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arxiv: 2604.21262 · v2 · submitted 2026-04-23 · 📡 eess.SY · cs.SY

Frequency Security Assessment in Power Systems With High Penetration of Renewables Considering Spatio-Temporal Frequency Distribution

Changjun He , Hua Geng , Xiuqiang He , Chen Shen , Yushuang Liu This is my paper

Pith reviewed 2026-05-09 21:20 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords frequency securityrenewable energy integrationnodal inertiaeffective nodal frequencypower system stabilityspatio-temporal frequencyinertia assessmentsystem security index
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The pith

The effective nodal inertia is the dominant factor determining frequency security at individual nodes in power systems with high renewable penetration.

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

The paper develops a simplified model for frequency behavior at each node in a power grid by distilling dynamics down to effective nodal inertia, damping, and regulation terms. This model yields closed-form expressions for how frequency evolves after a disturbance and for the security limits that must be met. Quantitative checks show that effective nodal inertia has the strongest effect on whether those limits are breached, which in turn permits an exact calculation of the smallest inertia value needed at each node. With those critical values in hand, the authors define a system-level security index and an efficient evaluation procedure that uses precomputed tables for rapid checks.

Core claim

Under temporary power disturbances, the effective nodal inertia emerges as the parameter with greatest influence on nodal frequency security. This leads to an analytical derivation of the critical nodal inertia required to maintain security at every node. A system frequency security index is then constructed from the ratio of actual to critical inertia, supporting an offline-online assessment workflow.

What carries the argument

The effective nodal frequency model, which approximates nodal frequency dynamics using constant effective nodal inertia, damping, and primary regulation parameters.

Load-bearing premise

That keeping only the dominant constant component in the effective nodal frequency model is enough to capture the essential dynamics and security behavior during actual disturbances.

What would settle it

Running a full dynamic simulation on the modified IEEE 39-bus system with a known temporary disturbance and checking whether the observed nodal frequency nadir and rate of change match the predictions from the critical ENI formula; a consistent mismatch at multiple nodes would falsify the model's sufficiency.

Figures

Figures reproduced from arXiv: 2604.21262 by Changjun He, Chen Shen, Hua Geng, Xiuqiang He, Yushuang Liu.

Figure 1
Figure 1. Figure 1: Topology and frequency distribution of a power system integrated [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The actual nodal inertia/frequency, the effective nodal inertia/frequency, and the remaining components. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Transient process under the temporary power disturbances scenario. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Flow of data-based calculation for nodal frequency. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The actual nodal frequency and ENF at nodes (a) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Unit-less sensitivity of (a) the maximum RoCoF and (b) the maximum [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Modified IEEE 39-bus system. A. Critical Nodal Inertia and ENI Results Take case II as an example, the critical nodal inertia constraint by the frequency security is plotted with the red line in Fig.9. It can be seen that the critical nodal inertia ranges [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Maximum RoCoF under (a) case I and (b) case II. [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The maximum frequency deviation under (a) case I and (b) case II. [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: SFSA results of (a) cases I and (b) case II. [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
read the original abstract

The increasing integration of renewable energy sources exacerbates the spatial and temporal differences in frequency across the power system, posing a serious challenge to the accurate and efficient assessment of system frequency security. To address this issue, a generic effective nodal frequency (ENF) model is first established to concisely characterize nodal frequency dynamics. This model is featured by the effective nodal inertia (ENI), damping, and primary regulation parameters, which retain only the dominant constant component governing nodal frequency dynamic performance. This model enables the tractable analytical formulation of nodal frequency trajectory and the key frequency security indicators. Quantitative analysis under the temporary power disturbance condition reveals that the ENI is the most influential parameter governing frequency security. Consequently, the critical nodal inertia for ensuring nodal frequency security is analytically derived. A system-level frequency security index based on the actual ENI and critical nodal inertia is proposed. On the basis of the proposed index, the system frequency security assessment is carried out with the procedure of ``offline calculation and online evaluation'', which is achieved using a lookup table approach and an interpolation method. Simulations on the modified IEEE 39-bus system verify the effectiveness of the proposed assessment method.

