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A Small-Gain-Like Framework for Large-Signal Stability Evaluation of Multi-Converter Systems

Qiannan Qu, Kaiwen Chen, Xin Xiang, Wuhua Li, Yunjie Gu

arXiv:2608.29570v1eess.SY

TL;DR

The paper addresses the difficulty of evaluating large-signal stability in multi-converter systems with diverse control algorithms and coupled angle dynamics. It models interconnected relative-angle motions and establishes a small-gain-like framework that constructs Lyapunov functions and an ellipsoidal stability region. The method explicitly computes stability boundaries for converter systems and its estimates are validated experimentally.

  • Problem

    Diverse converter controls and complex circuit-coupled angle dynamics make large-signal stability evaluation of multi-converter systems challenging.

  • Method

    The framework models interconnected relative-angle motions, establishes a small-gain-like property within angle limits, and uses it to construct Lyapunov functions and an ellipsoidal forward-invariant region.

  • Results

    The method explicitly computes large-signal stability boundaries for multi-converter systems, with estimated critical clearing results offering effective approximations of actual values in experimental validation.

  • Takeaways & Limitations

    The framework enables quantitative stability assessment for systems containing different converter types and may support scalable stability evaluation and parameter design.

Abstract

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The increasing penetration of grid-connected converters has greatly altered the large-signal behavior of power systems. Their angle dynamics, shaped by diverse control algorithms and coupled through complex circuit interactions, pose substantial challenges to large-signal stability evaluation of multi-converter systems. To resolve this issue, the small-gain theorem, which characterizes the dissipation capability of interconnected systems (the small-gain-like property) via the individual dissipation capabilities of subsystems, is introduced to investigate transient angle motions in multi-converter systems. A large-signal model involving a set of interconnected relative angle motions is first developed, and the small-gain-like property is then established for multi-converter systems within certain angle limits, which further enables the construction of Lyapunov functions. Based on this, an ellipsoidal forward-invariant region is identified inside the angle-limit region, which serves as an effective estimate of the large-signal stability region for multi-converter systems. The method is further applied to a paralleled system and a four-converter system, where the large-signal stability boundaries are explicitly computed and subsequently validated through experiments. The proposed small-gain-like based large-signal stability evaluation method enables quantitative stability assessment in multi-converter systems with diverse control algorithms, which may provide a scalable framework for large-signal stability evaluation and parameter design in modern power systems.

NOMENCLATURE

The paper models multi-converter systems whose control-governed dynamics and circuit interactions complicate large-signal stability analysis. It introduces relative-angle-based modeling and a small-gain-like framework for assessing such systems.

  • Grid-connected converters are governed predominantly by software-defined control strategies, creating challenges for large-signal stability analysis of multi-converter systems.
  • The proposed framework targets rigorous and scalable large-signal stability analysis without relying on physical energy-conservation properties.
  • The overall angle dynamics are represented using interconnected relative angle motions among converters with different control structures.
  • The system model classifies converters as grid-following or grid-forming according to their distinct control structures.
  • GFL converters synchronize through a first-order PLL and track a constant current reference, while converter terminal quantities are determined through network relationships.

B. Relative Angle Motion Characterization

Relative-angle dynamics form a high-dimensional interconnected structure because each converter-pair motion is influenced by other relative angles. This coupling makes intuitive and quantitative large-signal stability assessment difficult.

  • Relative angles are defined between GFL-GFL, GFL-GFM, and GFM-GFM converter pairs.
  • Each relative-angle motion is influenced by all other relative angles, producing a large-scale interconnected structure with complex interactions.
  • A system containing n converters has n(n −1)/2 relative-angle motions between converter pairs.

C. Large-signal Modeling of the Multi-converter System

The model represents multi-converter dynamics through interconnected relative angle motions and selects a nonredundant subset with stronger self-damping. An invertible transformation shows that these selected motions fully represent the system angle dynamics.

