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Decentralised Plug-and-Play Stability Conditions for AC Grids-Part I

Liam Hallinan, Ioannis Lestas

arXiv:2609.07315v1eess.SYcs.SY

TL;DR

Small-signal stability assessment for increasingly distributed AC grids needs approaches that do not depend on centralised, system-wide models. This paper develops frequency-domain local-subsystem conditions for stability certification, and a modified 9-bus case study demonstrates reduced conservatism from full-block multipliers.

  • Problem

    Classical small-signal techniques applied to fixed system-wide models are no longer scalable as inverter-based resources replace synchronous generation and grids become more distributed.

    Source: §I · p. 1

  • Method

    The paper formulates quadratic constraints on local grid subsystems’ frequency-domain input–output properties that collectively ensure the multivariable Nyquist criterion.

    Source: §I · p. 1

  • Results

    The modified 9-bus case study demonstrates reduced conservatism from full-block multipliers and the framework’s suitability as a foundation for grid codes in fixed networks.

    Source: §VI · p. 12

  • Takeaways & Limitations

    The framework generalises prior conditions and offers flexibility across bus representations and, in impedance form, extended subsystems containing connecting lines and loads.

    Source: §VI · p. 12

    Two bus impedances in the case study were unstable, preventing use of the impedance representation and its extended-subsystem formulation; without a stiff voltage source, no decentralised condition was feasible at low frequencies.

    Source: §VI · pp. 12–13

Abstract

from arXiv · show

The rise of renewable generation in AC power systems calls for small-signal stability assessment methods that do not rely on centralised, system-wide models. In this two-part paper, we present a decentralised stability framework based on frequency-domain conditions imposed on local subsystems (many admitting equivalent graphical interpretations) which collectively imply the stability of the entire grid. The framework generalises many previously reported results, with added flexibility and plug-and-play compatibility provided by the additional degrees of freedom of the imposed conditions, the ability to mix conditions across frequency ranges, conditions on extended subsystems that couple buses to their connecting lines and loads, and a method to incorporate unstable subsystems. Detailed, heterogeneous device and line models are accommodated in the frequency domain using impedance, admittance, and power-flow models, making the framework suitable as a foundation for grid codes that enable plug-and-play functionality in inverter-dominated grids. Part I focuses on stable impedance and admittance representations and Part II extends the results to unstable subsystems, which are shown to be fundamentally present in many grid models. The results of Part I are validated on a modified 9-bus case study, demonstrating the reduced conservatism and significance of the proposed approach.

I. INTRODUCTION

The paper develops a frequency-domain framework for decentralised small-signal stability certification in AC grids, addressing limitations of centralised analysis and conservative or simplified local conditions. It extends local conditions across subsystem representations and frequency ranges, then demonstrates reduced conservatism in a modified 9-bus case study.

I. INTRODUCTION

  • Earlier approaches can impose conservative conditions or simplify network dynamics, and many require centralised knowledge of network parameters and interconnection structure.The paper also notes that impedance-, admittance-, and power-based subsystem representations can have unstable open-loop poles in common scenarios.Source: §I · p. 1
  • Quadratic constraints on local subsystem input–output properties collectively ensure the multivariable Nyquist criterion and grid stability.The framework generalises prior results and extends them to alternative subsystem representations.Source: §I · p. 1
  • The framework supports impedance, admittance, and power representations, with different criteria mixed across frequency regions to reduce conservatism.In impedance and power representations, conditions can also cover extended subsystems coupling a bus with its connecting lines and loads.Source: §I · p. 2
  • Designed appropriately, local constraints can be verified without global network parameters or topology, supporting scalable, plug-and-play integration.Adding a bus or line can require minimal disruption to non-local subsystems.Source: §I · p. 2
  • The framework accommodates heterogeneous dynamic line models and full-order bus models, including black-box device models specified in the frequency domain.Many constraints also admit graphical interpretations that restrict frequency responses or their Davis–Wielandt shell to specified regions.Source: §I · p. 2
  • The modified 9-bus case study shows proposed full-block conditions holding over frequency ranges where previously reported conditions fail.Part I assumes a stable impedance or admittance representation; unstable subsystem poles motivate extensions in Part II.Source: §I · p. 2

A. Notation and Definitions

  • The numerical range is a convex, compact set containing a matrix’s spectrum, and the Davis–Wielandt shell generalises it to a higher-dimensional set.The section also defines standard number sets and matrix notation used in the paper.Source: A. Notation and Definitions · p. 2

B. Nyquist Theory

The section uses the generalized Nyquist theorem to connect feedback stability with characteristic-locus encirclements of −1 on an indented contour. It then develops sufficient quadratic constraints and multiplier conditions to certify stability through continuous deformations of those loci.

