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Frequency Measurement Practices for Inertia Assessment in Inverter Based Resources Dominated Power Systems

Christos M. Nikolakakos, Hassan Haes Alhelou, Nikos Hatziargyriou

arXiv:2609.34531v1eess.SYcs.SY

TL;DR

TSOs and relays infer inertia from frequency measurements averaged over finite windows, but response delivered within a window can alter the apparent inertia. The paper defines a windowed estimator and predicts it using a linear response model; at 500 ms, fast response makes estimates across systems converge or exceed physical inertia.

  • Problem

    Filtered RoCoF measurements can make fast frequency response appear inertial, while systematic analysis of how IBRs appear under TSO measurements is limited and grid codes leave measurement methods unspecified.

    Source: §I · p. 1

  • Method

    The paper defines a windowed inertia estimator, derives an identity using delivered response fractions, and applies it in closed form to trajectories from a linear response model.

    Source: B. Definition and Analytical Framework · p. 6, §VII · p. 14

  • Results

    At 500 ms, GFL with 20% less synchronous inertia reports 40.8 s versus SG’s 41.5 s, while GFM reads 52.8 s.

    Source: §VII · p. 14

  • Takeaways & Limitations

    Moving-average measurements can misclassify fast IBR frequency response as inertia, making reported inertia depend on the measurement method and response mix.

    Source: §I · p. 1, §VII · p. 14

    Comparisons use COI frequency; single-bus estimates vary by measurement location and provide no usable unfiltered value.

    Source: 2) Base Case Response (All Synchronous… · p. 8

Abstract

from arXiv · show

Transmission system operators and protective relays compute the rate of change of frequency (RoCoF) from frequency averaged over tens to hundreds of milliseconds, as standards and grid codes specify. Any response delivered inside that window lowers the measured RoCoF, so the inferred inertia depends on the window and the responses in service. This paper defines the windowed inertia estimate and derives an exact identity the system inertia divided by one minus the fraction of the disturbance energy delivered inside the window and a closed-form prediction that applies the same estimator to the trajectory of a linear response model. Eleven IEEE 9-bus RMS runs synchronous, grid-following and grid-forming configurations, disturbance sizes and parameter sweeps are reproduced by that model, and the identity closes on the measured response powers. Fast frequency response inflates the estimate most: at 500 ms a grid-following system with 20 % less synchronous inertia reports the same value as the all-synchronous system, the grid-forming system a higher one. On a reduced Greek transmission system the 500ms estimate exceeds the inertia by 17-26%, and a scenario with 10% less synchronous inertia reports more than the all-synchronous scenario's inertia. Operators can thus compute what their measurement reports.

I. INTRODUCTION

The paper addresses how filtered frequency measurements shape the apparent contribution of inverter-based resources, then develops a windowed inertia estimate and a model-based prediction. Its framing distinguishes physical inertia from control-based support and relates inertia to initial frequency change.

I. INTRODUCTION

  • Filtered RoCoF measurements can mask fast frequency response or make it appear inertial, while grid codes leave measurement methods insufficiently specified.The paper identifies measurement definitions and signal processing as important to assessing apparent IBR support.Source: §I · p. 1
  • Across eleven IEEE 9-bus RMS runs spanning SG-only, GFL, and GFM configurations, the unfiltered estimate returned reference inertia within 2.5% in every run; 100–500 ms windows raised it by up to 1.36× (SG), 1.66× (GFL), and 1.71× (GFM).Source: §I · p. 2

A. Traditional Frequency Response Framework

  • Synchronous generators provide passive inertia through rotating masses, whereas IBR frequency support is control-designed and can be faster than synchronous response.IBR support depends on converter headroom and control design.Source: §I · p. 1, A. Traditional Frequency Response… · p. 2

B. Inertia-Frequency Relationship in Power Systems

  • The swing-equation relation links inertia to instantaneous RoCoF, but does not account for filtering that shapes the RoCoF used to infer inertia.The reference inertia includes SGs and GFM inverters, but excludes delayed GFL response at the disturbance instant.Source: B. Inertia-Frequency Relationship in… · p. 2

C. Low-Inertia Phenomena, Time Scales and Distributed Resources

Low-inertia systems combine faster frequency changes with control responses operating on overlapping timescales, so measurement windows and filtering affect the RoCoF observed. Standards and grid codes specify performance or withstand requirements but leave measurement choices open; GFM response can also enter windowed inertia estimates.

