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Performance Limits and Tradeoffs of Power Systems Synchronization Recovery in Complex-Frequency Representation

Xiaoyu Peng, Zelin Sun, Zhongze Li, Feng Liu

arXiv:2610.02719v1eess.SYcs.SY

TL;DR

The paper asks how well post-disturbance synchronization recovery can be shaped when phase-angle and voltage dynamics are coupled. It constructs a complex-frequency recovery residual and establishes a conservation law whose budget constrains recovery, yielding frequency- and time-domain limits.

Abstract

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Power-system synchronization studies mainly ask whether synchronization can be achieved, whereas a complementary question is how well post-disturbance synchronization recovery can be shaped. In inverter-dominated systems, this recovery involves coupled phase-angle and voltage-magnitude dynamics, for which complex frequency provides a unified representation. This paper establishes a conservation law for synchronization recovery in this representation. For a given system and its initial state, the logarithmic integral of the proposed recovery residual is fixed by the initial contraction rate and the right-half-plane (RHP) zero--pole structure. Hence, synchronization recovery performance cannot be improved uniformly over frequencies and variables unless the budget set by these terms changes; controller tuning that preserves them can only redistribute the residual response across frequencies. The conservation law applies to both linear and nonlinear recovery responses satisfying the mild analyticity conditions. It further yields finite-frequency peak--bandwidth limits and a lower bound on the time-domain integral of squared complex frequency recovery. A system-level extension recovers classical Bode-type sensitivity conservation as an LTI special case. Studies on a heterogeneous IEEE 39-bus system verify the theory.

I. INTRODUCTION

In inverter-dominated systems, phase-angle and voltage dynamics are coupled, motivating a complex-frequency view of post-disturbance recovery. The paper develops a conservation-based framework for asking how recovery can be shaped and what limits remain.

A. Motivation and Background

  • Coupled phase-angle and voltage dynamics mean phase angle or frequency alone may not describe recovery; complex frequency combines normalized voltage rate and angular frequency,,.The motivating question is how well recovery can be shaped for a given system and initial state.Source: A. Motivation and Background · p. 1

B. Literature Review

  • Prior complex-frequency work studies generation, propagation, and decomposition, while this paper constrains the free recovery response rather than a prescribed disturbance-rejection channel,,.Under linearization and additional assumptions, the result reduces to.Source: B. Literature Review · pp. 1–2

C. Main Contribution

  • The proposed residual characterizes the remaining coupled angle–voltage synchronization error after a disturbance, enabling direct analysis of its performance limits.The formulation differs from analyses centered on stability, modal behavior, or controller-specific metrics.Source: C. Main Contribution · p. 2
  • A logarithmic recovery budget depends on initial contraction and non-minimum-phase structure, distinguishing a changed limitation from performance redistribution within a fixed budget.The theory derives peak–bandwidth and complex-frequency-activity limits.Source: C. Main Contribution · p. 2

D. Paper Organization

  • The paper proceeds from the model and residual to conservation, recovery limits, a system-level extension, numerical validation, and conclusions.The extension recovers earlier Bode-type sensitivity conservation as a special case.Source: D. Paper Organization · pp. 2–3

A. General Power-System Model

  • The general model includes a network and heterogeneous devices, with bus voltage represented as a function of dynamic states and algebraic variables.Under well-posedness, the model and initial state uniquely determine the synchronization response and recovery budget.Source: A. General Power-System Model · p. 3

B. Complex Angle, Complex Frequency, and Synchronization

Complex synchronization requires phase angle and voltage to converge together. The paper defines a system- and initial-condition-dependent residual to represent the remaining error and its correction during recovery.