B. Nyquist Theory

  • Stability requires the characteristic loci to avoid −1 and make no net encirclements of that point on the indented Nyquist contour, subject to the stated closed-loop pole condition.The contour follows the imaginary axis, indents into the right half-plane around the origin, and closes with an infinite right-half-plane semicircle.Source: B. Nyquist Theory · p. 3, Figure 1 · p. 3
  • A continuous deformation from zero loci to the system’s characteristic loci certifies stability if the loci avoid −1 throughout the deformation on the contour.The closed-loop system must be well-posed and the inverse return difference may have at most one pole at the origin.Source: B. Nyquist Theory · p. 3
  • Sufficient quadratic constraints on subsystem input–output properties can ensure that the deformed characteristic loci avoid −1 at each contour frequency.The conditions must hold for every deformation parameter and frequency; a multiplier structure can reduce the parameter check to its endpoint.Source: B. Nyquist Theory · p. 3
  • If the multiplier blocks satisfy Π11(s) ≤ 0 and Π22(s) ≥ 0, checking the constraint at τ = 1 suffices for all τ in [0, 1].The proof uses convexity in τ and the negative-semidefinite condition at τ = 0.Source: B. Nyquist Theory · p. 3
  • The Nyquist-based approach can extend to improper systems and use alternative continuous locus deformations; its reconfigurations and multipliers can be mixed across frequency ranges.The passage contrasts this extension with classical IQC results, which apply only to bounded systems.Source: B. Nyquist Theory · pp. 3–4

III. GRID MODEL

The grid is represented as a directed graph of buses, lines, and shunt connections, with signals expressed in common and local DQ reference frames. Each bus is modeled by linearized dynamics mapping current injection to voltage.

A. Topology and Reference Frames

  • The grid topology is a directed graph whose nodes comprise buses and ground, with power lines and shunt connections represented as edges.The bus–line subgraph is assumed connected.Source: A. Topology and Reference Frames · p. 4
  • Balanced three-phase signals are projected into common and local DQ frames, with the local d-axis aligned to the bus voltage vector.At equilibrium, the local and common frame frequencies coincide, and the bus voltage in its local frame is [Vi, 0]^T.Source: A. Topology and Reference Frames · p. 4

B. Buses

  • Each bus is modeled by linearized dynamics with voltage as output and negative net current injection as input; the ground node has zero voltage deviation.Linearization about the network equilibrium yields the bus impedance Zi(s).Source: B. Buses · p. 4

C. Network

Line and load dynamics define branch admittances, which assemble into the network admittance matrix. The matrix is positive-real for the stated branch models, though capacitive shunts can change that property.

C. Network

  • Each line or load subsystem maps the voltage difference across the element to its branch current.Series RL models use nonnegative resistance and reactance, while capacitive shunts are represented by nonnegative shunt susceptance.Source: C. Network · p. 4
  • The network admittance matrix is assembled from branch admittances, with its diagonal entries incorporating local and neighboring-bus admittances.The network can also be written using the bus–line incidence matrix and block-diagonal branch and shunt admittances.Source: C. Network · p. 5
  • With the stated RL line and load models and capacitive branch models, YN(s) is positive-real, but capacitive shunts can make the relevant sign definiteness depend on network configuration.Non-positive-real loads can instead be included in bus impedances; the branch property used in Section IV-C does not hold for capacitive shunts.Source: Properties · p. 5

D. Closed-Loop Grid

The grid’s small-signal stability can be analyzed through an impedance feedback interconnection or its admittance inverse. The framework combines local frequency-domain conditions into stability certificates, but Part I’s assumptions and the non-sparsity of the inverted network model constrain some plug-and-play conditions.