  • Relative-angle dynamics: GFL-GFL relative motions have limited self-damping, whereas GFL-GFM and especially GFM-GFM motions exhibit stronger self-damping.The GFL-GFL terms oppose each other; GFM-GFM terms reinforce, while GFL-GFM terms are approximately in quadrature.
  • Relative-angle selection: For p GFL and q GFM converters, the selected variables comprise p GFL-GFM motions and q −1 GFM-GFM motions.The GFL-GFM motions include every GFL converter, while the GFM-GFM motions connect all GFM converters without redundancy.
  • Relative-angle selection: Each GFL converter is paired with its nearest GFM converter, defined by the maximum self-damping amplitude among candidate GFM converters.A weighted graph uses corresponding self-damping amplitudes as edge weights for selecting GFM-GFM connections.
  • Relative-angle selection: The GFM-GFM motions are selected from a maximum spanning tree, connecting all GFM converters without redundancy while preserving relatively strong self-damping.The tree uses GFM converters as nodes and self-damping amplitudes as edge weights.
  • Coordinate equivalence: The transformation matrix T_r is invertible, so the selected relative-angle coordinates are equivalent to conventional reference-based relative-angle coordinates.Its block structure has an identity upper-left block, zero lower-left block, and a full-rank tree incidence block.
  • Coordinate equivalence: Consequently, p GFL-GFM and q −1 GFM-GFM relative motions completely depict the large-signal dynamics of systems with p GFL and q GFM converters.The selected set provides a complete description despite the original system containing relative angles between converter pairs.

III. SMALL-GAIN-LIKE FRAMEWORK FOR LARGE-SIGNAL STABILITY EVALUATION

The framework establishes a small-gain-like property for interconnected systems and extends it to multi-converter angle dynamics within specified angle limits. This supports Lyapunov-based stability characterization and an ellipsoidal estimate of the large-signal stability region.

  • Framework overview: The small-gain-like property is verified for multi-converter relative angle motions within certain angle limits.The framework analyzes individual dissipation capabilities and their interconnections before identifying a stability region inside those limits.
  • Framework overview: An ellipsoidal forward-invariant region inside the angle-limit region provides a straightforward estimate of the multi-converter large-signal stability region.The region follows from the Lyapunov characterization enabled by the small-gain-like framework.

A. Small-gain-like Property of Interconnected Systems

The small-gain-like framework combines subsystem dissipation inequalities through a dissipation matrix and a weighted sum of Lyapunov functions. A nonsingular M-matrix condition provides a sufficient stability criterion for the interconnected system.

  • Subsystem dissipation: Each subsystem satisfies a dissipation inequality combining intrinsic damping with supply-rate terms from interconnected inputs.This inequality characterizes an input-to-state stability-like property of each subsystem.
  • Interconnection representation: The interconnected model sets each input u_ij equal to the state x_j of the influencing subsystem.The resulting inequalities are collected into a dissipation matrix representing subsystem and connection capabilities.
  • Lyapunov construction: A whole-system Lyapunov function is constructed as the positive weighted sum V = c^T V̄, with all coefficients c_i positive.Its construction depends on both subsystem dissipation properties and interconnection structure.
  • Small-gain-like condition: The small-gain-like property requires a positive coefficient vector for which the sum-type Lyapunov function satisfies the system dissipation inequality.This definition makes the property depend on the existence of an appropriate weighted Lyapunov function.
  • Small-gain-like condition: If the dissipation matrix E is a nonsingular M-matrix, the interconnected system possesses the small-gain-like property.Because E is a Z-matrix with nonpositive off-diagonal entries, the M-matrix condition yields a positive vector supporting the Lyapunov inequality.
  • Stability implication: The resulting Lyapunov dissipation inequality characterizes interconnected-system stability and enables large-signal stability evaluation of multi-converter systems.For a single-loop connection, the condition reduces to a loop-gain-based small-gain stability criterion.

B. Dissipation Capability Analysis for Multi-converter Systems

The paper models selected relative angle motions as interconnected subsystems and quantifies their dissipation capabilities through subsystem inequalities and a dissipation matrix.