B. Complex Angle, Complex Frequency, and Synchronization

  • Complex synchronization requires both phase angle and voltage to converge to their post-disturbance equilibrium.The complex-frequency components are normalized voltage rate and angular-frequency deviation from the synchronous frame.Source: B. Complex Angle, Complex Frequency,… · p. 3

C. Complex Synchronization Residual

  • The system, initial state, and observation bus determine the residual constructed from the Laplace transform of the complex-angle synchronization error.Its unit-step response has a physical interpretation in the recovery representation.Source: C. Complex Synchronization Residual · p. 3
  • The residual R represents the remaining synchronization-error fraction, while C represents the corrected fraction relative to the initial error.This complementary representation is analogous to sensitivity and complementary sensitivity functions.Source: C. Complex Synchronization Residual · p. 3

III. INTEGRAL CONSERVATION FOR COMPLEX SYNCHRONIZATION

The section establishes a conservation law for synchronization recovery: once the system and initial state are specified, the logarithmic residual budget is fixed. Its value depends on the initial contraction or expansion and RHP zero–pole structure, so tuning that preserves these terms redistributes performance rather than changing the budget.

III. INTEGRAL CONSERVATION FOR COMPLEX SYNCHRONIZATION

  • The conservation law applies to residuals satisfying Assumption 3 and to residuals induced by synchronized trajectories satisfying Assumption 2.Assumption 3 requires meromorphic RHP behavior, continuous imaginary-axis boundary values, no imaginary-axis poles or zeros except a finite-order structural zero at s = 0, and the stated regularity conditions.Source: §III · p. 4
  • For each specified system and initial state, the synchronization-recovery budget has a unique value, although different initial states can yield different budgets.The identity is universal in form, not in its value.Source: §III · p. 4
  • The budget is determined by the initial complex-angle error contraction or expansion rate and the RHP-zero term, which captures non-minimum-phase cancellation.Tuning that preserves these terms can reshape the response but only redistributes constrained performance across frequencies or physical variables.Source: §III · p. 4

IV. INTERPRETATION AND COMPUTATION OF RESIDUAL ZEROS AND POLES

This section interprets residual poles as response singularities and RHP zeros as cancellation among recovery components and propagation paths. For LTI systems, it relates residual poles and zeros to modes and transmission zeros of the relevant observed or disturbance-driven channels.

A. Poles as Response Singularities

  • A pole of a meromorphic continuation of Φ_i(s) or R_i(s) is a singularity of the selected response transform.For stable synchronized responses satisfying Assumption 2, Φ_i is analytic in the open RHP and has no RHP pole.Source: A. Poles as Response Singularities · p. 4

B. Zeros as Exponentially Weighted Recovery Cancellation

  • An RHP residual zero represents exact cancellation of the demodulated complex-angle error within an exponentially weighted window of scale 1/σ.The cancellation arises among response components and propagation paths seen through the selected channel; disturbance location or initial state can change the residual zeros.Source: B. Zeros as Exponentially Weighted… · p. 4

C. Characterization for Linear-Time-Invariant Systems

1) Unspecified Disturbance Directions:

  • For an unspecified initial-state direction x0, the residual poles are modes both excited by x0 and observed by C_i, while its nonzero zeros are transmission zeros of the initial-condition channel.The structural residual zero is at s = 0.Source: 1) Unspecified Disturbance Directions · p. 5

2) Prescribed Disturbance Directions:

  • For a prescribed impulse direction q with D_id = 0 and C_iB_dq ≠ 0, the residual's nonzero poles and zeros match those of the disturbance-to-observation transfer function after minimal realization.The residual additionally has a structural synchronization zero at s = 0.Source: 2) Prescribed Disturbance Directions · p. 5

D. Computation from Nonlinear System Models

The section describes contour-moment and sampled-response procedures for estimating the nonlinear residual budget without linearization. It also gives finite-frequency peak–bandwidth constraints: low-frequency recovery requirements and high-frequency roll-off can force an intermediate peak or require a minimum effective recovery bandwidth.

