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On the Achievable Inertia Constant of Inverter Based Resources

Christos M. Nikolakakos, Hassan Haes Alhelou, Nikos Hatziargyriou

arXiv:2609.34580v1eess.SYcs.SY

TL;DR

As inverter-based resources displace synchronous generators, their achievable inertia is constrained by available energy, converter power and VRT obligations, and control activation. The paper combines closed-form bounds into an envelope; it finds that VRT can eliminate active-current headroom below a loading-dependent critical voltage and that GFL activation delay reduces inertia observed in short measurement windows.

  • Problem

    IBR inertia is constrained by available DC-side energy, converter overcurrent rating and VRT obligations, and the control scheme with its activation delay.

    Source: §I · p. 1

  • Method

    The paper derives closed-form bounds and combines them as Heff = min(Henergy, Hpower, Hcontrol), validating the framework on the IEEE 9-bus system.

    Source: §X · p. 10

  • Results

    VRT reactive-current priority can reduce active-power headroom to zero below a loading-dependent critical voltage, while GFL activation delay discounts inertia observed in short measurement windows.

    Source: F. Measurement Window Results · p. 8, §X · p. 10

  • Takeaways & Limitations

    The inertia envelope quantifies achievable IBR inertia by loading, grid voltage, and time using parameters accessible to the system operator.

    Source: §X · p. 10

Abstract

from arXiv · show

As inverter-based resources (IBRs) displace synchronous generators, the inertia they can actually contribute to the grid becomes a critical planning parameter. Unlike synchronous machines, this contribution is bounded simultaneously by the available energy reserve, the converter power rating combined with voltage ride-through (VRT) obligations, and the inertia-emulation control scheme with its activation delay. This paper derives each bound in closed form and combines them into a unified envelope $\Heff(t,\lambda,V_g)=\min(\HE,\HP,\HC)$, which uses parameters accessible to the system operator to quantify how much inertia a plant can provide at a given loading $\lambda$, grid voltage $V_g$, and time $t$ after a disturbance. The analysis shows that below a loading-dependent critical voltage, VRT reactive-current priority does not leave active-current headroom for inertial power injection; that the converter overload ratio $\kappa$ matters mainly during voltage dips; and that de-loading yields a linear, quantifiable inertia gain, while collocated energy storage (ESS) removes the source-coupling constraint, making grid-following (GFL) and grid-forming (GFM) plants with storage equivalent post-activation, up to the GFL activation-delay discount within short RoCoF measurement windows. Simulations on the IEEE 9-bus system confirm the framework and motivate a two-dimensional inertia capability curve $H(\lambda,V_{g,\min})$ as a grid-code instrument analogous to the P--Q diagram of synchronous generators.

I. INTRODUCTION

As synchronous generation is displaced, IBRs face energy, converter-power, and control limits that constrain their actual inertial response. The paper combines these bounds into an operating-state-dependent framework and distinguishes achievable inertia from programmed or measured inertia.

I. INTRODUCTION

  • IBR inertia is constrained by available DC-side energy, converter overcurrent capability and VRT obligations, and the emulation control scheme with its activation delay.These limits constrain what a plant can deliver beyond the inertia value programmed into its controller.Source: §I · p. 1

A. Literature Gap

  • Earlier studies often treated energy, control, hardware ratings, or estimation separately, leaving no direct map from operating condition to achievable IBR inertia.The limiting constraint can change during a disturbance when VRT reactive-current priority reduces active-power headroom.Source: A. Literature Gap · pp. 1–2

B. Contributions

  • The paper proposes a TSO-accessible envelope Heff(t, λ, Vg) = min(Henergy, Hpower, Hcontrol) and a capability curve H(λ, Vg,min) for grid-connection agreements.Its other contributions include a critical voltage for VRT headroom, analysis of κ, and a link between GFL activation delay and the RoCoF measurement window.Source: B. Contributions · p. 2

II. INERTIA DEFINITIONS

  • Achievable inertia is what a plant can deliver under energy, power, and control limits; apparent inertia is the value inferred from system-operator frequency measurements.The measured value can differ from achievable inertia because the RoCoF window may include the GFL activation ramp.Source: §II · p. 2

III. OPERATING-STATE DEPENDENCE OF IBR INERTIA

  • The effective inertia bound is the minimum of three coupled limits and varies with time since disturbance, pre-disturbance loading, and post-disturbance grid voltage.Unlike synchronous-generator inertia, the IBR envelope depends on operating state.Source: §III · p. 2

IV. ENERGY BOUND — Henergy

The energy bound depends on the usable DC-link energy or, for a plant with collocated storage, the available battery reserve. The DC-link contribution is small compared with the reserve available from the illustrated battery systems.