D. Closed-Loop Grid

  • The grid’s small-signal stability is analyzed through the negative-feedback interconnection of bus impedances and network admittance, or through their inverted admittance representation.The impedance representation is [ZB(s), YN(s)]; the alternative is [bYB(s), bZN(s)].Source: D. Closed-Loop Grid · p. 5, Figure 2 · p. 5, §IV · p. 6
  • Part I assumes stable bus impedances, while the inverted network impedance is generally non-sparse and introduces non-local coupling that obstructs local extended-subsystem conditions.The paper states that unstable poles in ZB(s) occur in many common scenarios and violate Assumption 2.A.Source: D. Closed-Loop Grid · p. 6

IV. STABILITY FRAMEWORK

  • Local quadratic constraints that hold pointwise across regions collectively certify stability when those regions cover the Nyquist contour.The framework presents results using the impedance representation and also discusses the admittance representation.Source: §IV · p. 6
  • The framework distinguishes independently verifiable bus conditions, extended subsystems coupling buses to neighboring lines and loads, and conditions requiring fixed-network information.Conditions from different locality levels can be combined into frequency-domain certificates and used to design local grid codes.Source: §IV · p. 6

A. Decentralised Bus Conditions

  • A bus condition is fully decentralised when its multiplier is independent of other buses’ dynamics and network-level parameters.When the network constraint holds, verifying the corresponding local condition at each bus is sufficient for the stated stability condition.Source: A. Decentralised Bus Conditions · p. 6
  • Full-block multipliers allow decentralised constraints beyond those captured by three-dimensional separating-plane tests.The cited discussion contrasts their additional degrees of freedom with standard positive-real, small-gain, and small-phase conditions.Source: A. Decentralised Bus Conditions · pp. 6–7

1) Positive-Real Condition:

The positive-real condition yields a locally checkable sufficient test for grid stability, but is typically not satisfied across all frequency ranges. The section also notes an infinity-norm condition for AC grids with plug-and-play functionality.

1) Positive-Real Condition:

  • At a particular frequency, Corollary 9 can be viewed as a positive-real constraint on the bus dynamics.The condition is frequency-wise.Source: 1) Positive-Real Condition · p. 7
  • A positive-real network admittance lets each bus check a sufficient stability condition independently, without other-bus dynamics or the full network admittance matrix.The condition is conservative and is typically not satisfied at all frequency ranges.Source: 1) Positive-Real Condition · p. 7
  • A matrix infinity-norm condition can also be derived for AC grids while allowing plug-and-play functionality.Source: 2) Infinity-Norm Condition · p. 7

2) Infinity-Norm Condition:

The infinity-norm condition gives each bus a local impedance test based on adjacent lines and local loads or shunts. The section also reformulates the grid as extended bus subsystems, making stability conditions based on local dynamics and network sparsity possible.

2) Infinity-Norm Condition:

  • Each bus can check the infinity-norm condition over a specified frequency range using its impedance, adjacent-line parameters, and any local load or shunt, without full-network knowledge.Source: 2) Infinity-Norm Condition · p. 7

B. Decentralised Extended-Subsystem Conditions

1) Reformulated System:

  • Stability of the original grid is equivalent to stability of the feedback system formed by the block-diagonal extended subsystems and the static interconnection matrix.Source: 1) Reformulated System · p. 8

2) Stability Conditions on Reformulated System:

The reformulated-system conditions reduce stability verification to local checks using multiplier terms associated with each bus’s incident edges. Example conditions offer additional flexibility, including a graphical feasibility interpretation and constraints that limit the effect of adding a line to its connected buses.