C. Low-Inertia Phenomena, Time Scales and Distributed Resources

  • Synchronous inertia acts within 0–0.5 s, synthetic inertia and FFR within 50–500 ms, primary response within 1–10 s, and secondary control beyond tens of seconds.The passage also notes that distributed-resource protection and RoCoF settings can affect system-wide frequency stability.Source: C. Low-Inertia Phenomena, Time Scales… · p. 2

1) International Measurement Standards:

  • IEC/IEEE 60255-118-1:2018 defines synchrophasor, frequency, and RoCoF measurement, with performance classes and test waveforms for dynamic conditions.The standard harmonises and supersedes the measurement component of the earlier IEEE C37.118.1 specification.Source: 1) International Measurement Standards · pp. 2–3

2) Grid-Code Practice and the Windows Studied:

  • European and Australian guidance commonly uses a 500 ms RoCoF window, while NERC gives 3–6 cycles (50–100 ms at 60 Hz) as typical; measurement methods remain unspecified.Thus, protection relays and operational studies can see different RoCoF values for the same event.Source: 2) Grid-Code Practice and the Windows… · p. 3

A. Grid-Forming Inverter Characteristics

  • GFM VSG control emulates synchronous-machine swing dynamics, and its voltage-source behavior enables immediate power injection; moving-average estimates can exceed designed virtual inertia.Power injected within the measurement window contributes to the perceived response.Source: A. Grid-Forming Inverter Characteristics · p. 3

B. Grid-Following Inverter Behavior

GFL response passes through PLL-based synchronization and control delays, so it cannot constrain initial RoCoF in the studied setup even though filtering can make its FFR appear inertial. Moving-average RoCoF measurements accumulate support delivered inside the window, making the inferred inertia window-dependent.

B. Grid-Following Inverter Behavior

  • GFL FFR is proportional to measured frequency deviation but arrives behind PLL synchronization; its windowed contribution grows with longer windows and higher droop gains.The filtered signal makes this FFR indistinguishable from inertial response.Source: B. Grid-Following Inverter Behavior · pp. 3–4
  • At Td = 40 ms and Tw up to 500 ms, the GFL converter contributes nothing to initial RoCoF, while the 500 ms window reclassifies its FFR as overstated physical inertia.The passage states that any response behind a delay cannot limit initial RoCoF.Source: B. Grid-Following Inverter Behavior · p. 4

A. Filtered RoCoF

  • Moving-average RoCoF averages frequency change over Tw, and typical protection and monitoring windows range from tens to several hundred milliseconds.In the inertia-only step-disturbance case, the measured RoCoF rises during the window and converges to the instantaneous value only after Tw elapses.Source: A. Filtered RoCoF · p. 4
  • Any frequency support delivered during 0 ≤ t < Tw reduces measured RoCoF, regardless of whether it comes from physical inertia or fast control.The paper expresses the windowed estimate as inertia divided by one minus the delivered response fractions.Source: A. Filtered RoCoF · p. 4

B. Definition and Analytical Framework

The framework defines windowed inertia from filtered RoCoF and relates its bias exactly to the response energy delivered within the measurement window. A linear response model supplies a closed-form prediction, while a planning form separates response contributions by delivery dynamics.

