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Decentralised Plug-and-Play Stability Conditions for AC Grids-Part II: Unstable Subsystems

Liam Hallinan, Ioannis Lestas

arXiv:2609.07331v1eess.SYcs.SY

TL;DR

Decentralised stability analysis of AC grids is complicated when subsystem representations contain unstable poles, even if the aggregate system is stable, and low-frequency checks may require centralised analysis. Part II links impedance and power representations to construct stable hybrid subsystems and derives low-frequency Nyquist conditions. The approach certifies stability on the Kundur two-area system, combining decentralised high-frequency conditions with low-frequency branch analysis.

  • Problem

    Unstable poles can occur in common subsystem representations even when the aggregate grid is stable, complicating decentralised stability analysis.

    Source: §I · p. 1, C. Unstable Poles in IV and PQ Models · p. 5

  • Method

    The paper links IV and PQ models, uses a stable hybrid representation with PQ properties at low frequencies and IV properties at high frequencies, and derives branch conditions for low-frequency Nyquist analysis.

    Source: §I · pp. 1–2

  • Results

    The Kundur two-area system’s stability is certified using decentralised plug-and-play conditions at high frequencies and low-frequency analysis of bounded and unbounded branches.

    Source: §I · p. 2, §VIII · p. 12

  • Takeaways & Limitations

    The low-frequency bounded branches are governed by a matrix closely related to the reduced power-flow Jacobian constrained during generator dispatch, allowing grid codes to focus on higher frequencies.

    Source: §I · p. 2, §VIII · p. 12

    The results were verified on the Kundur two-area system; further validation studies remain future work.

    Source: §VIII · p. 12

Abstract

from arXiv · show

Part I of this paper presented a decentralised framework for certifying small-signal stability in AC grids using frequency-domain quadratic constraints on individual grid subsystems. Part II extends the framework to a broader class of networks that contain unstable subsystems. In particular, we show that such unstable subsystems arise in many common scenarios, even when the aggregate system is stable and well behaved. Such subsystems must be stabilised by the closed-loop network interconnection, which complicates decentralised stability analysis. A stable hybrid representation resembling the power (PQ) model at low frequencies and the impedance (IV) model at high frequencies is then defined, to which the stability framework of Part I can be applied, allowing plug-and-play compatible conditions to be formulated. Furthermore, we show that at low frequencies, the characteristic loci in the Nyquist plot of the return ratio in this hybrid representation decouple into unbounded and bounded branches along the classical $P$-$\delta$ / $Q$-$V$ separation, and give sufficient conditions on each branch for ensuring system stability. The results are validated on the Kundur two-area system, where stability is certified, with a decentralised, plug-and-play compatible grid code covering the frequency ranges in which electromagnetic interactions arise.

I. INTRODUCTION

Part II addresses gaps in Part I’s decentralised AC-grid stability framework: common subsystem representations can be unstable, and low-frequency certification may require centralised analysis. It links impedance and power models, develops a stable hybrid representation, and gives conditions for stability certification across frequency ranges.

I. INTRODUCTION

  • Part I’s framework lost plug-and-play capability when unstable bus impedances required a nonsparse admittance representation, while low-frequency decentralised checks also required a stiff voltage source.Without that source, a centralised low-frequency check was needed.Source: §I · p. 1
  • Unstable subsystem poles arise in common configurations: droop-controlled buses may have low-frequency unstable IV poles with positive equilibrium reactive power, while series passive impedances may produce high-frequency unstable PQ poles.The network interconnection must stabilise these open-loop poles, complicating decentralised analysis.Source: §I · p. 1
  • A dynamic loop transformation creates stable hybrid subsystems that use PQ-model properties at low frequencies and IV-model properties at high frequencies.The network modification is additive, so it remains stable and sparse.Source: §I · p. 1
  • The Part I framework applies to the hybrid model with a modified multiplier, and bus-level constraints need checking only at the two homotopy endpoints.This recovers extended-subsystem conditions and their plug-and-play capability.Source: §I · pp. 1–2
  • At low frequencies, hybrid-model Nyquist loci split into bounded and unbounded branches aligned with P–δ and Q–V dynamics; sufficient conditions on both branches combine with high-frequency decentralised conditions to certify stability.The bounded branches are governed by a matrix closely related to the reduced power-flow Jacobian constrained during dispatch.Source: §I · p. 2
  • The Kundur two-area case study certifies stability using decentralised plug-and-play conditions at high frequencies and centralised checks of low-frequency branches.The case study complements Part I’s inverter-based example.Source: §I · p. 2

III. GRID MODEL

The section presents the grid in both impedance (IV) and power (PQ) representations, then links the bus transfer functions through a feedback transformation. The PQ model expresses active and reactive powers in relation to phase angle and voltage magnitude.