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

3 major / 2 minor

Summary. The paper proposes a generic effective nodal frequency (ENF) model that simplifies nodal frequency dynamics in high-renewable power systems by retaining only the dominant constant components of effective nodal inertia (ENI), damping, and primary regulation parameters. This enables analytical derivation of nodal frequency trajectories and security indicators (nadir, RoCoF, settling). Quantitative analysis under temporary power disturbances identifies ENI as the most influential parameter, from which critical nodal inertia is analytically derived. A system-level frequency security index is then defined, supporting an offline-calculation/online-evaluation procedure via lookup tables and interpolation. The approach is verified through simulations on a modified IEEE 39-bus system.

Significance. If the ENF reduction is shown to be sufficiently accurate, the work provides a tractable analytical framework for assessing spatio-temporal frequency security, which could support practical offline-online tools for system operators. The explicit identification of ENI as dominant and the derivation of critical inertia are potentially useful contributions, as is the emphasis on nodal-level security in renewable-heavy grids. However, the significance is moderated by the absence of quantitative validation of the model approximation.

major comments (3)
  1. [ENF model] ENF model section (as described in the abstract): The reduction to dominant constant components for ENI, damping, and regulation is presented as enabling tractable analytics, but no error bounds, residual analysis, or direct comparison to the full-order dynamic model are provided. This is load-bearing because the subsequent ranking of ENI influence and the closed-form critical inertia expression are derived directly from the reduced model.
  2. [Quantitative analysis and critical inertia] Quantitative analysis and critical inertia derivation (abstract and results): The claim that ENI is the most influential parameter and the analytical critical nodal inertia formula lack sensitivity checks across disturbance locations, durations, or renewable variability levels, as well as comparison of predicted vs. simulated frequency trajectories. Without these, the system-level index and the offline/online procedure rest on an unquantified approximation.
  3. [Simulations] Verification on modified IEEE 39-bus system (simulation section): The effectiveness claim is supported only by the modified 39-bus case; no full-order model benchmarks, error metrics (e.g., nadir/RoCoF deviation), or tests under varied renewable penetration scenarios are reported, weakening confidence that the index reliably captures security under realistic conditions.
minor comments (2)
  1. [Abstract] The abstract and introduction could more explicitly state the scope of the ENF approximation (e.g., valid disturbance types) to help readers assess applicability.
  2. [Notation] Notation: Ensure consistent use of 'ENI' versus 'effective nodal inertia' and define all security indicators (nadir, RoCoF) with equations in the main text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive review of our manuscript. The comments have identified key areas where additional validation can strengthen the presentation of the ENF model and its applications. We provide point-by-point responses below and will revise the manuscript to incorporate the suggested analyses.

read point-by-point responses

  1. Referee: [ENF model] ENF model section (as described in the abstract): The reduction to dominant constant components for ENI, damping, and regulation is presented as enabling tractable analytics, but no error bounds, residual analysis, or direct comparison to the full-order dynamic model are provided. This is load-bearing because the subsequent ranking of ENI influence and the closed-form critical inertia expression are derived directly from the reduced model.

    Authors: We acknowledge that the original manuscript did not include explicit error bounds or residual analysis for the ENF approximation. The reduction is motivated by retaining only the dominant constant terms that govern the primary frequency dynamics, based on the separation of time scales between inertia, damping, and slower regulation effects. To directly address this concern, we will add a dedicated subsection in the revised manuscript that compares the ENF-predicted nodal frequency trajectories against the full-order dynamic model. This will include quantitative error metrics (e.g., maximum deviation in nadir and RoCoF) across multiple disturbance scenarios, thereby quantifying the approximation accuracy and supporting the subsequent analytical derivations. revision: yes

  2. Referee: [Quantitative analysis and critical inertia] Quantitative analysis and critical inertia derivation (abstract and results): The claim that ENI is the most influential parameter and the analytical critical nodal inertia formula lack sensitivity checks across disturbance locations, durations, or renewable variability levels, as well as comparison of predicted vs. simulated frequency trajectories. Without these, the system-level index and the offline/online procedure rest on an unquantified approximation.

    Authors: The original quantitative analysis focused on temporary power disturbances to establish ENI dominance and derive the closed-form critical inertia. We agree that broader sensitivity checks would improve robustness. In the revision, we will expand the results section to include sensitivity analyses with respect to disturbance location, duration, and varying renewable penetration levels. We will also add direct comparisons of ENF-predicted frequency trajectories versus full simulations, reporting deviation metrics for nadir, RoCoF, and settling frequency. These additions will provide quantitative support for the system-level security index and the offline-calculation/online-evaluation procedure. revision: yes

  3. Referee: [Simulations] Verification on modified IEEE 39-bus system (simulation section): The effectiveness claim is supported only by the modified 39-bus case; no full-order model benchmarks, error metrics (e.g., nadir/RoCoF deviation), or tests under varied renewable penetration scenarios are reported, weakening confidence that the index reliably captures security under realistic conditions.