B. Definition and Analytical Framework

  • The planning form attributes estimate corrections to delivered fractions: undelayed response contributes linearly with window length, while a first-order-lag response contributes quadratically.These short-time terms assume linear, unsaturated delivery and a window before the nadir that meets the alignment condition; the quadratic form additionally uses the initial trajectory.Source: B. Definition and Analytical Framework · pp. 5, 7
  • The event-start window maximizes averaged slope over the initial decline only when its slope magnitude is non-increasing and the window ends before the nadir; a later window may still be larger.The reported runs checked the full record for the maximizing window, which ended at the event plus the window length in every reported run.Source: B. Definition and Analytical Framework · p. 5
  • Responses are classified by driving signal and delivery dynamics: undelayed proportional action, lags on actual frequency, and lags on measured frequency.The studied GFL droop is delayed by PLL-based measurement and power-control dynamics; GFM virtual inertia is counted in H, while its proportional damping is a response.Source: B. Definition and Analytical Framework · pp. 4–5
  • The exact identity makes the estimate H divided by one minus the total delivered response-energy fraction, and holds for linear or nonlinear dynamics within the aggregate swing balance.The disturbance magnitude cancels; the estimate is independent of disturbance size only when delivered fractions are, as with linear responses from zero initial conditions.Source: B. Definition and Analytical Framework · p. 6
  • The closed-form prediction applies the estimator to the modeled trajectory, with the estimate set by closed-loop poles and their mode coefficients.The common rule uses the predicted frequency drop at the window end; for linear responses, the drop per unit disturbance is independent of disturbance size.Source: B. Definition and Analytical Framework · p. 6
  • The model prediction uses operator-held parameters and the measuring device’s window to estimate the inertia and RoCoF it will report for a contingency.The model assumes zero initial conditions, linear unsaturated responses, a first-order-lag representation of GFL measurement delay, and neglects post-event load-side response.Source: B. Definition and Analytical Framework · pp. 4, 7

V. RESULTS

The IEEE 9-bus results compare closed-form predictions with RMS measurements, then extend the analysis to a reduced Greek system and different converter-penetration levels.

  • The IEEE 9-bus results compare closed-form predictions with RMS measurements, while the reduced Greek system tests window dependence across converter-penetration levels.Source: §V · p. 7

1) Test System and Methods:

The study evaluates three IEEE 9-bus configurations under a bus-5 load increase, using a shared estimation procedure and a closed-form response model.

1) Test System and Methods:

  • Three RMS configurations use a +62.5 MW bus-5 load step, with a sensitivity study varying disturbance size.Source: 1) Test System and Methods · p. 7
  • The response model uses one parameter set per configuration across windows, combining declared settings with retrospectively estimated governor and GFL power-loop equivalents.Source: 1) Test System and Methods · p. 7
  • Predictions apply the same COI-frequency estimator to a modeled trajectory; GFL measurement delay is represented as a lag, while droop deadband is omitted in the linear model.With the deadband retained, the model lies within 2.4% of every GFL measurement; the deadband accounts for 75–83% of the 500 ms excess.Source: 1) Test System and Methods · p. 8
  • Measurements use rotor-speed COI frequency, with unfiltered slopes and trailing averages over 100, 300, and 500 ms.Source: 1) Test System and Methods · p. 8

2) Base Case Response (All Synchronous Generators):

For the all-synchronous base case, closed-form estimates closely match RMS measurements, while the inferred inertia rises with the averaging window.

2) Base Case Response (All Synchronous Generators):

3) Grid-Following Inverter Response:

The inverter cases show that windowed estimates can exceed reference inertia, and the GFL estimate at 500 ms nearly matches the synchronous case despite lower synchronous inertia.

3) Grid-Following Inverter Response:

  • The GFL measured estimate at 500 ms reaches 1.66 times the synchronous reference inertia.The closed form gives 25.59/31.60/42.54 s at 100/300/500 ms, compared with RMS measurements of 26.11/31.53/40.78 s; the 500 ms prediction is 4.3% above measurement, largely because it omits droop deadband.Source: 3) Grid-Following Inverter Response · p. 8

4) Grid-Forming Inverter Response:

  • The GFM measured estimate at 500 ms is 1.71 times its reference, consistent with governor and damping response delivered inside the window.The closed form predicts 33.45/41.37/53.96 s at 100/300/500 ms, compared with RMS estimates of 33.59/40.13/52.76 s, within 3.1%.Source: 4) Grid-Forming Inverter Response · p. 8

5) Comparison Across Configurations:

  • At 500 ms, GFL and SG estimates differ by only 1.7%, despite 20% less synchronous inertia in the GFL system.Unfiltered estimates agree with reference inertias within 2.5%.Source: 5) Comparison Across Configurations · p. 8

6) Agreement with the Response Model:

The response model closely reproduces measured windowed inertia estimates, while response-power shares close the energy identity and support a model-based consistency check. Sensitivity results show that disturbance size and controller settings affect the reported estimate, with deadband omissions particularly relevant to GFL predictions.