A. Review of IV Model

  • The IV model represents the grid as bus impedances interconnected in negative feedback with a sparse network admittance matrix.The network is built from line, load, and shunt admittances.Source: A. Review of IV Model · pp. 3–4

B. PQ Model

  • The PQ model couples bus active and reactive powers to electrical phase angle and voltage magnitude.Its network dynamics inherit stability and sparsity from the IV model's network admittance.Source: B. PQ Model · p. 4
  • The PQ and IV bus transfer functions are related by a feedback loop transformation involving U† and Wi.The IV bus impedance results from placing Gi(s) in negative feedback with Wi, with pre- and post-multiplication by the stated matrices.Source: B. PQ Model · p. 4

C. Unstable Poles in IV and PQ Models

The section explains that unstable open-loop subsystem poles complicate decentralised stability analysis because the network interconnection may supply their stabilisation. For low-frequency IV impedances, Nyquist loci split into bounded and unbounded branches, and an example shows an unstable bus impedance.

C. Unstable Poles in IV and PQ Models

  • Open-loop unstable poles complicate decentralised analysis because network interconnection may have to stabilise the subsystems.This can reintroduce centralised elements and prevent use of extended-subsystem conditions that support plug-and-play analysis.Source: C. Unstable Poles in IV and PQ Models · pp. 4–5

1) Low-frequency poles in IV impedances:

  • As s approaches zero, the IV return-ratio characteristic loci split into bounded and unbounded branches, with the unbounded branch's imaginary-axis direction set by equilibrium reactive power.For positive equilibrium reactive power, the branch makes a clockwise half-turn through the left half-plane.Source: 1) Low-frequency poles in IV impedances · p. 5
  • In the example, the unbounded branch encircles −1 and the IV bus impedance has a right-half-plane pole at s = 0.7684.The passage attributes the instability to the Nyquist encirclement.Source: 1) Low-frequency poles in IV impedances · p. 5

2) High-frequency poles in PQ systems:

Converting a passive series impedance to a PQ representation can introduce high-frequency unstable poles near resonance. The Nyquist analysis identifies this mechanism in common series-component configurations, though real bus dynamics require checking.

2) High-frequency poles in PQ systems:

  • A passive series impedance can acquire two unstable poles in its PQ transfer function, with two clockwise Nyquist encirclements near resonance.For the illustrated circuit, the poles are at s = 1.1 × 10^3 and s = 1.6 × 10^4.Source: 2) High-frequency poles in PQ systems · pp. 5–6
  • The instability mechanism is tied to the resonant band of the passive series component, rather than its behavior across all frequencies.In the illustrated example, the Nyquist loci are far from −1 outside the resonance band.Source: 2) High-frequency poles in PQ systems · p. 6
  • The result can apply to generators or inverters with series transformers or RLC filters, whose high-frequency dynamics may contribute poles requiring interconnection stabilisation.The conclusion applies when the bus dynamics are well approximated at low frequencies or the series-component condition holds near resonance.Source: 2) High-frequency poles in PQ systems · p. 6

D. Hybrid Model

The hybrid model uses PQ-like bus behavior at low frequencies and IV-like behavior at high frequencies to avoid the subsystem instabilities of either representation. The Part I stability framework extends to this stable hybrid model, including its plug-and-play conditions.

D. Hybrid Model

  • A frequency-dependent loop transformation makes each hybrid bus resemble the PQ model at low frequencies and the IV model at high frequencies.A high-pass filter cutoff can usually be placed between the low-frequency IV instability dynamics and the higher-frequency PQ instability dynamics so each hybrid bus is stable.Source: D. Hybrid Model · p. 6
  • The hybrid network approximates the PQ network below the cutoff and the transformed IV network at high frequencies.The bus transformation uses high-pass and complementary low-pass filters.Source: D. Hybrid Model · p. 6

IV. STABILITY FRAMEWORK FOR HYBRID MODEL

  • The Part I decentralised stability conditions extend to the hybrid model, whose network retains the sparsity needed for extended-subsystem and plug-and-play analysis.The framework's conditions are sufficient for stability certification of the hybrid grid.Source: §IV · p. 7

A. Bus-Level Conditions

The Part I bus-level and extended-subsystem conditions carry over to the hybrid model. Its sparse network structure supports local, plug-and-play stability certification.