  • Relative-angle model: The equilibrium of each selected relative angle motion is shifted to zero before constructing the large-signal state model.The shifted variables are collected into the state vector xδ.
  • Interconnected dynamics: Each relative-angle subsystem is represented as ˙xδi = fδi(xδ), with dynamics influenced by the other relative angles through coupling functions gixj.Hadamard’s lemma is used to express the interconnected dynamics in this form.
  • Dissipation analysis: Positive definite, radially unbounded subsystem functions and Young’s inequality yield dissipation inequalities over a prescribed relative-angle domain.The coefficients ϵij quantify the Young’s inequality terms used in these bounds.
  • Dissipation analysis: The subsystem inequalities are assembled into a dissipation matrix Eδ that captures self-damping and coupling effects across the multi-converter angle dynamics.The matrix is constructed from the obtained subsystem dissipation bounds.

C. Large-signal Stability Evaluation with Small-gain-like Property

The framework verifies the small-gain-like property within prescribed angle limits, constructs a Lyapunov function, and derives an ellipsoidal forward-invariant stability estimate.

  • Angle limits: Because Eδ varies with transient relative angles, the analysis restricts trajectories to an angle-limit region Bx where the dissipation matrix can be explicitly evaluated.Within this region, the matrix is rewritten in constant form for direct verification.
  • Small-gain-like property: The small-gain-like property holds when the corresponding dissipation matrix is a nonsingular M-matrix.This condition provides the criterion used to establish the interconnected-system property.
  • Lyapunov region: For an admissible angle-limit set, a Lyapunov function is constructed, and the system is Lyapunov stable within the corresponding angle-limit region.The Lyapunov stability region is defined through a sublevel set of Vδ.
  • Lyapunov region: The ellipsoidal region BV is forward-invariant because Vδ is non-increasing along trajectories and trajectories cannot cross its defining level set.This region is therefore retained for all future time under the stated conditions.
  • Stability estimate: The nδ-dimensional ellipsoid provides an effective estimate of the large-signal stability region, while the Lyapunov-function decay characterizes convergence tendency and relative stability-recovery speed.Its volume can be computed using the standard n-dimensional ellipsoid formula.
  • Limitations: The estimate is conservative because interactions are uniformly treated as destabilizing and maximum-value approximations with Young’s inequality bound actual dissipation capabilities.Favorable interactions may instead contribute to synchronization restoration under specific transients.
  • Optimization procedure: The procedure searches multiple candidate angle-limit sets, optimizes σδ for ellipsoid volume, and can include control parameters for coordinated tuning.The candidate producing the largest ellipsoid volume is selected for stability evaluation.

IV. CASE STUDY

The case studies apply the framework to a paralleled GFL-GFM system and a two-area four-converter system, computing ellipsoidal stability regions as approximations of actual large-signal stability regions.

  • Case-study systems: The case-study systems include a paralleled GFL-GFM grid-connected system and a two-area four-converter system.For both systems, the small-gain-like property is assessed and corresponding Lyapunov functions are constructed.
  • Paralleled system: The paralleled system contains two converters: VSC1 operates as GFL and VSC2 as GFM, with both connected to the PCC through circuit impedances.The grid connection is represented through Zg and the converter connections through Z1 and Z2.
  • Stability-region assessment: For the studied systems, the resulting ellipsoidal stability regions provide effective approximations of their actual large-signal stability regions.The case-study section uses the framework to determine these regions after assessing the small-gain-like property.

A. Paralleled GFL-GFM Grid-connected System

The paralleled GFL-GFM system is modeled through two interconnected relative angle motions, enabling small-gain-like stability conditions and an ellipsoidal stability-region estimate. Optimizing control parameters enlarges the estimated region.

  • System modeling: The paralleled system contains a GFL converter and a GFM converter connected through impedances to the PCC and grid.The grid is modeled as an infinite-bus special case of a GFM converter.
  • System modeling: Two relative angle motions define the state vector: GFL-GFM and GFM-grid angles.These motions form a two-dimensional state for the paralleled system.
  • Stability analysis: The small-gain-like property holds when the dissipation conditions and associated inequalities are satisfied throughout the prescribed angle-limit set.Under these conditions, a Lyapunov function can be constructed.
  • Stability analysis: The resulting two-dimensional forward-invariant ellipsoid is selected by examining angle-limit candidates and choosing the one with the largest ellipse area.The ellipsoid approximates the actual large-signal stability region.
  • Numerical evaluation: [1.485, 1.322]T is the optimized angle-limit set for the studied paralleled system, whose ellipsoidal boundary lies within the angle-limit boundary.The boundary is visualized for the original parameters in Fig. 5(a).
  • Numerical evaluation: [Kp1; Dp1] = [0.5; 50] enlarges the estimated stability region, with optimized angle limits [2.182, 1.840]T.The optimized parameters are selected by maximizing ellipsoid volume within predefined bounds.