D. Computation from Nonlinear System Models

1) Contour-Moment Principle:

  • The contour-moment principle uses the zeroth moment to count enclosed zeros and the first moment to sum them.The zero count also serves as an integer-valued convergence check.Source: 1) Contour-Moment Principle · p. 5

2) Numerical Evaluation from Nonlinear Response Data:

  • For sampled nonlinear responses, the search contour is expanded and its boundary grid refined until the zero-count and zero-sum moments converge.The outer search range should match the adopted model's dynamical bandwidth; convergence provides evidence that all RHP zeros meaningful to that model are enclosed.Source: 2) Numerical Evaluation from Nonlinear… · p. 5

A. Frequency-Domain Limits: Peak–Bandwidth Trade-Off

  • Low-frequency attenuation under a finite high-frequency tail budget forces a compensating peak between ωd and ωc; specified recovery and peak limits can require a minimum effective recovery bandwidth.If the bandwidth feasibility quantity d_bw is nonpositive, the requirements are incompatible.Source: Figure 2 · p. 6, A. Frequency-Domain Limits:… · p. 6

B. Time-Domain Limits: Minimum Complex-Frequency Activity in Recovery

The residual integral sets a lower bound on time-domain complex-frequency activity during recovery. Finishing most correction within a shorter horizon requires a larger peak deviation.

B. Time-Domain Limits: Minimum Complex-Frequency Activity in Recovery

  • Theorem 3 bounds the total squared complex-frequency activity during recovery from below, reflecting a structural recovery cost rather than the full transient waveform.The activity combines voltage-rate and phase-angle-rate contributions.Source: B. Time-Domain Limits: Minimum… · p. 6
  • For fixed initial error and RHP-zero burden, a controller cannot make total complex-frequency activity arbitrarily small, though it may redistribute it between frequency and voltage or across time.The RHP-zero term represents a nonminimum-phase recovery path.Source: B. Time-Domain Limits: Minimum… · p. 6
  • Completing most of the correction before a finite horizon requires a larger peak complex-frequency deviation as the horizon decreases.The tail activity is defined for a finite recovery horizon T.Source: B. Time-Domain Limits: Minimum… · p. 6

VI. EXTENSION TO SYSTEM-LEVEL INTEGRAL CONSERVATION

The system-level extension constructs a residual from differential propagation of bus synchronization error and establishes a global integral law. In the LTI limit, the budget reduces to pole-zero structure, with full observation yielding an interpretation in terms of total available damping.

A. General System-Level Conservation

  • The system-level residual tracks normalized differential propagation of the initial bus synchronization error along the post-fault solution.The construction eliminates algebraic variations and uses the stateflow Jacobian.Source: A. General System-Level Conservation · p. 7
  • Theorem 4 gives a global synchronization residual integral under stated regularity and pole-zero conditions, with its RHP zero and pole terms evaluated at the initial time.The theorem requires the error and its derivatives to satisfy the specified integrability conditions.Source: A. General System-Level Conservation · p. 7
  • In LTI systems, the system-level construction reduces to the observed channel's pole-zero structure; with full observation, the budget reduces to −ℜ[tr(A)], interpreted as total available damping.Higher-order device and network dynamics enter through the DAE variational flow and bus complex-angle observation.Source: A. General System-Level Conservation · p. 7

B. Bode-Type Sensitivity Conservation as a Special Case

The paper relates its global recovery law to classical Bode-type sensitivity conservation under specified LTI feedback conditions. Numerical studies on a heterogeneous IEEE 39-bus system test the frequency- and time-domain limits across disturbance directions and observation buses.

B. Bode-Type Sensitivity Conservation as a Special Case

  • Under the stated stable LTI feedback, relative-degree, and no-RHP-cancellation conditions, the global law reduces to the classical sensitivity-conservation identity with zero recovery budget.The paper identifies the prescribed disturbance channel as a special mechanism for generating the synchronization dynamics considered here.Source: B. Bode-Type Sensitivity Conservation… · pp. 7–8

VII. NUMERICAL VALIDATION

  • The validation uses a small-signal-stable IEEE 39-bus system with synchronous generators and heterogeneous inverters; its first two experiments use a linearized model and the rest nonlinear DAE dynamics.Source: §VII · p. 8