A. Bus-Level Conditions

  • Existing Part I bus-level conditions, including their graphical interpretations, apply directly to each hybrid bus.The conditions are expressed at the endpoints of a homotopy between the hybrid and IV impedances.Source: A. Bus-Level Conditions · p. 7

B. Extended Subsystem-Level Conditions

  • The hybrid network retains the sparsity of the original network, allowing extended subsystems to include one bus, its incident lines and loads, and its loop transformation.The hybrid extended graph adds a shunt edge at every bus.Source: B. Extended Subsystem-Level Conditions · p. 8
  • Local edge constraints in the extended-subsystem formulation support plug-and-play stability certification.The edge-local multiplier conditions carry over from Part I to the hybrid model.Source: B. Extended Subsystem-Level Conditions · p. 8

V. LOW-FREQUENCY NYQUIST ANALYSIS

The low-frequency Nyquist analysis separates the hybrid return ratio into frequency- and voltage-related dynamics, recovering the classical P–δ / Q–V separation. The low-frequency hybrid network reduces to the power-flow Jacobian under stated network assumptions.

V. LOW-FREQUENCY NYQUIST ANALYSIS

  • At low frequencies, the characteristic polynomial separates into frequency and voltage factors, yielding unbounded and bounded loci, respectively.The unbounded branches are governed by local setpoints; the bounded branches by a matrix closely related to the reduced power-flow Jacobian.Source: §V · p. 8

A. Low-Frequency Network Model

  • Near s = 0, the hybrid network reduces to the power-flow Jacobian when the network admittance has the stated RLC-compatible block form and varies little over the low-frequency band.The assumption requires each block of YN(0) to take the form GI2 + BJ and YN(s) ≈ YN(0) for |s| ≤ ωl.Source: A. Low-Frequency Network Model · p. 9
  • The P–δ and Q–δ Jacobian blocks have zero row sums, although the individual blocks are generally non-symmetric.This property holds for |s| ≤ ωl.Source: A. Low-Frequency Network Model · p. 9

B. Decoupling of the Characteristic Loci

Near zero frequency, the return-ratio characteristic loci separate into unbounded frequency-related and bounded voltage-related branches. Sufficient conditions for these branches can be combined with higher-frequency criteria to certify stability across locality layers.

B. Decoupling of the Characteristic Loci

  • Near s = 0, the characteristic loci split into unbounded branches scaling as 1/s and bounded branches governed by the reduced matrix R(s).This recovers the classical P–δ / Q–V separation in a Nyquist setting.Source: B. Decoupling of the Characteristic Loci · p. 9

C. Behaviour at Typical Operating Points

  • At typical operating points, droop-controlled frequency-to-active-power dynamics and transmission-line structure place the nonzero eigenvalues of eLωδ(s) in the right half-plane.This satisfies the sufficient condition for the unbounded loci; for bounded loci, the reduced matrix is related to the reduced power-flow Jacobian.Source: C. Behaviour at Typical Operating Points · p. 10
  • For more complex operating cases, the bounded-locus conditions can be incorporated into power-flow analysis during generator dispatch.For many standard configurations, the low-frequency conditions can be omitted from grid codes, which can focus on higher frequencies.Source: C. Behaviour at Typical Operating Points · p. 10

VI. COMBINED STABILITY CRITERIA

  • Proposition 8 combines hybrid-model stability conditions spanning high-frequency IV behavior, low-frequency PQ behavior, and bus, extended-subsystem, and centralised locality layers.The centralised condition covers contour regions not covered by the transferred Part I conditions.Source: §VI · p. 10

VII. CASE STUDY: KUNDUR TWO-AREA SYSTEM

The Kundur two-area case study contains unstable subsystem poles at both low and high frequencies, but the hybrid representation supports a stability certificate. High-frequency conditions are plug-and-play compatible, while centralised analysis verifies low-frequency behavior.

