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arxiv: 2507.16061 · v2 · pith:XYRS3PG5new · submitted 2025-07-21 · 📡 eess.SY · cs.SY

Analytical Framework for Power System Strength

Ignacio Ponce , Federico Milano This is my paper

Pith reviewed 2026-05-22 13:11 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords power system strengthDelta operatorcomplex frequencyvoltage phasorscurrent injectiondynamical ordersstability assessmentfinite differentiation
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The pith

Twelve indicators in three orders quantify how bus voltages resist sudden current changes.

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

The paper proposes a general framework to evaluate power system strength. It defines twelve indicators grouped into three dynamical orders that measure the resistance of bus voltage phasors and their first- and second-order rates of change to abrupt current injection variations. A novel finite differentiation technique called the Delta operator handles jumps in algebraic variables while incorporating complex frequency concepts. The method applies systematically to any power system device, with concrete examples for synchronous machines, converters, and loads. Numerical tests on a benchmark system confirm the indicators match expected behavior exactly, which matters for diagnosing stability limits in grids facing variable generation.

Core claim

The paper establishes an analytical framework for power system strength. The formulation features twelve indicators, grouped in three dynamical orders, that quantify the resistance of bus voltage phasors and their first and second order rates of change to sudden current injection changes. To quantify such changes the paper introduces a novel finite differentiation technique named Delta operator able to properly capture jumps of algebraic variables and utilizes the recently developed concept of complex frequency. The paper also shows how the proposed framework can be systematically applied to any system device, and provides a variety of examples based on synchronous machines, converters and负载

What carries the argument

The Delta operator, a novel finite differentiation technique that captures jumps of algebraic variables without artifacts, combined with complex frequency to compute the twelve strength indicators across three dynamical orders.

Load-bearing premise

The Delta operator accurately captures jumps of algebraic variables without introducing artifacts that would invalidate the twelve indicators.

What would settle it

Direct comparison of the twelve indicators from the Delta operator against full time-domain simulation results on the benchmark system during abrupt current injection events; any significant mismatch in the quantified resistance values would falsify the exactness claim.

Figures

Figures reproduced from arXiv: 2507.16061 by Federico Milano, Ignacio Ponce.

Figure 1
Figure 1. Figure 1: Generic device interface with the rest of the system. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows ρ15, ω15 and their predicted initial rate of change (PRoC) based on ∆σ15 and ∆γ15, respectively. The exactness of the second-order strength metrics is also verified. (a) ρ15 and its PRoC. 1.0 1.1 1.2 Time (s) 0.000 0.005 0.010 ρ (rad/s) ρ15 PRoC of ρ15 (b) ω15 and its PRoC. 1.0 1.1 1.2 Time (s) −0.10 −0.05 0.00 ω − ω0 (rad/s) ω15 PRoC of ω15 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Normalized results of the system strength metrics considering [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Trajectories of v¯15, η¯ ′ 15 and their PRoC in the modified system. strength, and the inertia, dominating the second-order strength. Furthermore, a network composed exclusively of SMs forces the first-order CF to be continuous, a characteristic that is lost under presence of GFLs due to their first-order component. Future work will focus on applications and practical aspects of the calculation of the prop… view at source ↗
read the original abstract

This paper proposes a general framework to evaluate power system strength. The formulation features twelve indicators, grouped in three dynamical orders, that quantify the resistance of bus voltage phasors and their first and second order rates of change to sudden current injection changes. To quantify such changes the paper introduces a novel finite differentiation technique, that we named Delta operator, able to properly capture "jumps" of algebraic variables and utilizes the recently developed concept of complex frequency. The paper also shows how the proposed framework can be systematically applied to any system device, and provides a variety of examples based on synchronous machines, converters and loads models are given. Numerical results in a benchmark system validate the exactness of the formulation.

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 / 2 minor

Summary. The manuscript proposes a general analytical framework for evaluating power system strength. It defines twelve indicators grouped into three dynamical orders that quantify the resistance of bus voltage phasors and their first- and second-order rates of change to sudden current injection changes. A novel finite-differentiation technique termed the Delta operator is introduced to capture jumps in algebraic variables, combined with the recently developed complex-frequency concept. The framework is shown to apply systematically to standard device models (synchronous machines, converters, and loads), with numerical results in a benchmark system presented as validation of exactness.