B. Two-area Four-converter System

The two-area four-converter system is represented by three interconnected relative angle subsystems. The small-gain-like condition yields a three-dimensional ellipsoidal stability estimate whose volume increases after control-parameter optimization.

  • System structure: The system has two converter groups linked by a transmission line, with GFL and GFM converters assigned to separate PCCs.VSC1 and VSC3 are GFL converters, while VSC2 and VSC4 are GFM converters.
  • System structure: Self-damping comparisons select the relative motions (VSC1,VSC2), (VSC3,VSC4), and (VSC2,VSC4) for stability assessment.The grouping follows the nearest GFM converter connected to the same PCC for each GFL converter.
  • Stability analysis: The four-converter angle dynamics are represented by three interconnected subsystems with state vector [xb1, xb2, xb3]T.Their coefficient functions and dissipation matrix are constructed for the small-gain-like analysis.
  • Stability analysis: The small-gain-like property holds when all leading principal minors of the dissipation matrix are positive throughout the angle-limit set.A Lyapunov function and a three-dimensional ellipsoidal stability estimate then follow.
  • Numerical evaluation: [1.994, 1.753, 1.221]T yields the largest ellipsoid volume for the original four-converter parameters.The estimated boundary is shown as an orange ellipsoid within the angle-limit boundary.
  • Numerical evaluation: [0.471; 0.488; 18.282; 39.914] enlarges the estimated ellipsoid volume, with optimized angle limits [2.415, 2.355, 1.375]T.The parameters correspond to [Kp1; Kp2; Dp1; Dp2].

V. EXPERIMENTAL VALIDATION

Hardware-in-the-loop experiments test the proposed stability estimates under voltage sag, phase jump, and short-circuit faults. The estimated critical conditions correctly distinguish stable and unstable transients in both systems.

  • Experimental setup: RT-LAB hardware-in-the-loop experiments compare critical fault-clearing points from the proposed method with time-domain transient waveforms.Tests cover both the paralleled and four-converter systems under multiple fault types.
  • Paralleled system: For the paralleled system, CCTA and CCAA are obtained from intersections between fault-on trajectories and the ellipsoidal boundary.CCT is the maximum fault duration allowing return to the same SEP after clearance.
  • Paralleled system: 729 ms remains stable while 882 ms becomes unstable under voltage sag, matching the estimated CCTA of 729 ms.The unstable case reaches a different SEP after a one-cycle oscillation.
  • Paralleled system: 1.84 rad is correctly classified as stable, whereas 2.26 rad causes loss of grid synchronization and settling at a new SEP.The estimated maximum phase jump angle is 1.84 rad.
  • Four-converter system: 228 ms lies between the tested 226 ms stable and 405 ms unstable fault-clearing times for a short-circuit fault at Zg.The estimated CCTA therefore separates the two observed outcomes.
  • Four-converter system: A 40 ms fault-clearing time at Z1 remains stable, while 130 ms is unstable, consistent with the estimated CCTA of 40 ms.The results further validate the method for four-converter systems.

VI. CONCLUSION

The paper proposes a small-gain-like framework that models interconnected relative angle motions, constructs Lyapunov functions within angle limits, and estimates stability regions with ellipsoids. Applications to paralleled and four-converter systems are experimentally validated, while future work targets direction-dependent interactions and improved dissipation estimates.

  • Conclusion: The framework establishes small-gain-like properties within relative angle limits and identifies an ellipsoidal region as an estimate of the large-signal stability region.It supports stability assessment and control-parameter design optimization for multi-converter systems.
  • Conclusion: Applications to paralleled and four-converter systems directly solve stability boundaries and validate them through experiments.The conclusion presents these applications as demonstrations of the framework.
  • Conclusion: Future work will incorporate direction-dependent interaction effects and improved dissipation estimation strategies.These refinements are intended to further reduce limitations in the proposed framework.
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