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Quantifying Power to Voltage and Frequency Dynamics for Oscillation Propagation Assessment

Onur Alican, Dionysios Moutevelis, Marc Cheah-Mañe, Oriol Gomis-Bellmunt, Eduardo Prieto-Araujo

arXiv:2609.03929v1eess.SY

TL;DR

IBR-driven multi-timescale oscillations can propagate through interconnected power systems, making their sensitivity and pathways difficult to assess across voltage and frequency variables. The paper derives FSI and VSI from linear EMT-based PQ–V ω transfer functions and validates them against EMT simulations. The indices identify frequency sensitivity and propagation patterns, including resonance alignment and disturbance-dependent spatial amplification in modified IEEE 68-bus cases.

  • Problem

    IBR-driven oscillations can propagate through large interconnected systems, while their source, frequency, and pathways are difficult to identify across voltage and frequency variables.

  • Method

    The paper derives FSI and VSI from linear EMT-based PQ–V ω transfer functions linking bus power injections to bus voltages and generator rotor speeds.

  • Results

    The indices align with EMT time-domain behavior: FSI identifies the 74.5 Hz modal resonance, while VSI and frequency responses capture disturbance-dependent propagation and amplification patterns.

  • Takeaways & Limitations

    Global and frequency-based indices provide complementary tools for detecting vulnerable variables and assessing how oscillations propagate through the network.

Abstract

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The integration of Inverter-Based Resources (IBRs) into power systems introduces multi-timescale dynamics and oscillations which may propagate to distant areas and endanger the safe system operation. These oscillations pose a significant challenge as their source, frequency, and propagation pathways are often challenging to identify in large interconnected systems, comprising numerous synchronous machines and IBRs. For the above reasons, analytical tools that identify the sensitivity of the system to oscillations within a large frequency spectrum, affecting both voltage and frequency variables across various network locations, are of interest. This paper addresses this topic by introducing a frequency-domain framework based on linear analysis, which characterizes the sensitivity to oscillations of each network bus voltage and of each generation unit frequency. Within this framework, two quantitative indicators are proposed, namely the Frequency Sensitivity Index (FSI) and Voltage Sensitivity Index (VSI). These indexes are derived analytically from the transfer functions which relate the active and reactive power injections to each bus with the voltage and frequency variables across the network, derived from the linear Electromagnetic Transient (EMT) power system model. The proposed indices quantify the sensitivity of the system to oscillations of different type and frequency, providing insights for both oscillation detection and propagation analysis. The methodology is applied to a case study based on the modified IEEE 68-bus benchmark system under partial and full IBR penetration, while its accuracy is validated through EMT time-domain simulations using linear and nonlinear models developed in Matlab/Simulink environment.

I. INTRODUCTION

IBR integration and network expansion create a need for multi-timescale, bus-level analysis of oscillations and their propagation. The paper proposes frequency-domain indices derived from EMT-based transfer functions to assess voltage and frequency sensitivity across the network.

  • I. INTRODUCTION: IBR integration introduces broader-timescale dynamics, while network expansion increases the need for localized bus-level information about system behavior.
  • I. INTRODUCTION: Natural oscillations arise from system modes, forced oscillations from external periodic disturbances, and either type may remain local or propagate through the network.
  • I. INTRODUCTION: Existing measurement-based methods primarily identify oscillation sources, whereas this model-based framework assesses propagation, vulnerable locations, and transmission paths.
  • I. INTRODUCTION: FSI and VSI quantify the impact and propagation of voltage- and frequency-related oscillations across a wide frequency range.They are derived from the complete system PQ–V ω MIMO transfer function.
  • I. INTRODUCTION: The framework develops global and local formulations, connects the indices analytically to system eigenstructure, and supports general and detailed oscillation assessment.
  • I. INTRODUCTION: The methodology uses an EMT state-space model with active and reactive power injections as inputs and bus voltages and generator rotor speeds as outputs.

B. Transfer Function Derivation

The framework converts active and reactive power disturbances at every bus into virtual current injections and maps them to generator rotor speeds and bus-voltage magnitudes through a system transfer function. The resulting PQ−Vω transfer functions support frequency-domain extraction of voltage and frequency sensitivity indicators.

  • Active and reactive power disturbances are linearized through bus current and voltage relationships before entering the complete system state-space model.
  • Figure 1 summarizes the disturbance-to-output mapping from bus P/Q injections to generation-unit frequencies and bus-voltage magnitudes.
  • The model treats ΔP and ΔQ at every bus as inputs and generator rotor speeds plus bus-voltage magnitudes as outputs.
  • The state-space representation is converted to G(s) = C(sI − A)^−1B + D, yielding PQ−Vω transfer functions across the network.
  • FSI and VSI are extracted from these transfer functions in the frequency domain to characterize active and reactive power relationships with system voltage and frequency variables.