    Authors: The verification used the modified IEEE 39-bus system with high renewable penetration as a representative test case. We recognize the value of additional benchmarks. The revised manuscript will incorporate full-order model comparisons with explicit error metrics for nadir and RoCoF. We will also present results under multiple renewable penetration scenarios to demonstrate the index's behavior across different system conditions. These extensions will strengthen the evidence for the method's effectiveness in realistic renewable-heavy grids. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is model-based and self-contained

full rationale

The paper first defines the ENF model by retaining only dominant constant components of nodal inertia, damping, and regulation parameters, then analytically formulates nodal frequency trajectories and security indicators directly from this reduced model. Quantitative analysis under temporary disturbances identifies ENI influence and derives critical nodal inertia as an analytical consequence of the same model equations. The system-level index is constructed from the actual ENI values versus these derived critical values, with assessment performed via lookup tables and interpolation. No step reduces the final index or critical inertia formula to a fitted parameter by construction, no load-bearing self-citation is invoked for uniqueness or ansatz, and the chain remains independent of the target security claims rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the dominant-constant reduction for nodal dynamics and standard power-system swing-equation assumptions; no new physical entities are postulated.

axioms (1)
  • domain assumption Nodal frequency dynamics can be adequately captured by retaining only the dominant constant components of inertia, damping, and primary regulation.
    This reduction is the defining step of the ENF model stated in the abstract.

pith-pipeline@v0.9.0 · 5516 in / 1123 out tokens · 49840 ms · 2026-05-09T21:20:26.152988+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    A unified method for assessing frequency support capability of diverse renewable power generation units,

    G. Hu, Y . Yang, H. Xin, L. Huang, X. Liu, H. Ruan, and H. Gao, “A unified method for assessing frequency support capability of diverse renewable power generation units,”IEEE Transactions on Sustainable Energy, vol. 17, no. 1, pp. 377–392, 2026

    work page 2026

  2. [2]

    Review of frequency regulation requirements for wind power plants in international grid codes,

    L. Li, D. Zhu, X. Zou, J. Hu, Y . Kang, and J. M. Guerrero, “Review of frequency regulation requirements for wind power plants in international grid codes,”Renewable and Sustainable Energy Reviews, vol. 187, p. 113731, 2023

    work page 2023

  3. [3]

    What does the gb power outage on 9 august 2019 tell us about the current state of decarbonised power systems?,

    J. Bialek, “What does the gb power outage on 9 august 2019 tell us about the current state of decarbonised power systems?,”Energy Policy, vol. 146, p. 111821, 2020

    work page 2019

  4. [4]

    Analytic assessment of the power system frequency security,

    P. Ju, Y . Zheng, Y . Jin, C. Qin, Y . Jiang, and L. Cao, “Analytic assessment of the power system frequency security,”IET Generation, Transmission & Distribution, vol. 15, no. 15, pp. 2215–2225, 2021

    work page 2021

  5. [5]

    Trustworthy data-driven prefault frequency security assessment for power systems using disturbance-informed learning,

    Y . Lan, Y . Zhang, W. Yao, S. Wei, T. Chen, H. Shuai, and J. Wen, “Trustworthy data-driven prefault frequency security assessment for power systems using disturbance-informed learning,”IEEE Transactions on Industrial Informatics, vol. 22, no. 3, pp. 1950–1961, 2026

    work page 1950

  6. [6]

    Frequency stability of synchronous machines and grid-forming power converters,

    A. Tayyebi, D. Groß, A. Anta, F. Kupzog, and F. D ¨orfler, “Frequency stability of synchronous machines and grid-forming power converters,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 1004–1018, 2020

    work page 2020

  7. [7]

    Online frequency security assessment based on analytical model considering limiting modules,

    Y . Zhang, Q. Guo, Y . Zhou, and H. Sun, “Online frequency security assessment based on analytical model considering limiting modules,” CSEE Journal of Power and Energy Systems, vol. 8, no. 5, pp. 1363– 1372, 2022

    work page 2022

  8. [8]

    Online inertia allocation for grid-connected renewable energy systems based on generic asf model under frequency nadir constraint,

