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Relative-Degree Wall Restricts Passivity-Based Stability Analysis in Inverter-Dominant Grids
Xiaoyu Peng, Zhongze Li, Xi Ru, Xinghua Chen, Feng Liu
TL;DR
The paper asks why passivity-based distributed certification often fails for detailed inverter models despite their possible stability. It analyzes relative-degree compatibility in the standard formulation and identifies a fundamental exclusion of many electromagnetic models, while discussing dynamic-operator extensions. The resulting stop rule can reject uncertifiable cases before passivity-index computation, although the proposed extensions remain subject to relative-degree restrictions.
Problem
High-fidelity inverter dynamics, especially electromagnetic transients, lack a rigorous passivity-based distributed stability framework, and the structural reason has been unclear.
Method
The paper analyzes the relative degree imposed by standard passivity formulations and examines dynamic passivity operators as an extension.
Results
Standard formulations with n(P) = n(Q) = 0 apply only to device dynamics with no more than two relative degrees, excluding electromagnetic models with n ≥3.
Takeaways & Limitations
A relative-degree feasibility test provides a practical stop rule before passivity-index computation, while dynamic operators can extend applicability without removing relative-degree restrictions.
Takeaways & Limitations
The proposed dynamic-operator extensions still follow a relative-degree restriction, and more detailed models may require definitions beyond differential passivity.
Abstract
from arXiv · showhide
This letter reveals a fundamental limitation of passivity-based distributed stability analysis in power systems. Under the standard formulation, passivity certification inherently imposes a relative-degree compatibility constraint that excludes many high-fidelity inverter dynamic models (e.g., those that include electromagnetic transients). Potential extensions of passivity frameworks are discussed to break this limitation.
I. INTRODUCTION
The paper studies why passivity-based distributed stability analysis has not extended rigorously to high-fidelity inverter dynamics. It identifies a structural relative-degree constraint underlying this limitation.
- Inverter-based resources increase the need for scalable distributed stability analysis, for which passivity is attractive because of its modularity.
- Passivity methods have succeeded for electromechanical models, but rigorous extensions to dynamics including electromagnetic transients remain unavailable.
- Existing approaches usually construct storage functions without a general recipe, so failed computations do not distinguish technical difficulty from structural impossibility.
- The paper identifies an inherent relative-degree constraint that conflicts with high-fidelity inverter dynamics.
- A relative-degree feasibility test can precede passivity-index computation: violation rules out certification in the considered framework, while satisfaction is necessary but not sufficient.
B. Generalized Passivity-Based Formulation of Power Systems
The generalized formulation rewrites device-network interconnections using passivity indices and linear operators, yielding distributed stability criteria based on transformed subsystems. Strict passivity remains a sufficient, not necessary, certificate.
- Power-system stability analysis models devices and networks as a feedback interconnection to exploit passivity modularity.
- Passivity indices and operators P, Q, R transform the interconnection while allowing flexible, potentially non-causal analysis choices rather than hardware-control implementations.
- The formulation subsumes conventional, differential, rotated, and other passivity-based models through different operator choices.
- The transformed device condition depends only on local device information, making the resulting stability criterion distributed.
- Asymptotic small-signal stability follows when every transformed device, the transformed network, and residual coupling are strictly passive.
- The relative-degree obstruction persists even though strict passivity is used only as a sufficient certificate with a nonzero margin.
C. Passivity-Based Formulation under Different Coordinates
Distributed passivity analysis uses polar and Cartesian coordinate frameworks to represent device and network dynamics. The paper presents them as mainstream formulations and examines their shared limitation.
- Polar coordinates represent device inputs by active and reactive power increments and outputs by phase-angle and voltage-magnitude increments.
- In polar coordinates, the network is obtained by linearizing the power-flow equations around an operating point.
- Cartesian coordinates represent device dynamics from DQ-axis current increments to DQ-axis voltage increments.
- The Cartesian network satisfies ID+jIQ = (G+jB)(VD+jVQ), and this representation is also called admittance representation.
- The paper unifies mainstream distributed passivity formulations and states that the relative-degree limitation examined in polar coordinates also persists in Cartesian coordinates through a similar approach.
A. Detailed Dynamic Modeling Increases Relative Degree
High-fidelity inverter models use cascaded control and filtering stages whose relative degrees add. Retaining electromagnetic dynamics therefore raises relative degree beyond the range commonly associated with electromechanical models.
- Inverter models generally cascade outer reference, voltage, current, and lag dynamics, with outer loops generating angle and voltage references from power signals.
- The voltage dynamics are modeled as GV = −∆V/∆Q = GRL · GV L · GIL · GDL · · ·.
- The model stages include a droop or reference loop, voltage and current PI controllers, and an LC filter with parameters Lf, Rf, and Cf.
- The aggregate lag GDL represents filtering, inner-loop tracking, modulation, and implementation effects, while pure time delay precludes full-frequency passivity because of unbounded phase lag.
- Relative degree follows an additive rule across the cascaded stages: n(GV) = n(GRL) + n(GV L) + n(GIL) + n(GDL) + · · ·.