1) Singular Value Decomposition:

SVD decomposes each relevant transfer-function block into amplification and input/output directions. The FSI uses the maximum gain and its output-direction weights to map how disturbances at each bus affect generator frequencies across the network.

  • SVD separates each complex transfer matrix into singular values and orthonormal input and output directions.
  • For an active-power injection at bus n, the frequency transfer block is evaluated at s = j2πf before its singular components are computed.
  • With a scalar input, the maximum singular value represents worst-case gain from the bus injection to generator-frequency responses.
  • The FSI combines this maximum gain with the absolute dominant left singular vector to quantify each generator’s sensitivity to the bus disturbance.
  • Repeating the calculation for every bus produces a network-wide map of disturbance locations that most influence frequency dynamics.
  • The same procedure extends to reactive-power injections and quantifies their effects on generator rotor speeds.

B. Voltage Sensitivity Indicator

The VSI applies the sensitivity-extraction procedure to bus-voltage outputs for active and reactive power injections. Its values distinguish attenuation from amplification, while operational thresholds provide a separate security interpretation.

  • VSI is obtained by selecting bus-voltage magnitudes as outputs for active- and reactive-power injection transfer functions.
  • Values below 1 indicate attenuation, whereas values above 1 indicate output amplification and high-sensitivity conditions.
  • Because VSI is unbounded above, values below 1 indicate attenuation but do not by themselves guarantee compliance with operational limits.
  • The adopted voltage security threshold is 10%, corresponding to VSI|V|,P or VSI|V|,Q values exceeding 0.1 as critical conditions.
  • The global VSI aggregates voltage behavior across selected frequency ranges to provide a first-level system-wide sensitivity assessment.

VSI|V |,P , VSI|V |,Q

The modal interpretation links local sensitivity to disturbance controllability, output observability, and pole proximity. When multiple poles cluster, their complex residue interactions can shift amplification peaks away from individual modal frequencies.

  • A high local FSI or VSI indicates that the injection bus has high modal controllability and can strongly excite a poorly damped natural mode.
  • The output direction reflects modal observability and mode shape, so a high index identifies generators or buses with strong modal participation.
  • Under single-mode dominance, the localized indicator depends on the selected output row, input column, and the mode’s frequency proximity.
  • When multiple control and electromechanical poles cluster, individual modal truncation is invalid and the indicator must sum all modal contributions.
  • Complex residue phase interactions can shift the maximum amplification frequency away from the imaginary part of any single natural mode.

IV. APPLICATION METHODOLOGY AND VISUALIZATION

The proposed indices are applied through frequency-focused analysis and 3-D heatmaps that expose modal sensitivity and disturbance propagation across network variables. Values above one indicate amplification, while logarithmic visualization uses zero as the corresponding threshold.

  • High FSI and VSI frequencies identify the aggregated influence of dominant system modes and correspond to the system’s natural modes.This correspondence is used to validate the screening methodology.
  • FSI or VSI values greater than 1 indicate disturbance amplification, and simultaneous exceedances across variables indicate propagation through the system.
  • 3-D heatmaps map power injections by bus against rotor speeds or voltage magnitudes, with oscillation frequency on the z-axis and index magnitude encoded by color.Very low values are omitted, and all plotted values use a logarithmic scale.
  • The heatmap color scale uses log10(1)=0 as the amplification threshold, so values below zero represent secure conditions and values above zero indicate potential amplification.

A. General Description of Case Studies

The case studies use modified IEEE 68-bus systems to examine index behavior across mixed and converter-only generation mixes. In Case Study 1, the indices identify GFL-related sensitivity, link a 76-Hz VSI peak to natural-mode analysis, and match a 74.5-Hz poorly damped mode and its propagation pattern.

  • General Description of Case Studies: Case Study 1 uses a modified IEEE 68-bus system with 65.2% SG, 23.3% GFL, and 11.5% GFM generation, while Case Study 2 uses 9 GFM and 7 GFL converters.
  • Case Study 1: SG–Dominant System with EMT model: The global screening gives FSI = 26.3 and VSI = 0.67, indicating high frequency sensitivity while voltage remains within critical operating margins.
  • Case Study 1: SG–Dominant System with EMT model: GFL-connected buses show substantially higher frequency sensitivity than GFM and SG buses, with GFL14 especially affected because of its electrical distance from the remaining units.Active- and reactive-power FSI provide complementary information about the dominant coupling mechanism at each bus.
  • Case Study 1: SG–Dominant System with EMT model: All voltage-based VSI values remain below 1, but bus-specific vulnerabilities approach the critical range near 100–300 Hz around GFL-connected or electrically proximate buses.Bus 7 is most sensitive to active-power disturbances at 200–250 Hz and to reactive-power disturbances near 300 Hz.
  • Case Study 1: SG–Dominant System with EMT model: At 76 Hz, the V SI|V |,Q14 value computed by the SVD-based method matches the alternative computation, supporting the natural-oscillation detection procedure.
  • Case Study 1: SG–Dominant System with EMT model: At 74.5 Hz, FSI sensitivity and the worst output-direction vector peak at the same nodes as the modal resonance, validating frequency and propagation identification.The corresponding modal analysis reports a pole pair at −38.2 ± j468.2 with an 8% damping ratio.