    Z. Peng, Q. Peng, Y . Zhang, H. Han, Y . Yin, and T. Liu, “Online inertia allocation for grid-connected renewable energy systems based on generic asf model under frequency nadir constraint,”IEEE Transactions on Power Systems, vol. 39, no. 1, pp. 1615–1627, 2024

    work page 2024

  9. [9]

    Long-term frequency stability assessment based on extended frequency response model,

    Y . Xie, C. Li, H. Zhang, H. Sun, and V . Terzija, “Long-term frequency stability assessment based on extended frequency response model,”IEEE Access, vol. 8, pp. 122444–122455, 2020

    work page 2020

  10. [10]

    Security as- sessment method for inertia and frequency stability of high proportional renewable energy system,

    G. Zhang, J. Ren, Y . Zeng, F. Liu, S. Wang, and H. Jia, “Security as- sessment method for inertia and frequency stability of high proportional renewable energy system,”International Journal of Electrical Power & Energy Systems, vol. 153, p. 109309, 2023

    work page 2023

  11. [11]

    Study on the necessity and role of inertia in power system frequency response,

    M. Sun, G. Xie, L. Chen, Y . Liu, X. Li, and Y . Min, “Study on the necessity and role of inertia in power system frequency response,” in2020 IEEE 4th Conference on Energy Internet and Energy System Integration (EI2), pp. 155–159, 2020

    work page 2020

  12. [12]

    Inertia security evaluation and application in low-inertia power systems,

    W. Zhang, Y . Wen, and C. Y . Chung, “Inertia security evaluation and application in low-inertia power systems,”IEEE Transactions on Power Systems, vol. 40, no. 2, pp. 1725–1737, 2025

    work page 2025

  13. [13]

    Extracting spatial-temporal characteristics of frequency dynamic in large-scale power grids,

    N. Ma and D. Wang, “Extracting spatial-temporal characteristics of frequency dynamic in large-scale power grids,”IEEE Transactions on Power Systems, vol. 34, no. 4, pp. 2654–2662, 2019

    work page 2019

  14. [14]

    Analysis and control of frequency stability in low-inertia power systems: A review,

    C. He, H. Geng, K. Rajashekara, and A. Chandra, “Analysis and control of frequency stability in low-inertia power systems: A review,” IEEE/CAA Journal of Automatica Sinica, vol. 11, no. 12, pp. 2363–2383, 2024

    work page 2024

  15. [15]

    Frequency security assessment of power systems based on dynamic inertia con- stants,

    J. Fu, Q. Zhang, W.-L. Tao, X.-T. Wang, and H.-W. Wang, “Frequency security assessment of power systems based on dynamic inertia con- stants,” in2025 10th International Symposium on Advances in Electrical, Electronics and Computer Engineering (ISAEECE), pp. 119–123, 2025

    work page 2025

  16. [16]

    Response characteristics and regulation feasibility of dc charging station controlled by gfm/gfl virtual inertia for grid frequency stability,

    S. Ke, L. Ding, X. Shi, P. Fan, H. Wang, L. Chen, J. Yang, and C. Y . Chung, “Response characteristics and regulation feasibility of dc charging station controlled by gfm/gfl virtual inertia for grid frequency stability,”IEEE Transactions on Transportation Electrification, vol. 11, no. 3, pp. 7743–7758, 2025

    work page 2025

  17. [17]

    Frequency support strategy for fast response energy storage systems,

    A. E. Leon, ´A. R. del Nozal, and J. M. Mauricio, “Frequency support strategy for fast response energy storage systems,”IEEE Transactions on Power Systems, vol. 39, no. 3, pp. 5439–5442, 2024. 11

    work page 2024

  18. [18]

    A new fault type classification method in the presence of inverter-based resources,

    M. M. Mobashsher, A. A. Abdoos, S. M. Hosseini, S. M. Hashemi, and M. Sanaye-Pasand, “A new fault type classification method in the presence of inverter-based resources,”International Journal of Electrical Power & Energy Systems, vol. 147, p. 108793, 2023

    work page 2023

  19. [19]

    Inertia matching principle: Improving transient synchronization stability in hybrid power systems with vsgs and sgs,

    C. He, L. Zhang, Q. Liu, and R. Zou, “Inertia matching principle: Improving transient synchronization stability in hybrid power systems with vsgs and sgs,” 2026

    work page 2026