2) Stability Conditions on Reformulated System:

  • Local multiplier matrices can be chosen for each line, load, or shunt, then used at each bus to check the corresponding stability inequality.When the multipliers are local to each line, the condition supports plug-and-play functionality.Source: 2) Stability Conditions on… · p. 8

3) Example Conditions:

  • The numerical-range interpretation makes multiplier selection graphical: feasibility requires the numerical range of C_E,i(s) to lie to the right of a line through −1.The line angle depends on η̄(s), and choosing η̄(s) = 0 gives a vertical line.Source: 3) Example Conditions · p. 8
  • The edge-wise condition is less conservative than applying the numerical-range condition directly to E_i(s), because Y_E,i(s) is typically closer to Hermitian.Source: 3) Example Conditions · p. 9
  • Allowing edge-specific scalar multipliers makes each local constraint depend only on nearby lines, loads, and shunts; adding a line affects only the conditions at its connected buses.The scalars may differ by edge and can be selected by solving a local linear matrix inequality.Source: 3) Example Conditions · p. 9
  • Extended-subsystem plug-and-play conditions rely on sparse network admittance; they cannot be used in the admittance representation when its inverse is generally non-sparse.Source: 3) Example Conditions · p. 9

C. Local Bus Conditions for Fixed Networks

For fixed network topologies, the section presents local bus conditions that recover established tests, while noting their dependence on global network properties. Conic combinations of these conditions reduce conservatism and can certify stability where individual conditions fail.

C. Local Bus Conditions for Fixed Networks

  • The section treats existing results as special cases of a general condition, but their multiplier parameters depend on global network parameters or centralized knowledge and suit fixed topologies only.Source: C. Local Bus Conditions for Fixed… · p. 9
  • The property used in one condition may fail when shunts are modeled as in (14), so its applicability depends on the exact network configuration.Source: C. Local Bus Conditions for Fixed… · p. 9
  • The combined multipliers add freedom beyond the specific identity- or J-proportional blocks used by the listed conditions.The general condition permits arbitrary 2 × 2 blocks.Source: C. Local Bus Conditions for Fixed… · p. 9
  • Conic combinations of the specified multiplier conditions reduce conservatism and can certify stability in frequency ranges where every individual condition fails.The combination can satisfy the network constraint even when an individual condition does not.Source: C. Local Bus Conditions for Fixed… · p. 10

D. Combined Stability Criteria

The combined criteria certify grid stability by checking local subsystem conditions across the Nyquist contour, with a centralised condition available where decentralised checks are infeasible. Mixing feasible conditions across frequency ranges and using extended subsystems adds flexibility beyond bus-level checks.

D. Combined Stability Criteria

  • Implementable Corollaries 9, 11, 14 and 16 to 18 form the basis of grid codes; when decentralised conditions are infeasible, Proposition 19 includes centralised condition (21) to cover problematic Nyquist-contour regions.Source: D. Combined Stability Criteria · p. 10
  • The admittance alternative requires stable bus admittances and loses extended-subsystem checks because the inverse network matrix is generally non-sparse.Part II proposes a dynamic loop transformation that can stabilise bus impedances at low frequencies, allowing extended conditions in such cases.Source: D. Combined Stability Criteria · p. 10
  • Impedance and admittance criteria can be mixed across frequency ranges, since no single condition necessarily holds over the entire spectrum.Full-block multipliers generalise earlier conditions, and extended-subsystem criteria add flexibility beyond bus-level conditions.Source: 1) Reduced conservatism · p. 10

1) Reduced conservatism:

  • Local inequalities allow bottom-up certification, with per-bus verification effort independent of network size and limited need for centralised computation or global model knowledge.The approach accommodates heterogeneous device models; centralised coordination is needed only where centralised checks are required.Source: 2) Locality · p. 10

2) Locality:

In the modified 9-bus study, mixed decentralised conditions certify stability across all frequencies when an infinite bus is connected, but low-frequency checks without it require centralised analysis. The case study also illustrates how representation stability limits the use of plug-and-play conditions.

2) Locality:

  • Grid-code rules can assign frequency ranges for each condition, allowing device operators to verify compliance using local dynamics while retaining freedom to modify local controls.Suitable parameters can be computed by convex optimisation, and buses can verify conditions over frequency bands using discrete grids or the generalised KYP lemma.Source: 3) Grid code formulation · p. 11, 5) Practical implementation · p. 11

3) Grid code formulation:

  • Adding a bus requires local certification, while a new line or load triggers re-verification only at connected buses under the plug-and-play conditions.Conditions depending on global network parameters instead require network-wide re-verification after any modification.Source: 4) Plug-and-play operation · p. 11