A. DC-Link Capacitor

  • A typical utility-scale module yields Hdc energy ≈0.4 ms, orders of magnitude below synchronous-generator inertia.The per-unit DC-link energy bound depends on the per-module capacitance-to-rating ratio, not the number of parallel modules.Source: A. DC-Link Capacitor · p. 2

B. Collocated Battery Energy Storage System

  • Collocated BESS energy is typically sufficient to make the energy bound non-binding over the seconds-long durations relevant to inertial support.The bound remains dependent on the usable state-of-charge range and storage size.Source: B. Collocated Battery Energy Storage… · p. 3

C. De-Loaded / Curtailed Generation

For PV or wind plants without collocated storage, de-loading supplies short-term active-power headroom, while converter power and control can still limit inertia. The power bound depends on loading, overload capability, and the assumed design RoCoF.

C. De-Loaded / Curtailed Generation

  • Below maximum power point, available source headroom is ΔP = (λMPP − λ)Srated, while at maximum power point the DC-link capacitor sets the energy bound.Below maximum power point, the relevant limits shift to converter power and control; the “+∞” energy entry denotes that headroom, not unbounded stored energy.Source: C. De-Loaded / Curtailed Generation · p. 3

A. Inverter Current Limit and Loading

  • At full load (λ = 1.0, κ = 1.2) under a |ḟ| = 1 Hz/s event at 50 Hz, Hpower = 5 s.This bound is specific to the design RoCoF and can be reduced by VRT obligations and source–converter coupling.Source: A. Inverter Current Limit and Loading · p. 3

B. Voltage Ride-Through Interaction

During voltage dips, VRT reactive-current priority can reduce the active-current headroom available for inertial power injection. A loading-dependent critical voltage marks where that headroom is exhausted.

B. Voltage Ride-Through Interaction

  • When voltage falls below the VRT activation threshold, grid-code reactive-current priority competes with active current within the inverter current limit.The reactive-current request is determined by the VRT droop and the voltage at the point of injection.Source: B. Voltage Ride-Through Interaction · p. 3
  • At or below the loading-dependent critical voltage Vg*, power-limited inertia is zero because VRT leaves no active-current headroom for inertial injection.Vg* is the voltage at which the VRT reactive-current request exhausts active-current capacity, not a prediction of the disturbance voltage.Source: B. Voltage Ride-Through Interaction · pp. 3–4
  • As loading increases, a less severe voltage dip is sufficient to eliminate active-current headroom; at lower loading, a deeper dip is required.Thus, more heavily loaded inverters are more vulnerable to losing inertial-response capability during voltage depressions.Source: B. Voltage Ride-Through Interaction · p. 4

C. Source–Converter Coupling (Plants without Collocated Storage)

Without collocated storage, source–converter coupling limits sustained power headroom, while the commanded virtual inertia is only achievable when the plant’s energy and power bounds do not bind.

C. Source–Converter Coupling (Plants without Collocated Storage)

  • Without collocated ESS, the DC bus cannot buffer a deficit between converter capability and primary-source output, creating a source-coupled power bound.At normal voltage, plants with different overload ratios but the same loading deliver the same inertia; overload ratio matters during voltage dips when VRT reduces effective converter capability.Source: C. Source–Converter Coupling (Plants… · p. 4

A. Grid-Forming Inverters (VSM / Droop / dVOC)

  • A GFM inverter responds immediately to torque imbalance, though not identically to a synchronous generator.The cited architectures include VSM, synchronverter, and droop-based controls.Source: A. Grid-Forming Inverters (VSM / Droop… · p. 4
  • Hset is the commanded virtual inertia from the connection agreement, whereas achievable inertia is at most Hset and equals it only when energy and power bounds are non-binding.The commanded value can be verified through type-testing per,.Source: A. Grid-Forming Inverters (VSM / Droop… · p. 4

B. Grid-Following Inverters (df/dt Loop)

GFL inertia builds after frequency measurement and control activation, so short RoCoF windows can observe less than the post-activation capability. The apparent value rises with the measurement window until activation is complete.

