Participation Analysis in Impedance Models: The Grey-Box Approach for Power System Stability

I. INTRODUCTION

• IBR-induced instability mechanisms remain incompletely understood, while proprietary and nonstandard controls limit access to precise analytical models and black-box impedance models have limited interpretability in complex systems.White-box state-space models offer greater interpretability but require internal apparatus states and control details.Source: §I · pp. 2–3
• The proposed grey-box approach defines impedance participation using whole-system admittance residues, which equal eigenvalue sensitivity to apparatus impedance and support state- and parameter-level root-cause tracing without disclosing internal details.The residues can be measured in the field or estimated from black-box impedance spectra with little prior knowledge.Source: §I · p. 3

II. WHOLE-SYSTEM ADMITTANCE MODEL

• At each node, virtual series voltage injections produce current perturbations whose transfer matrix is the whole-system admittance; its elements share system eigenvalue poles but can show different modal resonance peaks.A diagonal element represents the admittance of a loop containing the apparatus impedance and the impedance of the rest of the grid as seen at that node.Source: §II · p. 3

III. IMPEDANCE PARTICIPATION FACTOR AND THE GREY-BOX APPROACH

• The review frames state participation analysis as the reference for revealing how impedance-based participation relates to the classic state-based approach.The paper introduces the review specifically to clarify that relationship.Source: §III · p. 3

A. State Participation Factor

• The state participation factor p_mn = ψ_nmφ_mn relates a state to a mode and, through eigenvalue sensitivity, indicates how local state dynamics affect whole-system dynamics and which states may warrant retuning.The state equation is transformed into modal coordinates, with eigenvalues determining system stability.Source: A. State Participation Factor · pp. 3–4
• A transfer-function residue equals the state participation factor only under a special input-output selection, so this residue relationship does not generally hold without that condition.The section uses a SISO case and requires the selected elements of B and C to be one, with all other elements zero.Source: A. State Participation Factor · p. 4

B. Impedance Participation Factor

• The residue of the whole-system admittance seen at an apparatus equals the system eigenvalue’s sensitivity to that apparatus impedance, defining the impedance participation factor.The result follows by applying a transfer-function residue relationship to the whole-system admittance at the apparatus node, with the rest-of-grid impedance unchanged during an apparatus impedance perturbation.Source: B. Impedance Participation Factor · p. 4
• An admittance participation factor can likewise be defined from the residue of the whole-system impedance, and is theoretically equivalent to the impedance participation factor.Either form may suit applications in which impedance or admittance is more readily available.Source: B. Impedance Participation Factor · p. 5
• The participation-factor chain rule links impedance participation to state and parameter participation, allowing black-box apparatus contributions to be traced to states and internal parameters without disclosure of the state equation.The paper treats a state-matrix coefficient as a virtual parameter to link state and parameter participation factors.Source: B. Impedance Participation Factor · p. 5

C. The Grey-Box Approach

• Greater transparency requires more prior knowledge but provides more useful information for root-cause tracing and troubleshooting.Source: C. The Grey-Box Approach · p. 5
• The figure relates impedance participation to state participation and shows the chain rule for propagating participation.Source: Figure 2 · p. 5
• The three layers progress from potential-participant estimates, through damping contribution, to apparatus-level root-cause identification and parameter retuning.Layer-1 uses an upper bound on ∆λ; Layer-2 uses its real part under impedance scaling; Layer-3 uses impedance-parameter sensitivity.Source: Figure 3 · p. 5

1) Grey-Box Layer-1:

• Layer-1 requires apparatus impedances and the network nodal admittance; impedance participation can be calculated from transfer functions or estimated from frequency spectra.Source: 1) Grey-Box Layer-1 · p. 5
• Layer-1 uses a norm-based index to bound eigenvalue change and identify potential participants, who may not necessarily participate in the mode.The index is ∥pλ,Zk∥·∥Zk(λ)∥, the upper bound on |∆λ| under the specified perturbation.Source: 1) Grey-Box Layer-1 · pp. 5–6

2) Grey-Box Layer-2:

• Layer-2 aligns the impedance perturbation with the original impedance, modeling amplitude scaling while maintaining its angle.Examples include changing apparatus base power or connecting or disconnecting part of a farm of identical apparatus.Source: 2) Grey-Box Layer-2 · p. 6
• Layer-2 uses ⟨pλ,Zk, Zk(λ)⟩ to estimate damping impact; a positive real part means scaling up tends to stabilize the system.Source: 2) Grey-Box Layer-2 · p. 6

3) Grey-Box Layer-3:

• Layer-3 propagates internal-parameter changes through impedance sensitivity and the chain rule to identify retuning that moves an eigenvalue in a desired direction, without exposing detailed apparatus design.This supports root-cause tracing to a physical component or control loop.Source: 3) Grey-Box Layer-3 · p. 6

D. Practical Implementation:

• Implementation uses apparatus impedance, whole-system admittance, and impedance participation spectra; Layer-3 additionally needs impedance sensitivity to internal parameters.Frequency spectra avoid symbolic calculation, and the principal matrix operations have complexity O(N^3) or less.Source: D. Practical Implementation · p. 6, 2) Measurement-Based Route · p. 6

1) Model-based Route:

• The model-based route derives and linearizes apparatus differential equations, then transforms the resulting state-space matrices into transfer functions to obtain impedance spectra.This route is suited to manufacturers with detailed analytical models that preserve every state.Source: 1) Model-based Route · p. 6