A. Frequency-Domain Conservation Across Initial Conditions and Observation Buses

  • For three initial-state directions observed at bus 31, phase-angle and voltage transients are distinct and residual peaks differ in location and magnitude, yet the frequency integral equals the zero-pole expression in every case.Source: Figure 3 · p. 8, A. Frequency-Domain Conservation… · p. 8
  • For valid cases across observation buses, the frequency integral agrees with the zero-pole expression despite channel-dependent budgets and common nonzero RHP-zero burdens.The comparison spans phase-angle-only, voltage-only, and coupled disturbance directions.Source: Figure 4 · p. 8, A. Frequency-Domain Conservation… · p. 8

B. Time-Domain Performance Limitation

  • Across valid buses, the time-domain result bounds combined voltage-rate and frequency-deviation activity from below by an RHP-zero-dependent cost.The IEEE 39-bus study fixes a 45° coupled disturbance direction at bus 31 and varies the observation bus.Source: B. Time-Domain Performance Limitation · p. 8, Figure 5 · p. 9

C. Performance Redistribution Through Damping Tuning

Damping tuning redistributes synchronization-recovery residuals rather than removing their fixed budget when the initial contraction rate and RHP-zero burden are unchanged. The damping study validates finite-frequency peak lower bounds, and a nonlinear post-fault study checks conservation-budget closure.

C. Performance Redistribution Through Damping Tuning

  • Increasing damping lowers the intermediate-frequency peak but raises low-frequency residuals, shifting logarithmic area rather than removing the unchanged recovery constraint.The cumulative areas return to about zero at high frequency; tuning that preserves the initial contraction rate and RHP-zero burden preserves the logarithmic budget.Source: C. Performance Redistribution Through… · p. 9, Figure 6 · p. 9
  • For ρd = 1, 1.7, 2.5, and 4, eBth = 1.1507, 1.1380, 1.1216, and 1.0958, while actual Mpk = 3.1887, 2.4628, 2.0094, and 1.5874.All four actual peaks exceed their predicted lower bounds; the bounds are positive in every case.Source: Figure 7 · p. 9, C. Performance Redistribution Through… · p. 9

D. Validation with the Nonlinear DAE Model

VIII. CONCLUSION

  • The recovery budget is determined by system structure and initial state, so tuning generally redistributes degradation unless it changes that underlying budget.The paper’s conclusion recommends directing the constrained recovery cost toward less critical frequencies or channels, or changing the structural budget.Source: §VIII · p. 10

APPENDIX A PROOFS OF MAIN RESULTS

The appendix collects proofs connecting the recovery residual to Laplace-domain responses and establishing the conservation and finite-frequency bounds. It also derives zero–pole interpretations for state-space and impulse responses.

APPENDIX A PROOFS OF MAIN RESULTS

  • The appendix gathers proofs of the main-text results.The following proofs establish transform identities, the conservation law, and its corollaries and extensions.Source: Appendix A · p. 11

A. Proof of Proposition 1

  • Integration by parts links the Laplace transforms of angle, complex frequency, and residual, while decay and integrability give their zero- and infinite-frequency limits.The proof obtains R(0) = 0, C(0) = 1, and R(s) → 1 as |s| → ∞ in the closed right half-plane.Source: A. Proof of Proposition 1 · p. 11

B. Proof of Theorem 1

  • Theorem 1 follows by factoring R with a zero-free analytic function, taking its analytic logarithm, and applying a right-half-plane contour integral.The proof relates the imaginary-axis magnitude integral to the initial contraction rate and residual zero–pole terms.Source: B. Proof of Theorem 1 · p. 11

C. Proof of Corollary 1

  • For an L1 signal, its Laplace transform is analytic in the right half-plane, leaving the pole sum empty in the resulting corollary.The proof substitutes this into the preceding conservation identity.Source: C. Proof of Corollary 1 · p. 11