6) Agreement with the Response Model:

  • Reconstructed inertia is within 0.7% of reference for SG and 1.2% for GFM, while GFL retains an unattributed residual of 2% of disturbance energy.The GFL residual remains unattributed because load relief was not exported for that run.Source: 6) Agreement with the Response Model · p. 9
  • The model-based correction agrees with reference H within 3.0% for SG/GFM, 4.1% for the GFL base case, and 5.8% for reduced droop; uncorrected estimates meet a 5% bias criterion only up to about 150 ms for SG and 60 ms for GFM.The correction uses response-model ρ and is a consistency check, not an independent identification of H.Source: 6) Agreement with the Response Model · pp. 9–10

7) Sensitivity to Event Size and Control Parameters:

  • Halving and doubling the SG load step changes measured effective deficit from 28.2 to 105.0 MW, while measured estimates remain within 2.3% of the linear-model prediction.The closed-form estimate is independent of disturbance size for this linear model.Source: 7) Sensitivity to Event Size and… · p. 10
  • Raising GFL Kf from 10 to 40 raises measured 500 ms estimates from 40.18 to 41.96 s; the no-deadband model overstates this sensitivity by starting droop at the event instead of 135 ms later.With the deadband retained, the three predictions are 40.41, 41.11, and 42.57 s.Source: Figure 6 · p. 10, 7) Sensitivity to Event Size and… · p. 10, Figure 7 · p. 11
  • For the 500 ms prediction, the passage reports 41.33 s, against 39.62 s measured, giving a direct predicted-versus-measured comparison for that stated prediction interval.Source: 7) Sensitivity to Event Size and… · p. 10
  • Halving GFM damping from 58 to 29 lowers the predicted 500 ms estimate from 53.96 to 47.37 s, versus measured values falling from 52.76 to 47.09 s; both cases agree within 3.1% across windows.The unfiltered estimates change by less than 0.5%.Source: 7) Sensitivity to Event Size and… · pp. 10–11

6) Agreement with the Response Model:

  • Across base cases, predictions deviate by at most 1.9% for SG, 3.1% for GFM, and 2.4% for GFL at 100/300 ms; at 500 ms, GFL predictions are 2.3–8.7% above measurements.The 500 ms GFL discrepancy grows with droop gain, mostly because the linear model omits the deadband.Source: 6) Agreement with the Response Model · pp. 8–9, Table IV · p. 11

7) Sensitivity to Event Size and Control Parameters:

  • At 500 ms, predictions for Kg = 0/159/623 are 34.74/34.69/41.80 s versus measurements of 35.03/36.71/41.47 s; the reduced-gain case deviates by up to 5.5%.The shortfall is associated with the transferred governor equivalent; no parameter was re-estimated for this case.Source: 7) Sensitivity to Event Size and… · p. 10, Table IV · p. 11

8) Reference PMU Filter Results:

Reference-filter inertia estimates do not always correspond to a moving-average window, and the paper also applies the estimator to a reduced Greek transmission-system model. The supplied Greek-system passages establish the application scope and model composition, but do not report its estimated outcomes.

B. Reduced Greek System Results

1) Greek System Model Description:

  • The Greek model contains 218 equivalent buses and 61 equivalent synchronous generators, with equivalent AC ties representing neighbouring systems.Its baseline connected system includes 45 machines.Source: 1) Greek System Model Description · pp. 11–12

8) Reference PMU Filter Results:

  • P-class and 10 fps M-class estimates match moving-average estimates at 37–82 ms and 54–244 ms, respectively; marked 50/25 fps M-class values fall below the unfiltered value and have no positive moving-average equivalent.The signed kernels overshoot the RoCoF peak, so the comparison matches reported inertia values where possible without implying filter equivalence.Source: 8) Reference PMU Filter Results · p. 11, Table VI · p. 12

2) Scenario Definition:

The reduced Greek system shows that longer RoCoF windows raise reported inertia estimates, sometimes above the connected-machine inertia. Across the studied cases, window choice also affects how inertia procurement and windowed-RoCoF requirements should be interpreted.

