter should adapt under instability.

    Source: §1 · p. 2

  • Method

    The AEI combines a Nyquist Stability Margin from frequency-domain screening with modal sensitivity estimated through fitted state-space synthesis from black-box impedance data.

    Source: §1 · p. 3, §4 · p. 13

  • Results

    In the tested GFM/GFL system, the same proportional gain reduction restored stability with a much smaller AEI and significantly higher NSM for the GFL converter.

    Source: §4 · p. 13

  • Takeaways & Limitations

    The GFL provided the least intrusive path to stability in the tested system, and the AEI offers a tool for identifying which converter to reconfigure in black-box systems.

    Source: §4 · p. 13

    For the unstable case, unstable frequency-domain identification meant fitss could not capture the simulated initial steady state or transient response.

    Source: §3.1 · p. 9

Abstract

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The increasing use of converter based resources has made stability assessment in modern power systems more challenging, particularly because many devices are available only as black box models. Conventional eigenvalue based methods require full dynamic representations, while frequency domain impedance techniques based on the Generalized Nyquist Criterion (GNC) provide stability insight but do not quantify how much a converter must be retuned to restore stability. This paper introduces the Adaptation Effort Index (AEI), a metric that measures the sensitivity of dominant modes to parameter variations and quantifies the effort required for a converter to reconfigure itself under unstable operating conditions. The AEI combines the Nyquist Stability Margin obtained from frequency domain screening with a modal sensitivity index derived from fitted state space models synthesized from black box impedance data. The method is applied to a system comprising a grid forming and a grid following converter. Results show that, although both units can be tuned to stabilize the system, the grid following converter requires significantly less adaptation effort. The AEI therefore provides a practical and model agnostic tool for identifying which converter should be reconfigured in converter dominated grids, offering valuable guidance for system operators and manufacturers.

1 Introduction

Converter-based resources create stability and interaction challenges, while black-box models limit assessment and mitigation. The paper addresses how operators can identify which converter should adapt when stability is at risk.

1 Introduction

  • High converter penetration introduces stability and interaction challenges associated with low inertia, complex controls, and high switching frequencies.New converters are expected to adapt their controls to local grid codes and avoid adverse interactions.Source: §1 · p. 2
  • Operators and manufacturers need to determine whether a newly connected or neighboring converter can adapt, while respecting grid-code and device operating constraints.The introduction cites a Balearic Islands case where a study identified commutation-failure risk but did not provide detailed mitigation strategies.Source: §1 · p. 2
  • Intellectual-property restrictions often leave systems with black-box converter models, limiting conventional small-signal assessment and effective mitigation.This creates particular challenges for transmission system operators.Source: §1 · p. 2
  • Eigenvalue methods require detailed dynamic models, whereas impedance methods can assess black-box systems but do not identify which components contribute to instability.The two approaches therefore leave gaps in comprehensive assessment and in deciding which converter should adapt.Source: §1 · p. 2
  • The paper proposes combining a Generalized Nyquist Criterion stability margin with an index estimating a converter’s maximum dynamic effort, using fitted state-space synthesis from black-box impedance data.The approach is applied to grid-forming and grid-following converters connected to a grid equivalent.Source: §1 · p. 3

2 Proposed Approach for the Adaptation Effort Index (AEI)

The proposed AEI combines frequency-domain stability screening with modal sensitivity estimated from black-box impedance data. It is intended to quantify adaptation effort and support stability decisions for connected converters.

2 Proposed Approach for the Adaptation Effort Index (AEI)

  • The AEI quantifies modal sensitivity to operational-parameter changes to estimate the effort required to maintain stability, including in pre-connection impact studies.The approach combines frequency-domain screening with a fitted state-space method for estimating the maximum dominant-mode change needed to move toward stability.Source: §2 · p. 3

2.1 Nyquist Stability Margin (NSM)

  • Frequency scanning identifies the new converter’s admittance and the existing system’s impedance; the GNC tests the resulting closed-loop trajectory against the critical point.The NSM measures the trajectory’s minimum distance from (-1,j0) at each operating point; the SIaD-Tool is used for matrix identification.Source: §2.1 · pp. 3–4

2.2 AEI Based on Fitted State-Space (fitss)

  • The fitss procedure synthesizes black-box state-space models from identified impedance or admittance matrices using vector fitting, model-order reduction, and pole expansion.It fits SISO subsystems independently before assembling the reduced MIMO representation; the reported average modal error is approximately 1.5%.Source: §2.2 · pp. 4–5
  • Although the fitted model’s states lack explicit physical meaning, its characteristic equation provides system modes; the dominant mode is selected by largest real part and tracked across operating points.Tracking uses minimum distance in the complex plane to prevent spurious jumps or incorrect assignments.Source: §2.2 · p. 5
  • The AEI is the maximum rate of dominant-mode change per unit parameter variation, evaluated separately for real and imaginary components to distinguish damping and frequency sensitivity.Large values indicate that small parameter changes produce significant modal shifts; mode-wise measures can be compared to prioritize corrective action.Source: §2.2 · p. 5
  • The AEI supplies a conservative condition for keeping the dominant mode from crossing the imaginary axis under a parameter change, with a safety factor available for identification error and nonlinearities.A new converter is accepted at an operating point only when the condition holds for the expected parameter change with the chosen margin and the NSM exceeds its minimum threshold.Source: §2.2 · p. 6

3 Study Case: Grid-Connected Grid-Forming and Grid-Following

In a two-converter black-box test system, the study compares reconfiguration effort using impedance screening and fitted state-space modal analysis. Both converters can restore stability, but the GFL requires less adaptation effort.