V. CASE STUDY

  • Two unstable bus impedances prevent use of the impedance representation and its extended-subsystem formulation, restricting this case study to fixed-network conditions.The study instead uses the admittance representation; Part II addresses unstable impedance poles to restore use of extended-subsystem criteria.Source: §V · p. 11, §VI · pp. 12–13
  • A modified 9-bus network replaces three synchronous generators with grid-forming inverters and adds a switchable connection to an infinite bus.The study models heterogeneous lines, loads, and shunts as admittances and uses Kron reduction to form the network admittance.Source: Figure 4 · p. 11, §V · p. 11

1) Infinite Bus Connection:

  • All-frequency decentralised checks certify stability with the infinite bus connected, using a conic combination where individual conditions do not cover the full spectrum.With the switch closed, the conic combination holds across all frequencies; positive-real, rotated positive-real, and DW shell separation conditions hold over different bands.Source: 1) Infinite Bus Connection · pp. 11–12, Figure 5 · p. 12

2) No Infinite Bus Connection:

  • With the switch open, a condition of the form (40b) held at all buses above 8.4 rad s−1; centralised analysis was required below 8.4, with rotated positive-real holding between 8.4 and 92, conic combination between 92 and 315, and positive-real above 315.Source: Figure 6 · p. 12, 2) No Infinite Bus Connection · p. 12

VI. CONCLUSION

  • The framework generalises earlier decentralised conditions and extends impedance-based criteria to subsystems containing a bus and its connected lines and loads.The case study demonstrates reduced conservatism from full-block multipliers and supports the framework as a foundation for grid codes in fixed networks.Source: §VI · p. 12

APPENDIX

The appendix supplies proofs connecting contour conditions and local inequalities to stability criteria, including norm bounds, eigenvalue equivalences, and edge-wise subsystem tests.

A. Proof of Lemma 3

  • The proof uses homotopy invariance to preserve the number of encirclements and concludes stability when the contour avoids −1 and the origin assumption holds.It also checks the imaginary axis away from zero and uses the assumption that the closed-loop inverse has at most one pole at zero.Source: A. Proof of Lemma 3 · p. 13

B. Proof of Lemma 4

  • The proof of Lemma 4 rules out a singularity by contradiction from the multiplier inequalities.Assuming singularity yields a nonzero vector relation that contradicts satisfaction of the inequalities.Source: B. Proof of Lemma 4 · p. 13

C. Proof of Corollary 11

  • Scaling each network-admittance block row to unit absolute row sum bounds the spectral radius and yields the required matrix inequality.The induced infinity norm bounds the spectral radius, which supplies the condition used by Corollary 5 and Lemma 7.Source: C. Proof of Corollary 11 · p. 13

D. Proof of Lemma 12

  • The Lemma 12 proof reduces the network condition to an equivalent condition using matrices with the same non-zero eigenvalues.The passage identifies the shared non-zero eigenvalues of the network loop matrix and its reformulation.Source: D. Proof of Lemma 12 · p. 13

E. Proof of Lemma 13

  • The Lemma 13 proof combines block-diagonal bus multipliers with edge-wise inequalities to establish the network condition.Because the relevant system matrix is block diagonal, the local condition follows from the edge inequalities.Source: E. Proof of Lemma 13 · pp. 13–14

F. Proof of Corollary 15

The appendix proves graphical and multiplier-based criteria through matrix inequalities, then lists the inverter parameters used in the case study.

F. Proof of Corollary 15

  • The Corollary 15 proof connects the matrix inequality to inclusion of the numerical range in the specified region.It shows the inclusion for all unit vectors, then reverses the argument to recover the inequality.Source: F. Proof of Corollary 15 · p. 14

G. Proof of Corollary 17

  • The Corollary 17 proof expresses norm, positive-real, rotated positive-real, and multiplier conditions in the network constraint form.Under the stated hypotheses, these conditions imply the result through Corollary 5 and Lemma 7.Source: G. Proof of Corollary 17 · p. 14

H. Case Study Parameters

  • Appendix H lists the filter, droop, voltage-control, current-control, and feedforward parameters for each of the three grid-forming inverters.The values are reported in the order of buses ν1, ν2, and ν3.Source: H. Case Study Parameters · p. 14
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