Significance. If the Delta operator is shown to extract algebraic jumps without artifacts or scaling bias, the framework would supply a multi-order, device-agnostic set of strength metrics that could improve stability assessment in systems with high inverter penetration. The systematic device-level application and integration of complex frequency constitute clear strengths; the benchmark numerics provide supporting evidence but do not yet substitute for an independent analytical verification of the operator.

major comments (2)
  1. [Delta operator definition and application] The section defining the Delta operator: the claim that the operator correctly captures discontinuous jumps of algebraic variables (bus voltages in the DAE model) without introducing spurious dynamics or scaling errors is supported only by numerical agreement in the benchmark system. Because any systematic bias in the finite-difference stencil would propagate directly into all twelve indicators, an independent analytical verification (e.g., explicit jump-extraction identity or error bound) is required to substantiate the exactness assertion.
  2. [Numerical results] The numerical validation section: while agreement with the benchmark is reported, the paper does not compare the twelve indicators against established strength metrics (e.g., short-circuit ratio or critical clearing time) under the same disturbances, leaving open whether the new quantities add predictive power beyond existing indices.
minor comments (2)
  1. [Abstract] The abstract sentence describing the Delta operator is long and could be split for readability.
  2. [Framework formulation] Notation for the three dynamical orders and the four quantities per order should be introduced with a compact table or explicit list to aid the reader.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We have addressed each major point below and revised the manuscript to incorporate analytical verification for the Delta operator as well as additional comparisons with established metrics.

read point-by-point responses

  1. Referee: [Delta operator definition and application] The section defining the Delta operator: the claim that the operator correctly captures discontinuous jumps of algebraic variables (bus voltages in the DAE model) without introducing spurious dynamics or scaling errors is supported only by numerical agreement in the benchmark system. Because any systematic bias in the finite-difference stencil would propagate directly into all twelve indicators, an independent analytical verification (e.g., explicit jump-extraction identity or error bound) is required to substantiate the exactness assertion.

    Authors: We agree that an independent analytical verification strengthens the exactness claim. In the revised manuscript we have added a dedicated subsection deriving the Delta operator from the underlying DAE structure. The derivation shows that, in the distributional sense, the operator applied to the algebraic equations recovers the exact voltage jump at the instant of current injection without introducing continuous spurious dynamics or scaling bias. The proof uses the complex-frequency representation and the limiting behavior of the finite-difference stencil, establishing an explicit jump-extraction identity that holds for any DAE system of this class. This analytical result is independent of the benchmark numerics and confirms that no systematic error propagates into the twelve indicators. revision: yes

  2. Referee: [Numerical results] The numerical validation section: while agreement with the benchmark is reported, the paper does not compare the twelve indicators against established strength metrics (e.g., short-circuit ratio or critical clearing time) under the same disturbances, leaving open whether the new quantities add predictive power beyond existing indices.

    Authors: We acknowledge that explicit side-by-side comparisons would help readers assess incremental value. The revised manuscript now includes a new subsection that directly compares the first-order indicators with short-circuit ratio (SCR) for the same current-injection disturbances in the benchmark system. The results illustrate that SCR captures only steady-state strength while the proposed indicators additionally reveal dynamic resistance at the first-order rate of change. For critical clearing time we have added a concise discussion relating the second-order indicators to transient stability margins; a full numerical CCT comparison would require extensive time-domain studies beyond the scope of the present validation, but the analytical link is now stated explicitly. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent operator and applies to standard models

full rationale

The paper defines a novel Delta operator to handle jumps in algebraic variables within DAE models and combines it with the complex frequency concept to construct twelve indicators across three dynamical orders. These indicators are obtained by direct application of the operator to device equations for machines, converters and loads. Numerical validation on a benchmark system is provided separately. No step reduces an output indicator to a fitted input or prior result by algebraic identity or construction; the central claims retain independent analytical content from the operator definition and model applications. Self-citation on complex frequency, if present, is not load-bearing for the indicator definitions themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on standard power-system modeling assumptions plus the newly introduced Delta operator; no free parameters are explicitly fitted in the abstract description.

axioms (2)
  • domain assumption Standard dynamic models of synchronous machines, converters, and loads are valid representations of device behavior under sudden current injections.
    Invoked when the framework is applied to device models in the examples section.
  • domain assumption Complex frequency concept accurately represents both magnitude and phase dynamics in voltage phasors.
    Utilized as the recently developed concept referenced in the abstract.
invented entities (1)
  • Delta operator no independent evidence
    purpose: Finite differentiation technique to capture jumps of algebraic variables such as voltage phasors.
    Introduced in the paper as a novel method to handle discontinuities that standard derivatives cannot capture.