B. Grid-Following Inverters (df/dt Loop)

  • Before activation, the GFL effective-inertia deficit is independent of loading and voltage, so operators should discount Kinertia in inertial-response calculations.The GFL loop first measures grid frequency, computes df/dt, and commands a power-reference change; PLL-based estimation introduces τact.Source: B. Grid-Following Inverters (df/dt Loop) · p. 4

C. Link to the TSO’s RoCoF Measurement Window

  • The apparent-inertia relation links the power observed at the TSO’s measurement-window endpoint to moving-average RoCoF, assuming injection does not appreciably alter the frequency trajectory.The derivation treats RoCoF as approximately constant and models GFL output as a linear ramp during activation.Source: C. Link to the TSO’s RoCoF Measurement… · p. 4
  • At 50 ms and 100 ms windows, a GFL plant with Hcap = 6.0 s and τact = 150 ms appears as 2.0 s and 4.0 s, respectively; at windows of at least 150 ms it appears as 6.0 s.GFM delivers Hcap immediately, while GFL ramps linearly from zero over [0, τact]. Fig. 3 compares the step response, apparent inertia versus window, and discounted envelope against the GFM+ESS reference.Source: C. Link to the TSO’s RoCoF Measurement… · p. 5, Figure 3 · p. 5

D. Measurement Window Impact: GFM+ESS vs. GFL+ESS

With collocated ESS, GFL and GFM plants have the same post-activation inertia envelope, although short measurement windows discount GFL because of its activation delay. The GFM+ESS example shows how loading and voltage determine whether power or control limits the envelope.

D. Measurement Window Impact: GFM+ESS vs. GFL+ESS

  • At 50 ms and 100 ms windows, a GFL+ESS plant with Hcap = 6.0 s appears as 2.0 s and 4.0 s, while an identical GFM+ESS plant receives full credit.After activation, both storage-equipped architectures deliver the same Hcap(λ, Vg).Source: D. Measurement Window Impact: GFM+ESS… · p. 5

A. Complete Formulation

  • The complete formulation uses λ, Vg, κ, λMPP, EESS, Hset or Kinertia, and τact, all available to the TSO through SCADA, connection agreements, and type-test certificates.It requires no proprietary internal firmware parameters.Source: A. Complete Formulation · p. 5

B. GFM with Collocated ESS Example

  • For the 100 MVA GFM inverter with BESS, Hpower ranges from 17.5 s at λ = 0.5 and Vg = 1.0 pu to zero below the loading-dependent critical voltage; Hcontrol = 6 s binds across most of normal operation.The power-limited bound reduces to zero as VRT consumes the available current.Source: B. GFM with Collocated ESS Example · p. 5
  • In the GFM+ESS example, the control-to-power transition occurs at λ*=0.96, 0.55, and 0.21 for Vg = 1.0, 0.7, and 0.5 pu, respectively.Fig. 4 presents the envelope magnitude, binding-constraint map, and loading and voltage slices.Source: B. GFM with Collocated ESS Example · p. 5, Figure 4 · p. 6

C. GFL without Collocated ESS — DC-Link Only

Without collocated ESS, GFL inertia is limited by source headroom and the DC link at maximum-power-point operation; de-loading increases available inertia until the control limit binds. Collocated ESS removes source coupling, making GFL+ESS and GFM+ESS envelopes identical at the same overload ratio.

C. GFL without Collocated ESS — DC-Link Only

  • At 24% de-loading (λ = 0.76), the GFL source-coupled power bound reaches the 6 s control limit, and further de-loading adds no inertia.The envelope ramps linearly with de-loading below MPP and saturates at Hcontrol = 6 s for λ ≤ 0.76.Source: C. GFL without Collocated ESS —… · p. 6
  • At MPP, the DC-link energy bound is 0.4 ms, approximately 15,000× below GFM+ESS, and binds regardless of voltage or control setting.Below MPP, source power headroom sustains inertial support over the relevant intervals.Source: C. GFL without Collocated ESS —… · p. 6

D. GFL with Collocated ESS — Decoupling the Source Constraint

  • Collocated ESS removes source coupling, so GFL+ESS and GFM+ESS have identical envelopes at the same κ, with a non-binding energy bound of 1,710 s.At κ = 1.2, ESS shifts the power-control boundary from λ ≈ 0.76 to λ ≈ 0.96.Source: D. GFL with Collocated ESS —… · p. 7
  • At λ = 0.85, ESS adds 2.25 s (+60%); at λ = 0.90 it adds 3.50 s, and at λ = 0.95 it adds 4.75 s.For λ ≤ 0.76, both GFL configurations remain control-limited at 6 s, so ESS adds no inertia in that range.Source: D. GFL with Collocated ESS —… · p. 7
  • For source-coupled GFL, κ does not constrain inertia at normal voltage but matters under VRT, where reactive-current priority reduces effective converter headroom.The effect of increasing κ from 1.1 to 1.2 is stronger at Vg = 0.7 pu than at 0.85 pu.Source: D. GFL with Collocated ESS —… · p. 7

E. Contrast with Synchronous Generators

Unlike a synchronous generator’s fixed inertia, achievable IBR inertia varies with loading, voltage, and time. Collocated storage can remove the GFL source-coupling penalty, but short measurement windows discount GFL’s apparent inertia; IEEE 9-bus simulations test the predicted storage-supported response.