C. Case Study 2: Fully Inverter-Based Resource System

The fully inverter-based system uses FSI/VSI heatmaps to identify frequency and voltage sensitivity, then validates selected propagation patterns with linear and nonlinear EMT simulations. Frequency disturbances amplify mainly across nearby GFL units, while GFM units remain comparatively protected.

  • Scenario 1: EMT modeling: FSI values exceed 1 for multiple GFL units between 40–50 Hz, indicating propagated frequency amplification from both active and reactive power injections.The affected buses include 1, 4, 5, 7, 8, 10, 13, 52, 53, and 55.
  • Scenario 1: EMT modeling: All VSI values remain below 1, but voltage magnitudes exceed safe limits at selected buses under active or reactive disturbances.Active-power violations occur mainly in the 150–300 Hz range, while reactive-power violations affect buses 5, 7, and 49.
  • Scenario 1: EMT modeling: Linear time-domain responses match the frequency-domain indices: the 42 Hz disturbance amplifies to 2.52, 1.42, 1.40, and 2.24 pu at GFL buses 1, 4, 5, and 8.GFM frequency deviations remain below 0.01 pu under both tested disturbances.
  • Scenario 1: EMT modeling: The 42 Hz active-power injection at Bus 55 propagates most strongly to nearby GFL units at buses 1 and 8, followed by buses 4 and 5.The 47 Hz reactive-power injection at Bus 8 produces the largest responses at buses 8 and 1.
  • Scenario 1: EMT modeling: The nonlinear validation reproduces the same response ordering as the FSI plots and linear simulations, with the largest deviation at GFL 8 and progressively smaller responses at more distant GFL units.The nonlinear results support the indices’ diagnosis of propagation through the fully inverter-based system.

2) Scenario 2: IBR–Dominant System with RMS modeling:

The RMS-network scenario compares a 42 Hz active-power disturbance at Bus 55 with the EMT case. RMS modeling shows weaker and more localized effects, indicating that omitting electromagnetic dynamics can underestimate propagation.

  • Scenario 2: RMS modeling: The RMS response is significantly smaller than the EMT response, which captures broader oscillation propagation through the network.The comparison uses the same FSIω,P55 disturbance frequency and upper and lower margins.
  • Scenario 2: RMS modeling: RMS models are computationally more efficient for large systems but neglect network electromagnetic dynamics, potentially producing optimistic propagation assessments.The conclusion positions EMT simulations as the more accurate representation for this comparison.

APPENDIX A GENERATION UNIT DATA

The appendix connects the proposed sensitivity indicators to state-space transfer functions and natural modes. Under single-mode dominance, singular-value analysis yields the maximum amplification and directional local indices.

  • Generation unit data: The transfer matrix is expanded through system eigenvalues and left and right eigenvectors, with output and input participation vectors describing each modal contribution.This modal form provides the basis for relating sensitivity indicators to natural modes.
  • Generation unit data: Near resonance, a single dominant mode permits computation of maximum amplification from the largest singular value and its primary left singular vector.The maximum singular value is obtained from G(j2πf)G(j2πf)^H, while the dominant direction aligns with the normalized output participation vector.
  • Generation unit data: The local FSI/VSI formulation uses the directional amplification product |u1|σmax, which simplifies when a single-bus injection makes the input participation vector scalar.The output norm cancels algebraically in the resulting expression.

B. Multi-Mode Interaction Analysis

When several modes contribute within the same frequency range, the single-mode approximation no longer applies. The sensitivity indicators must instead reflect the combined modal response, including possible modal interaction.

  • Multi-Mode Interaction Analysis: Multiple modes require summing the modal terms in the transfer matrix rather than retaining one isolated eigenmode.The combined response is represented as M1 + M2 + ··· + Ml.
  • Multi-Mode Interaction Analysis: High FSI/VSI values can result from modal interaction as well as resonance with an individual natural mode.Under multimodal conditions, dominant singular vectors are determined by the collective modal contribution.
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