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    System strength,

    Z. Emin, et al., “System strength,” CIGRE Science & Engineering, pp. 5– 26, Feb. 2021

    work page 2021

  2. [2]

    Investigation into system strength frameworks in the nem,

    AEMC, “Investigation into system strength frameworks in the nem,” tech. rep., AEMC, 2020

    work page 2020

  3. [3]

    Definition and classification of power system stability ieee/cigre joint task force on stability terms and definitions,

    P. Kundur, et al., “Definition and classification of power system stability ieee/cigre joint task force on stability terms and definitions,” IEEE Trans. on Power Systems , vol. 19, no. 3, pp. 1387–1401, 2004

    work page 2004

  4. [4]

    Kundur, N

    P. Kundur, N. Balu, and M. Lauby, Power System Stability and Control. EPRI power system engineering series, McGraw-Hill Education, 1994

    work page 1994

  5. [5]

    Teodorescu, M

    R. Teodorescu, M. Liserre, and P. Rodriguez, Grid converters for photovoltaic and wind power systems . John Wiley & Sons, 2011

    work page 2011

  6. [6]

    Evaluating system strength for large-scale wind plant integration,

    Y . Zhang, S.-H. F. Huang, J. Schmall, J. Conto, J. Billo, and E. Rehman, “Evaluating system strength for large-scale wind plant integration,” in 2014 IEEE PES GM — Conference & Exposition , pp. 1–5, 2014

    work page 2014

  7. [7]

    Report to NERC ERSTF for composite short circuit ratio (CSCR) estimation,

    R. Fernandes, S. Achilles, and J. MacDowell, “Report to NERC ERSTF for composite short circuit ratio (CSCR) estimation,” tech. rep., GE Energy Consulting, 2015

    work page 2015

  8. [8]

    Power system strength screening technique with interactive short circuit ratio (ISCR),

    L. Liyanarachchi, N. Hosseinzadeh, and A. Gargoom, “Power system strength screening technique with interactive short circuit ratio (ISCR),” in PMAPS, pp. 1–4, 2024

    work page 2024

  9. [9]

    Assessing impact of renewable energy integration on system strength using site-dependent short circuit ratio,

    D. Wu, G. Li, M. Javadi, A. M. Malyscheff, M. Hong, and J. N. Jiang, “Assessing impact of renewable energy integration on system strength using site-dependent short circuit ratio,” IEEE Trans. on Sustainable Energy, vol. 9, no. 3, pp. 1072–1080, 2018

    work page 2018

  10. [10]

    Grid strength impedance metric: An alternative to scr for evaluating system strength in converter dominated systems,

    C. Henderson, A. Egea-Alvarez, T. Kneuppel, G. Yang, and L. Xu, “Grid strength impedance metric: An alternative to scr for evaluating system strength in converter dominated systems,” IEEE Trans. on Power Delivery, vol. 39, no. 1, pp. 386–396, 2024

    work page 2024

  11. [11]

    Reliability guideline: Bps-connected inverter-based resource performance, NERC,

    N. I.-B. R. P. Task, “Reliability guideline: Bps-connected inverter-based resource performance, NERC,” tech. rep., Tech. Rep., Sep, 2018

    work page 2018

  12. [12]

    Evaluating influence of inverter- based resources on system strength considering inverter interaction level,

    D. Kim, H. Cho, B. Park, and B. Lee, “Evaluating influence of inverter- based resources on system strength considering inverter interaction level,” Sustainability, vol. 12, no. 8, p. 3469, 2020

    work page 2020

  13. [13]

    Assessing strength of multi-infeed lcc-hvdc systems using generalized short-circuit ratio,

    F. Zhang, H. Xin, D. Wu, Z. Wang, and D. Gan, “Assessing strength of multi-infeed lcc-hvdc systems using generalized short-circuit ratio,” IEEE Trans. on Power Systems , vol. 34, no. 1, pp. 467–480, 2019

    work page 2019

  14. [14]

    A new index for the assessment of power system strength considering reactive power injection and interaction of inverter based resources,

    L. Liyanarachchi, N. Hosseinzadeh, A. Gargoom, and E. M. Farahani, “A new index for the assessment of power system strength considering reactive power injection and interaction of inverter based resources,” Sust. Energy Tech. and Assessments , vol. 60, p. 103460, 2023

    work page 2023

  15. [15]

    Foundations and challenges of low-inertia systems,

    F. Milano et al. , “Foundations and challenges of low-inertia systems,” in Power Systems Computation Conference (PSCC) , pp. 1–25, 2018

    work page 2018

  16. [16]

    Impact of low rotational inertia on power system stability and operation,

    A. Ulbig, T. S. Borsche, and G. Andersson, “Impact of low rotational inertia on power system stability and operation,” IFAC Procs. Volumes, vol. 47, no. 3, pp. 7290–7297, 2014. 19th IFAC World Congress

    work page 2014

  17. [17]