- Electromagnetic dynamics increase n(GV) beyond 2 because each realizable stage contributes a non-negative relative degree.
B. Relative-Degree “Wall” of Passivity-Based Analysis
The standard passivity-based paradigm imposes relative-degree bounds on transformed device, network, and coupling dynamics. With zero-relative-degree operators, it excludes device models with three or more relative degrees, including many electromagnetic-transient models.
- Relative-degree constraints: Theorem 1 requires the transformed device, network, and coupling transfer functions to have relative degree between -1 and 1.This follows from the necessary condition that a passive transfer function has absolute relative degree no greater than one.
- Relative-degree constraints: Existing passivity operators with n(P) = n(Q) = 0 apply only to device dynamics with no more than two relative degrees.The restriction follows after substituting the standard operator degrees into the transformed-system bound.
- Model-fidelity consequence: More detailed device models typically increase relative degree because added realizable dynamics contribute non-negative relative degrees.The paper uses this modeling-fidelity relationship to connect electromagnetic dynamics with the passivity limitation.
- Model-fidelity consequence: Under standard formulations, electromagnetic models with n ≥3 fall outside the certifiable class regardless of storage-function search efforts.The result is a no-go implication for existing methods, not merely a difficulty in finding a suitable storage function.
C. Origins of Relative-Degree “Wall”
The relative-degree wall originates in the positive-real conditions underlying passivity, which constrain Nyquist behavior across the full frequency range. Finite-frequency screening cannot replace full-frequency verification for modular stability certification.
- Positive-real origin: Passivity requires the Nyquist contour to remain within the positive-real region −90° ≤ ∠T ≤ 90° across the relevant frequency range.The constraint comes from the real-part conditions on the finite-frequency response and the infinite-radius semicircle.
- Positive-real origin: Fig. 2(a) compares conventional and differential passivity regions with inverter Nyquist contours spanning relative degrees n = 1–4.The figure frames the limitation geometrically through passivity regions and contours of inverter dynamics.
- Full-frequency requirement: Finite-frequency passivity methods cannot replace full-frequency verification because low-frequency passivity can coexist with distinct stable and unstable responses at higher frequencies.The paper attributes the different closed-loop outcomes to higher-frequency behavior and interconnection effects.
D. Extension of Applicability
The relative-degree obstruction persists in multidimensional systems and grid-following models, but dynamic operators can extend passivity to at least some higher-order inverter dynamics. A demonstrated n(Gi) = 3 model becomes passivizable through an approximate inverse operator.
- Multiple-dimension systems: The relative-degree obstruction still applies to MIMO systems with crosscoupling.The paper extends the argument beyond single-dimensional channels and notes that increased model fidelity still raises device relative degrees.
- Multiple-dimension systems: For a nontrivial passive realization, assuming every nonzero element has relative degree at least two contradicts positive definiteness in the KYP-based realization.The contradiction establishes that element-wise n(˜Gi) ≥2, corresponding to n(Gi) ≥3, remains prohibited for multidimensional device dynamics.
- Grid-following dynamics: The demonstrated grid-following model preserves causal [θ, V] inputs to [P, Q] outputs and PLL-generated angle dynamics while retaining the relative-degree restriction.The restriction is presented as independent of the specific GFL model used for demonstration.
IV. MOVING BEYOND RELATIVE-DEGREE “WALL”
Dynamic passivity operators can partially extend the relative-degree range covered by passivity analysis, but the relative-degree restriction remains. A third-order inverter model can be passivized under this extension, while more detailed models require broader definitions.
- Extending the passivity framework: Dynamic operators with n(P), n(Q) < 0 can partially circumvent the existing relative-degree obstacle by extending the applicable range through (7).The paper frames this as an extension of static passivity operators rather than removal of the restriction.
- Extending the passivity framework: Differential passivity with dynamic operators extends the applicable relative degree up to n ≤4 when n(P), n(Q) ≠ 0.This follows from the stated bound |n(QP)| ≤2 combined with (7).
- Example: A third-order model Gi with n(Gi) = 3 can be passivized using dynamic operators, unlike with existing methods.The construction uses (P, Q, R) = (I, GcI, sI) and yields passivity index σi = 1/DV.
- Remaining boundary: The dynamic-operator approach can similarly derive n(Gi) = 4, but more detailed models require definitions beyond differential passivity.The extension increases the applicable relative degree without eliminating the restriction in (7).
- Interpretation: The dynamic-operator formulation is essentially equivalent to dynamic passivity indices through (P, Q, R) = (I, σ, σ−1).The dynamic-index viewpoint is also reported in.
V. CONCLUSION
The letter identifies a relative-degree requirement as a practical stop rule for passivity-based distributed certification. Within the existing paradigm, this requirement excludes many detailed models and explains why certificates can fail for stable electromagnetic inverter dynamics.
- Conclusion: Passivity-based distributed certification requires a relative-degree condition that excludes many detailed inverter models within the existing paradigm.The paper presents this requirement as a practically actionable stop rule.
- Conclusion: The relative-degree condition explains why distributed passivity certificates can fail for electromagnetic inverter dynamics even when the systems themselves are stable.