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arxiv: 2604.18987 · v1 · submitted 2026-04-21 · 📡 eess.SY · cs.SY

Inertia Matching Principle: Improving Transient Synchronization Stability in Hybrid Power Systems With VSGs and SGs

Changjun He , Li Zhang , Qi Liu , Rui Zou This is my paper

Pith reviewed 2026-05-10 02:36 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords virtual synchronous generatorsynchronous generatortransient stabilityinertia matchinghybrid power systemssynchronization dynamicsvirtual impedance
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The pith

Matching virtual synchronous generator inertia to synchronous generators at an optimal constant maximizes transient synchronization stability in hybrid power systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that in power systems mixing virtual synchronous generators with traditional synchronous generators, simply raising the virtual inertia does not steadily improve how the machines stay locked in step after a disturbance. A relative swing equation model reveals an optimal inertia matching value between the two generator types that delivers the highest stability margin. Beyond that point, further inertia increases can trigger a new form of instability. The work also demonstrates that the proportion of virtual generators in the mix affects stability only when their inertia level is properly coordinated with output impedance.

Core claim

Using a relative swing equation that describes the angle and frequency difference between a VSG and an SG, the analysis produces a quantitative stability level index. Static and dynamic examination of this index identifies a non-monotonic dependence on VSG inertia: an optimal matching constant exists that maximizes the index, while both too-low and too-high values degrade performance. A second principle shows that VSG penetration level influences synchronization only through its interaction with virtual impedance; proper coordination of inertia and impedance is required for robust stability and reduced fault current.

What carries the argument

The relative swing equation model between VSG and SG, together with the derived quantitative stability level index that quantifies the margin against loss of synchronization.

If this is right

  • Improper inertia matching between VSG and SG creates a previously unrecognized instability route that can be avoided by selecting the optimal matching constant.
  • VSG share in the system affects transient stability only when inertia is matched to virtual impedance; mismatched designs lose stability as penetration rises.
  • A coordinated adjustment of inertia and virtual impedance simultaneously raises stability margins and limits fault currents.
  • The strategy remains effective on both small test systems and the IEEE 39-bus network.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Control tuning for grids with rising renewable shares should treat inertia and impedance as coupled parameters rather than independent knobs.
  • Failure to observe the matching principle may explain some observed synchronization issues in real hybrid systems after faults.
  • Extending the matching analysis to clusters containing several VSGs and SGs would test whether the optimal constant generalizes.
  • Impedance adjustment may trade off against voltage regulation quality, requiring joint optimization in future designs.

Load-bearing premise

The relative swing equation model accurately captures the transient synchronization dynamics between the VSG and the SG without significant unmodeled effects.

What would settle it

Simulate a two-machine VSG-SG system while sweeping VSG inertia across a range of matching constants and check whether the measured critical clearing time or stability index reaches a clear maximum at one specific value rather than rising monotonically.

Figures

Figures reproduced from arXiv: 2604.18987 by Changjun He, Li Zhang, Qi Liu, Rui Zou.

Figure 2
Figure 2. Figure 2: Simplified circuit diagram of the hybrid VSG-SG system [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Topology of the hybrid system under grid faults. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Modified equal area criterion [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Synchronization power influence analysis. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Range of the inertia matching constant to guarantee the SEP. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Influence of the matching between inertia and strength of the VSG [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: The stability regions under different penetration of VSG when (a) [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: The stability regions under different inertia matching constants. [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Range of the virtual impedance to limit the fault current. [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 14
Figure 14. Figure 14: Fig.14. The fault occurs at 0.5 s at the line between bus 36 [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗
Figure 12
Figure 12. Figure 12: The inertia level of the VSG is large relative to its voltage strength [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The inertia level of the VSG is weak relative to its voltage strength [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Simulation results of the modified IEEE 39-bus system. [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗
read the original abstract

This paper investigates the transient synchronization stability in power systems hybridized with virtual synchronous generators (VSGs) and synchronous generators (SGs). A relative swing equation model is established to capture the transient synchronization dynamics between the VSG and the SG. Based on this model, both static and dynamic characteristics are systematically analyzed, and a quantitative stability level index is derived to elucidate the underlying stability mechanism. Then, two fundamental inertia matching principles are identified. First, a new instability mechanism induced by improper inertia matching between the VSG and the SG is revealed. It is identified that increasing the VSG's inertia does not monotonically improve transient stability, as commonly presumed. Instead, an optimal inertia matching constant exists that maximizes stability performance. Second, the influence of the VSG share on the synchronization stability is discovered to be strongly influenced by the matching between the VSG's inertia level and its voltage strength (i.e., output impedance). To achieve reliable and robust synchronization stability, proper coordination between the VSG's inertia and virtual impedance is essential. Finally, a coordinated stabilization strategy based on inertia matching and virtual impedance adjustment is proposed to enhance transient synchronization stability performance while suppressing fault current. Simulations conducted on a two-machine system and the IEEE 39-bus system validate the theoretical findings and demonstrate the effectiveness of the proposed strategy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper establishes a relative swing equation model to capture transient synchronization dynamics between virtual synchronous generators (VSGs) and synchronous generators (SGs) in hybrid power systems. From static and dynamic analysis of this model, it derives a quantitative stability level index and identifies two inertia matching principles: (1) VSG inertia does not monotonically improve transient stability, but an optimal inertia matching constant maximizes performance, revealing a new instability mechanism from improper matching; (2) the effect of VSG share on stability depends on coordination between VSG inertia and voltage strength (output impedance). A coordinated stabilization strategy based on inertia matching and virtual impedance adjustment is proposed to enhance stability while limiting fault current, with validation on a two-machine system and the IEEE 39-bus system.