D. Proof of Proposition 2

  • Away from the origin, a residual zero occurs exactly where the Laplace transform of the complex-angle response vanishes.The proof uses R(z) = zΦ(z)/φ(0) for z in the right half-plane.Source: D. Proof of Proposition 2 · p. 11

E. Proof of Proposition 3

  • In a linear state-space model, residual zeros match complex-angle transform zeros, while poles follow from the dynamics after unobservable, uncontrollable, and exactly cancelled modes are removed.The residual relation is derived from x(t) = exp(At)x0.Source: E. Proof of Proposition 3 · p. 11

F. Proof of Proposition 4

  • For an impulse response, scaling leaves zeros and poles unchanged, multiplication by s adds the structural zero at the origin, and exact cancellations are removed by minimal realization.The proof identifies the impulse response through Gi,q(s) = Ci(sI − A)−1Bdq.Source: F. Proof of Proposition 4 · p. 11

G. Proof of Theorem 2

  • Theorem 2’s finite-frequency bound follows by splitting the conservation integral into low-, intermediate-, and high-frequency bands and bounding each contribution.The high-frequency tail uses the envelope bound; combining the three inequalities yields the stated peak bound.Source: G. Proof of Theorem 2 · pp. 11–12

H. Proof of Corollary 2

These proofs extend the recovery bounds to time-domain energy and system-level responses, then simplify the system-level result for LTI models. The bandwidth corollary uses the prescribed bounds to exclude a negative bandwidth coefficient when J is positive.

H. Proof of Corollary 2

  • When J > 0, a negative bandwidth coefficient would contradict the resulting inequality, so the proof establishes the stated corollary for positive bandwidth.The supplied proof excerpt specifically rules out dbw < 0 under J > 0.Source: H. Proof of Corollary 2 · p. 12

I. Proof of Theorem 3

  • Theorem 3 integrates a logarithmic inequality over frequency and applies Parseval’s identity to obtain a lower bound on squared complex-frequency recovery.When ΓR < 0, the proof uses the positive-part form of the nonnegative frequency-domain quantity.Source: §I · p. 12

J. Proof of Theorem 4

  • Theorem 4 derives a matrix residual expansion from the system response, then applies Theorem 1 to the log determinant of that residual.The proof uses log |det(R)| = Σi log |λi(R)|.Source: J. Proof of Theorem 4 · p. 12

K. Proof of Corollary 3

  • For LTI models, the system-level result reduces to a Bode-type sensitivity relation; in the full-state case, only structural zeros at the origin and poles λi(A) remain.The proof derives the LTI expression from the general system-level identities.Source: K. Proof of Corollary 3 · p. 12

ON NONLINEAR TRAJECTORIES

The nonlinear-trajectory procedure estimates the frequency-side residual budget from a sampled post-fault response. It combines direct integration over resolved frequencies with an independently estimated high-frequency tail.

ON NONLINEAR TRAJECTORIES

  • The nonlinear budget evaluation starts from a sampled post-fault trajectory and treats positive- and negative-frequency branches separately because the complex trajectory need not have conjugate symmetry.The finite-horizon Laplace transform is computed by composite trapezoidal quadrature after the response has decayed sufficiently.Source: ON NONLINEAR TRAJECTORIES · p. 12, A. Direct Evaluation over the Resolved… · p. 12

A. Direct Evaluation over the Resolved Frequency Range

  • The resolved spectrum is split into low-, medium-, and high-frequency ranges, with the medium-band contribution integrated on a logarithmic grid by trapezoidal quadrature.The low-frequency approximation uses the structural zero behavior as ω → 0+.Source: A. Direct Evaluation over the Resolved… · p. 12

B. High-Frequency Tail Estimate

  • The unresolved high-frequency tail is estimated from initial trajectory derivatives, with paired frequency branches cancelling first-order terms, then checked for convergence as the upper limit expands.A fourth-order polynomial fit to the first 2 ms estimates the derivatives; acceptance uses successive-estimate and resolved-sample criteria.Source: B. High-Frequency Tail Estimate · p. 13
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