2) Scenario Definition:

3) Windowed Estimates on the Greek System:

  • At 500 ms, the Greek-system inertia estimates exceed connected-machine inertia by 17–26%, and G2b reports 72.6 GW s despite having 10% less synchronous inertia than G1’s 69.2 GW s.The figure compares centre-of-inertia frequency and windowed RoCoF across G1, G2b, and G3b.Source: 3) Windowed Estimates on the Greek… · pp. 12–13, Figure 10 · p. 13

A. Impact of Measurement Window on Perceived Response

  • The windowed estimate grows with window length and blurs system differences: at 500 ms, the GFL estimate is only 1.7% below the SG base case, while GFM reports the highest value.In the 9-bus runs, unfiltered estimates returned reference inertia within 2.5% across eleven runs; the study used RMS simulations and did not include emergency schemes in the studied windows.Source: A. Impact of Measurement Window on… · p. 13

B. Implications for System Planning and Operation

  • For the stated planning condition, the 300 ms model gives 35.00 s and a 0.40 Hz/s windowed RoCoF, versus 0.46 Hz/s unfiltered.The example illustrates the calculation and is not an additional validation.Source: B. Implications for System Planning… · p. 13
  • For recovering physical inertia, uncorrected 500 ms estimates exceed H by a factor of 1.36–1.71; a windowed RoCoF criterion instead constrains the estimate at its specified interval.Requirements should identify the measurement window and response classes counted; a response that withdraws after the window also requires a stated holding time.Source: B. Implications for System Planning… · pp. 13–14
  • The model’s short-window sensitivities are not universal exchange rates, and the study does not formulate the cost- and constraint-based optimisation needed to determine procurement quantities or a least-cost portfolio.A single-bus requirement also adds local effects not covered by the centre-of-inertia framework.Source: B. Implications for System Planning… · p. 14

C. Grid Code Evolution

The conclusion argues that measurement windows shape how inverter response is reported as inertia and recommends explicitly defining those windows. The authors report system-dependent estimate inflation, model agreement across most studied runs, and a need for further empirical validation.

C. Grid Code Evolution

  • The paper calls for grid codes to define measurement windows and filtering methods, use technology-neutral performance requirements, recognize FFR within defined frameworks, and standardize testing.The passage identifies IEEE Std 2800–2022 and emerging ENTSO-E grid-forming requirements as steps toward a harmonised, performance-based approach.Source: C. Grid Code Evolution · p. 14

D. Empirical Validation Opportunities

  • Future validation is proposed using GFM pilots and high-resolution event frequency data; attribution through the energy identity additionally requires delivered response-power records.The proposed comparisons use a specified operating condition, disturbance, and measurement window.Source: D. Empirical Validation Opportunities · p. 14

VII. CONCLUSION

  • The GFL base-case estimate rises from 24.62 s unfiltered to 40.78 s at 500 ms (1.66×), while the GFM windowed estimate reaches 1.71× at 500 ms.The GFM unfiltered estimate is 31.64 s against 30.87 s reference inertia (+2.5%); its windowed inflation is associated with damping and governor response accumulating inside long windows.Source: §VII · p. 14
  • At 500 ms, the GFL system with 20% less synchronous inertia reports an estimate only 1.7% below the SG system, while the reduced Greek-system case with 10% less reports 72.6 GW s against 69.2 GW s actual for the all-synchronous case.The Greek case with 34% less synchronous inertia reports 83% of the all-synchronous system’s inertia.Source: §VII · p. 14
  • Across eleven reported runs, the linear model reproduces measured estimates within 3.1% (5.5% for the reduced-droop run), except that 500 ms GFL predictions exceed measurements by 2.3–8.7%.The neglected droop deadband accounts for most of that excess.Source: §VII · pp. 14–15
  • For the studied models and 5% bias criterion, estimates should use short windows—about 150 ms for the synchronous case and 60 ms with the studied GFM damping—or include the model-based correction.Whether these thresholds transfer to other systems depends on response constants, measurement noise, and disturbance estimation, which the study does not address.Source: §VII · p. 15
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