E. Contrast with Synchronous Generators

  • IBR inertia varies across loading, voltage, and time, while synchronous-generator inertia remains fixed; GFL without ESS also has a zero-inertia region at MPP and during activation.With ESS, GFL’s zero-inertia region matches GFM+ESS, and at λ = 0.90 and 0.95 it recovers 3.50 s and 4.75 s, respectively.Source: E. Contrast with Synchronous Generators · p. 7

VIII. VALIDATION

  • The IEEE 9-bus validation model includes two synchronous generators and a 125 MVA IBR at Bus 10, with a 112.5 MW load-step contingency.The IBR configuration and control settings vary between scenarios.Source: §VIII · p. 8

F. Measurement Window Results

  • At λ = 0.8 and Vg = 1.0 pu, GFL+ESS provides 6.0 s physically but only 4.0 s apparent inertia at tma = 100 ms, a 2.0 s deficit.At tma = 50 ms, the deficit reaches 4.0 s; equivalent TSO credit requires tma ≥ τact or oversizing by τact/tma.Source: F. Measurement Window Results · pp. 7–8, Figure 10 · p. 8

A. Impact of Collocated Energy Storage on GFL Inertia Contribution

B. VRT Impact on Inertia Delivery

Under the tested voltage depression, VRT reactive-current demand exhausts active-current headroom, eliminating sustained inertia and slightly worsening RoCoF. The planning discussion frames guaranteed IBR inertia as dispatch- and voltage-contingency-conditioned.

B. VRT Impact on Inertia Delivery

  • At Vg = 0.62 pu with Kq = 5, VRT demand leaves approximately zero active-current headroom, so the inverter supplies no sustained inertia at any loading.The VRT case reaches peak RoCoF −1.79 Hz/s, compared with −1.33 Hz/s without VRT; PLL dynamics cause a modest transient degradation beyond the SG-only reference.Source: B. VRT Impact on Inertia Delivery · p. 9

A. Guaranteed Inertia for System Planning

  • Guaranteed IBR inertia is dispatch-dependent and conditioned on voltage contingencies, unlike the existing synchronous-generator inertia sum.The paper relates this framing to AEMO analysis of GFM BESS synthetic inertia, which varies with operating point, contingency size, and overload capacity.Source: A. Guaranteed Inertia for System… · p. 9

B. IBR Inertia Capability Curve

The proposed inertia capability curve makes guaranteed IBR inertia a function of loading and design-basis voltage for grid-connection and planning use. Online inertia estimates complement the bounds framework, while the conclusions emphasize operating-condition dependence and warn that ignoring it can overestimate effective inertia.

B. IBR Inertia Capability Curve

  • The proposed H(λ, Vg,min) curve specifies guaranteed inertia across loading and design-basis voltage for grid-connection agreements and planning, whereas the equivalent synchronous-generator curve is constant.The curve can be derived analytically or validated through type testing.Source: B. IBR Inertia Capability Curve · p. 9

C. Complementarity with Online Inertia Estimation

  • Online estimators provide current inertia but are backward-looking and do not identify its limiting constraint; the bounds framework instead gives worst-case guaranteed inertia and identifies the binding constraint.Differences between estimated and predicted guaranteed inertia can serve as an anomaly signal for unmodeled dynamics, model errors, or estimation artifacts.Source: C. Complementarity with Online Inertia… · p. 10

X. CONCLUSIONS

  • The framework expresses achievable inertia as the minimum of energy, power, and control bounds and is validated on the IEEE 9-bus system, with inertia varying by loading, voltage, and time.Unlike synchronous generators’ fixed inertia constant, the envelope uses parameters accessible to the system operator.Source: §X · p. 10
  • Below a loading-dependent critical voltage, VRT reactive-current priority can reduce active-power headroom to zero, while GFL inertia ramps to its full value over an activation delay of about 100–200 ms.For GFL plants without collocated ESS, each 1% of available power headroom adds approximately 0.25 s of inertia at 50 Hz and 1 Hz/s RoCoF, up to the control-limit crossover at λ ≈0.76.Source: §X · p. 10
  • Summing connection-agreement inertia values without accounting for operating-state dependence may overestimate effective inertia under adverse but credible conditions.The proposed curve provides loading and design-basis voltage dimensions for assessing that dependence.Source: B. IBR Inertia Capability Curve · p. 9, §X · p. 10
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