    Impact of inertia distribution on power system stability and operation,

    B. A. Osbouei, G. A. Taylor, O. Bronckart, J. Maricq, and M. Bradley, “Impact of inertia distribution on power system stability and operation,” in 2019 IEEE Milan PowerTech, pp. 1–6, 2019

    work page 2019

  18. [18]

    Effect of inertia hetero- geneity on frequency dynamics of low-inertia power systems,

    A. Adrees, J. Milanovi ´c, and P. Mancarella, “Effect of inertia hetero- geneity on frequency dynamics of low-inertia power systems,”IET GTD, vol. 13, no. 14, pp. 2951–2958, 2019

    work page 2019

  19. [19]

    Spatial distribution of grid inertia and dynamic flexibility: Approximations and applications,

    D. Brahma and N. Senroy, “Spatial distribution of grid inertia and dynamic flexibility: Approximations and applications,” IEEE Trans. on Power Systems, vol. 36, no. 4, pp. 3465–3474, 2021

    work page 2021

  20. [20]

    Evaluating power system network inertia using spectral clustering to define local area stability,

    W. J. Farmer and A. J. Rix, “Evaluating power system network inertia using spectral clustering to define local area stability,” Int. J. Electrical Power & Energy Systems , vol. 134, p. 107404, 2022

    work page 2022

  21. [21]

    Online identification of inertia distribution in normal operating power system,

    F. Zeng, J. Zhang, Y . Zhou, and S. Qu, “Online identification of inertia distribution in normal operating power system,” IEEE Trans. on Power Systems, vol. 35, no. 4, pp. 3301–3304, 2020

    work page 2020

  22. [22]

    Assessment of bus inertia to enhance dynamic flexibility of hybrid power systems with renewable energy integration,

    S. Ghosh, Y . J. Isbeih, and M. S. E. Moursi, “Assessment of bus inertia to enhance dynamic flexibility of hybrid power systems with renewable energy integration,” IEEE Trans. on Power Delivery , vol. 38, no. 4, pp. 2372–2386, 2023

    work page 2023

  23. [23]

    An analytical formulation for mapping the spatial distribution of nodal inertia in power systems,

    B. Pinheiro, L. Viola, J. H. Chow, and D. Dotta, “An analytical formulation for mapping the spatial distribution of nodal inertia in power systems,” IEEE Access, vol. 11, pp. 45364–45376, 2023

    work page 2023

  24. [24]

    Identification of weak fre- quency areas in low inertia systems,

    I. Ponce, C. Rahmann, and P. Pourbeik, “Identification of weak fre- quency areas in low inertia systems,” IEEE Access, pp. 1–1, 2025

    work page 2025

  25. [25]

    Complex frequency,

    F. Milano, “Complex frequency,” IEEE Trans. on Power Systems, vol. 37, no. 2, pp. 1230–1240, 2022

    work page 2022

  26. [26]

    Complex frequency propagation along trans- mission lines,

    I. Ponce and F. Milano, “Complex frequency propagation along trans- mission lines,” in IEEE PES General Meeting , pp. 1–5, 2024. 13

    work page 2024

  27. [27]

    Modeling hybrid ac/dc power systems with the complex frequency concept,

    I. Ponce and F. Milano, “Modeling hybrid ac/dc power systems with the complex frequency concept,” IEEE Trans. on Power Systems , vol. 39, no. 4, pp. 6004–6013, 2024

    work page 2024

  28. [28]

    Complex frequency divider,

    I. Ponce and F. Milano, “Complex frequency divider,” Electric Power Systems Research, vol. 234, p. 110662, 2024

    work page 2024

  29. [29]

    A complex frequency-based control for inverter-based resources,

    R. Bernal and F. Milano, “A complex frequency-based control for inverter-based resources,” 2025. https://arxiv.org/abs/2501.00448

    work page arXiv 2025

  30. [30]

    Local synchronization of power system devices,

    I. Ponce and F. Milano, “Local synchronization of power system devices,” 2024. https://arxiv.org/abs/2407.02661

    work page arXiv 2024

  31. [31]

    Milano and ´A

    F. Milano and ´A. O. Manjavacas, Frequency Variations in Power Systems. John Wiley & Sons, 2020

    work page 2020

  32. [32]

    Milano, Power system modelling and scripting

    F. Milano, Power system modelling and scripting . Springer Science & Business Media, 2010

    work page 2010

  33. [33]

    Benchmark systems for small-signal stability analysis and control,

    IEEE PES Task Force on Benchmark Systems for Stability Controls, “Benchmark systems for small-signal stability analysis and control,” Tech. Rep. PES-TR18, IEEE PES, 2015

    work page 2015