2) Measurement-Based Route:

• The measurement-based route perturbs apparatus inputs and measures output-to-input frequency-domain ratios to obtain impedance or admittance spectra.Pseudorandom or swept-frequency excitation can be used; measurements may be made in hardware or EMT simulation, but can be time-consuming and noise-sensitive.Source: 2) Measurement-Based Route · p. 6
• For Layer-3, impedance sensitivity to a parameter is calculated numerically using a small parameter perturbation and impedance values that account for the changed equilibrium point.The stated perturbation is ∆ρ = 10^-5(1+|ρ|).Source: 2) Measurement-Based Route · p. 6
• Participation factors are local sensitivities, so large parameter changes require iterative application; close or repeated poles can initially prevent separate residue attribution.The approach targets small-signal analysis and is complemented by EMT simulation for large-signal analysis.Source: 2) Measurement-Based Route · pp. 6–7

IV. CASE STUDY OF A COMPOSITE POWER SYSTEM

• The 68-bus case adds six Type-IV wind farms, replaces the bus-15 synchronous generator with a grid-forming inverter, and deliberately de-tunes selected controls to provoke oscillations.A15’s frequency droop gain was set high; A28 and A29 current-control bandwidths were de-tuned by −40% and −56%.Source: §IV · p. 7
• Three selected resonant modes comprise a standard 60 Hz winding-and-line flux mode, 1–2 Hz synchronous-generator rotor-swing modes, and a high-frequency grid-forming-inverter LCL-filter mode.The whole-system admittance plot displays one of four dq-frame matrix elements at source nodes.Source: §IV · p. 7, Figure 5 · p. 8
• Layer-1 and Layer-2 identify A11 as prominent in mode 1 and A15 in mode 2; mode 3 involves A9, A28, and A29, with A29 participating more than A28.For mode 3, the synchronous generator and IBRs have opposite signs in the eigenvalue-change components, and the analysis identifies the mode as IBR-induced.Source: §IV · p. 7, Figure 6 · p. 8
• Layer-3 identifies parameter-level tuning directions: increase A11’s AVR feedback gain KF(11), decrease A15’s droop gain KD(15), and increase current-control bandwidth fi in A28 and A29.For mode 3, increasing PLL bandwidth fPLL tends to destabilize the mode, while increasing current-control bandwidth tends to stabilize it; the authors attribute the mode to inner-loop and PLL coupling in a relatively weak grid.Source: §IV · p. 8, Figure 7 · p. 8
• Time-domain tests show the grey-box located oscillation root causes and indicated stabilization choices: tuning participating apparatus parameters stabilized or destabilized the system as predicted.The study applied a bus-61 load disconnection and a 5% bus-42 load increase; measured natural frequencies agreed with frequency-domain resonant peaks.Source: §IV · p. 9

V. CONCLUSIONS

• The conclusion presents the three-layer method as a way to trace root causes to apparatus and parameters using impedance information, without requiring manufacturers to disclose internal details.The approach is grounded in the proved relationship between whole-system-admittance residues and impedance participation, and a chain-rule for propagating sensitivity to states and parameters.Source: §V · p. 9

A. Residue

• A complex function’s residue is its g−1 Laurent-series coefficient; for a non-repeated pole, the appendix says the residue is found from a formula.Residues can also be applied element-wise to matrices of complex functions.Source: A. Residue · p. 9

B. Frobenius Inner Product

• The Frobenius inner product of complex matrices uses conjugation so that a matrix’s inner product with itself is a non-negative real number, inducing the Frobenius norm.The passage identifies h and l as row and column indices and conjugation as part of the definition.Source: B. Frobenius Inner Product · p. 9
• The appendix states that the Frobenius inner product and norm inherit common inner-product properties, including the Cauchy inequality, with equality only when the matrices are aligned in orientation.It notes that this equality property is used in Section III-C.Source: B. Frobenius Inner Product · p. 9

• A derivative of a complex function over a real parameter treats its real and imaginary parts separately.Source: C. Complex Function Derivative · p. 9
• A complex-to-complex derivative exists when the mapping from the input function to the output function is analytic.Source: C. Complex Function Derivative · p. 9
• For vectors and matrices, each element is treated as an independent variable, so the resulting derivative is also a vector or matrix.Source: C. Complex Function Derivative · p. 9

• The scalar proof considers a transfer function Gρ whose pole λ is a zero of Hρ = Gρ^-1.Source: Appendix B · p. 10
• A perturbation in ρ induces a change in λ, and analyticity of Hρ around λ permits a first-order Taylor expansion.Source: Appendix B · p. 10
• For a non-repeated pole, the residue is evaluated using L’Hôpital’s rule, yielding the scalar case of Lemma 1.Source: Appendix B · p. 10
• For square matrix Gρ, the proof applies the scalar result to det(Hρ), then expands the determinant and calculates the matrix residue element by element.Source: Appendix B · p. 10

• The example uses a three-node system whose whole-system admittance matrix has entries that are 2 × 2 transfer-function matrices in the dq frame.Source: Appendix C · p. 11
• The diagonal entry at the first node represents the whole-system admittance measured there, and its residue at a pole yields the connected apparatus’s impedance participation factor.Source: Appendix C · p. 11
• The direction of the eigenvalue shift Δλ depends jointly on the impedance participation factor and the impedance perturbation ΔZ1(λ).Source: Appendix C · p. 11