3 Study Case: Grid-Connected Grid-Forming and Grid-Following

  • The test system contains black-box GFM and GFL converters, a Thevenin grid equivalent with X/R = 10 and SCR = 10, and a 66.6 MW load.A separate fitss model is derived for each operating point.Source: §3 · p. 6, Figure 3 · p. 6, Table 1 · p. 7
  • Reducing the GFM droop gain by approximately 80% increases the NSM and removes the eigenvalue encirclement, restoring stability.The screening displays eigenvalue 1, identified as the mode involved in the instability.Source: §3 · p. 7, Figure 5 · p. 8

3.1 AEI for the Grid-Forming Converter

  • For the unstable GFM case, fitss fails to capture the simulated initial steady state and transient because frequency-domain identification is unstable.The resulting state-space model is automatically unstable.Source: §3.1 · p. 9, Figure 7 · p. 9
  • The GFM’s single unstable mode starts at 277 ± j6165 and shifts left as its gain is reduced, with no other mode entering the right-half plane.The reported modal evolution restores overall system stability.Source: §3.1 · p. 9, Figure 8 · p. 10
  • The GFM’s normalized AEI is 18.03; its NSM is about 0.75 through most of the sweep but falls near 0.2 at nominal gain.The AEI rises sharply near the stability boundary, and only the 100% gain operating point fails the local stability condition with a safety factor of 2.Source: §3.1 · p. 10, Figure 9 · p. 10, Figure 10 · p. 11

3.2 AEI for the Grid-Following Converter

  • Reducing the GFL droop gain from 100% to 30% leaves the unstable mode nearly unchanged, with about 1.2% variation, while the next dominant mode stays closer to its original position.The smaller mode-change rate corresponds to lower adaptation effort to achieve stability.Source: §3.2 · p. 11, Figure 11 · p. 12
  • At 30% GFL gain, the NSM is very large, while it drops below 0.3 as gain increases; the 100% case is the only unstable operating point.The normalized AEI is not a standardized threshold and should be compared across parameter variations.Source: §3.2 · p. 12, Figure 12 · p. 12, Figure 13 · p. 13
  • The normalized AEI is 17.91 for the GFL, below the GFM’s 18.03; this indicates a smaller dominant-mode displacement per unit droop-gain change and less adaptation effort for the GFL.Source: §3.1 · p. 10, §3.2 · p. 12

4 Conclusions

The AEI combines frequency-domain Nyquist margin with fitted-state-space modal sensitivity to quantify converter adjustment effort in black-box systems. In the tested GFM–GFL system, GFL retuning required less effort, and the AEI also supports a proposed local stability compliance criterion.

4 Conclusions

  • The AEI quantifies how strongly a converter must adjust control parameters to restore stability in black-box, converter-dominated systems.It combines Nyquist Stability Margin from frequency-domain screening with modal sensitivity from fitted state-space synthesis.Source: §4 · p. 13
  • The same proportional droop-gain reduction restored stability with much lower adaptation effort for the GFL than the GFM, alongside a significantly higher NSM.The method was applied to a system with one grid-forming and one grid-following converter; reducing the GFM gain also stabilized the system but required a large modal displacement.Source: §4 · p. 13
  • The results identify the GFL converter as the least intrusive path to stability and present the AEI as a tool for choosing which converter to reconfigure.The work also proposes a robust stability indicator for converter-based resource connection studies.Source: §4 · p. 13

APPENDIX

The fitss procedure takes frequency-domain admittance or impedance data and returns a fitted small-signal state-space model. It builds the model from fitted SISO sub-vectors, iteratively adjusts the fit when needed, and reduces model order.

Fitted State-Space (fitss)

  • fitss takes Y(s) or Z(s), frequency points, initial poles, tolerance, reduction mode, and input/output names, and returns a fitted small-signal state-space model.The procedure initializes the poles from the supplied initial poles and calls fitss with those settings.Source: APPENDIX · p. 14
  • The procedure validates input dimensions, fits SISO sub-vectors, assembles a MIMO state-space model, and retries with updated fitting settings if it is unstable or unconverged.The returned model is associated with the specified input and output channels.Source: APPENDIX · p. 14
  • redfit applies vector fitting until RMS error meets the pole tolerance or pole expansion stops, adding a pole in the maximum-error region when fitting does not improve.It then reduces model order using SVD or balanced realization and outputs convergence diagnostics.Source: APPENDIX · p. 14
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