VII. CASE STUDY: KUNDUR TWO-AREA SYSTEM

  • The case study applies the framework to the Kundur two-area system, whose stability for particular control structures is typically established through centralised analysis.The model includes four generators and eleven buses.Source: §VII · p. 10

A. Unstable Subsystems

  • Unstable poles occur in both Zi(s) and Gi(s) at all four buses: positive equilibrium reactive power predicts low-frequency Zi(s) poles, while Gi(s) poles arise near transformer-branch resonance.The Gi(s) encirclements occur over 2.0×10^4 to 2.1×10^4 rad s−1.Source: A. Unstable Subsystems · p. 11

B. Local Stability Analysis

  • For ω > 3.28 rad s−1, conditions supporting plug-and-play operation establish the required stability condition, unlike the Part I case with unstable bus impedances.Figure 10 reports the tested conditions across locality layers.Source: B. Local Stability Analysis · p. 11, Figure 10 · p. 11

C. Centralised Low-Frequency Analysis

  • At low frequencies, the bounded loci remain in the right half-plane and the unbounded loci diverge along the imaginary axis while sweeping the right half-plane.The bounded-locus behavior is governed by reduced power-flow Jacobian eigenvalues confirmed to lie in the right half-plane; under sensible operating points, this system-level check need not be repeated whenever a device connects.Source: C. Centralised Low-Frequency Analysis · pp. 11–12
  • The hybrid system is certified stable, with decentralised plug-and-play conditions at higher frequencies and centralised analysis at lower frequencies.The low-frequency bounded and unbounded loci do not pass left of −1 in the Nyquist plot.Source: C. Centralised Low-Frequency Analysis · pp. 11–12, Figure 12 · p. 12, Figure 13 · p. 12

VIII. CONCLUSION

  • Across Parts I and II, the framework accommodates unstable subsystems and networks without a stiff voltage source, using frequency-domain quadratic constraints on subsystem input–output properties.The paper reports validation of both results on the Kundur two-area system.Source: §VIII · p. 12

APPENDIX

The appendix passages outline proof steps for two lemmas and a corollary, including eigenvalue calculations and matrix-inequality manipulations.

A. Proof of Lemma 1

  • The proof of Lemma 1 uses a Laurent expansion of Gi(s) as s approaches zero and evaluates the resulting eigenvalues to obtain (27).The derivation omits bus subscripts and equilibrium superscripts to simplify notation.Source: A. Proof of Lemma 1 · pp. 12–13

B. Proof of Lemma 2

  • The proof of Lemma 2 computes the eigenvalues of LG,i(s) from its characteristic equation and solves for the expression in (32).Source: B. Proof of Lemma 2 · p. 13

C. Proof of Corollary 4

  • The proof of Corollary 4 uses congruence and a quadratic matrix polynomial in τ to establish positivity over τ ∈ [0, 1].It also rewrites the transformed network multiplier using the definitions in (35) and (36).Source: C. Proof of Corollary 4 · pp. 13–14

D. Proof of Lemma 6

The proof derives the low-frequency characteristic-polynomial reduction using a Schur complement and the projector associated with eLωδ(s).

D. Proof of Lemma 6

  • The proof begins from the characteristic polynomial and applies a Schur complement.Source: D. Proof of Lemma 6 · p. 14
  • Under Assumption 3, the projector P0 is rank one, with each column proportional to 1 and ξ the normalised left null vector of eLωδ(s).Source: D. Proof of Lemma 6 · p. 14
  • A resolvent expansion near s = 0 is substituted into the characteristic polynomial to obtain (58).Source: D. Proof of Lemma 6 · p. 14

E. Proof of Corollary 7

The proofs establish stability by checking characteristic-locus branches and extending the conditions across the relevant contours. The case-study controller passages specify model parameters.

E. Proof of Corollary 7

  • Condition (38) holds when every characteristic-locus branch avoids (−∞, −1]; the zero branch and bounded branch satisfy this directly, while the unbounded branches follow from (60a).For the unbounded branches, the proof uses ℜ(µ) > 0 to show the argument does not reach ±π.Source: E. Proof of Corollary 7 · p. 14

F. Proof of Proposition 8

  • The conditions imply stability because the determinant has a simple zero at the origin, leaving the inverse with at most one pole there.The proof uses Assumption 3 and conditions (60a) and (60b), then invokes [Part I, Lemma 3].Source: F. Proof of Proposition 8 · p. 14

G. Case Study Controllers

  • The case study specifies governor, PSS, and thyristor-exciter controller parameters for generators G1–G4.Governor time constants are Ta = 0.5 s, Tb = 2.1 s, Tc = 7 s, with droop R = {0.04, 0.05, 0.04, 0.06}; PSS KPSS = 20 and thyristor-exciter KA = 200.Source: G. Case Study Controllers · p. 14
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