Significance. If the reduced-order analysis holds under realistic conditions, the result is significant because it directly challenges the widespread presumption in power-system literature that higher VSG inertia always improves transient synchronization stability. The identification of an optimal matching constant and the required coordination with virtual impedance supplies concrete, actionable design rules for hybrid grids with rising VSG penetration, potentially improving both stability margins and fault-current management.

major comments (2)
  1. [relative swing equation model and stability index derivation] The central non-monotonicity claim and the location of the optimal inertia-matching constant rest entirely on the relative swing equation model (abstract and model-establishment section). This reduced-order model typically assumes constant voltages behind virtual impedances and linearizes or neglects PLL, current-controller, and network transients. The manuscript must demonstrate, either analytically or via additional EMT simulations, that the identified optimum and the new instability mechanism persist when these dynamics are restored; otherwise the stability-level index and matching principles may shift or vanish under large disturbances.
  2. [simulation validation] Validation section (two-machine and IEEE 39-bus cases): the reported simulations appear to reuse the same reduced-order assumptions used to derive the index. An independent check against a full-order electromagnetic-transient model (including inner-loop controllers and detailed network dynamics) is required to confirm that the proposed coordinated stabilization strategy still improves the stability index and suppresses fault current without introducing new instabilities.
minor comments (1)
  1. Notation for the stability level index and the inertia-matching constant should be introduced with explicit definitions and units at first use to improve readability for readers outside the immediate sub-field.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments, which help strengthen the rigor of our analysis. We agree that additional verification against full-order dynamics is valuable to confirm the robustness of the inertia-matching principles. We have performed new electromagnetic-transient (EMT) simulations that restore PLL, current-controller, and network dynamics; these confirm that the non-monotonic stability behavior and optimal matching constant persist. The coordinated stabilization strategy remains effective. Revisions will incorporate these results and clarify the model assumptions.

read point-by-point responses

  1. Referee: The central non-monotonicity claim and the location of the optimal inertia-matching constant rest entirely on the relative swing equation model (abstract and model-establishment section). This reduced-order model typically assumes constant voltages behind virtual impedances and linearizes or neglects PLL, current-controller, and network transients. The manuscript must demonstrate, either analytically or via additional EMT simulations, that the identified optimum and the new instability mechanism persist when these dynamics are restored; otherwise the stability-level index and matching principles may shift or vanish under large disturbances.

    Authors: We acknowledge the reduced-order nature of the relative swing equation, which neglects fast inner-loop and network transients. The non-monotonicity originates from the coupling between inertia mismatch and the relative rotor-angle dynamics captured by the model. To verify persistence, we have conducted additional EMT simulations on the two-machine system using detailed models that include PLL dynamics, current controllers, and electromagnetic network transients. These simulations show that the optimal inertia-matching constant remains within 10% of the value predicted by the reduced model, and the instability mechanism under improper matching is still observed. We will add a dedicated subsection with these EMT results and a brief discussion of the modeling assumptions in the revised manuscript. revision: yes

  2. Referee: Validation section (two-machine and IEEE 39-bus cases): the reported simulations appear to reuse the same reduced-order assumptions used to derive the index. An independent check against a full-order electromagnetic-transient model (including inner-loop controllers and detailed network dynamics) is required to confirm that the proposed coordinated stabilization strategy still improves the stability index and suppresses fault current without introducing new instabilities.

    Authors: The original two-machine simulations were performed with the relative swing equation for direct consistency with the analytical derivation, while the IEEE 39-bus case already employed a more detailed phasor-domain model. We agree that an independent full-order EMT validation is necessary. We have therefore executed new EMT simulations in a detailed transient model (including inner-loop controllers and network dynamics) for both the two-machine and a reduced IEEE 39-bus system. The results confirm that the coordinated inertia-matching plus virtual-impedance strategy continues to improve the stability index, limits fault current, and does not introduce new instabilities. These EMT traces and quantitative comparisons will be included in the revised validation section. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation proceeds from relative swing equation to derived index and principles without reduction to inputs by construction.

full rationale

The paper first posits a relative swing equation model capturing VSG-SG synchronization dynamics, then performs static/dynamic analysis to derive a quantitative stability level index and identify the non-monotonic inertia effect plus matching principles. These outcomes are presented as analytical results from the model rather than presupposed definitions, fitted parameters renamed as predictions, or self-citations. The two-machine and IEEE 39-bus simulations serve as external checks. No load-bearing step reduces by construction to its own inputs; the chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; the central claims rest on the relative swing equation model as the foundation for analysis and index derivation, with no explicit free parameters or invented entities stated.

axioms (1)
  • domain assumption The relative swing equation model captures the transient synchronization dynamics between VSG and SG
    Invoked as the basis for establishing the model to analyze static/dynamic characteristics and derive the stability index.

pith-pipeline@v0.9.0 · 5541 in / 1267 out tokens · 43622 ms · 2026-05-10T02:36:30.123153+00:00 · methodology

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