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An Energy-Based Framework for Transient Stability of Grid-Forming Converter Networks With Current Limiting

Wei Zhou, Yunjie Gu

arXiv:2609.36978v1eess.SYcs.SY

TL;DR

Network-level transient stability analysis under current limiting remains challenging, as prior analyses mainly address single converters. The paper constructs a dissipative network energy that incorporates current limiting, and its two-GFM example and EMT simulations support the stability analysis.

  • Problem

    Prior current-limiting transient-stability analyses mainly consider a single GFM, while network terminal voltages and power injections are jointly determined and do not yield a common energy directly.

    Source: §I · p. 1

  • Method

    The framework integrates inductive port current–voltage characteristics into a common network potential and combines it with synchronization dynamics to form a dissipative energy function.

    Source: §I · p. 2, A. GFM Port Characteristics · p. 2

  • Results

    The two-GFM example’s energy criterion guarantees convergence for five identified states, and all five corresponding EMT trajectories return to the target SEP.

    Source: B. Transient Stability Under Phase… · p. 7

  • Takeaways & Limitations

    The same energy supports stability analysis through normal operation, limiting, and recovery, and the formulation extends to arbitrary-size GFM networks under its modeling assumptions.

    Source: §V · p. 9

    The barrier-based estimate excludes energy dissipated along subsequent trajectories, so states outside the estimate may still return to the SEP.

    Source: C. Physical Insights into Transient… · p. 6

Abstract

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Existing analyses for the transient stability of grid-forming (GFM) converters under current limiting mainly address single-converter systems, while network-level transient stability analysis remains challenging. This paper develops a network energy construction that incorporates current limiting while retaining a provable dissipation property. This structure enables Lyapunov-based analysis consistent with classical direct method for synchronous-machine based systems, without switching between mode-dependent energy functions. A two-GFM example using a reactive circular current limiter and impedance insertion illustrates the proposed framework. Electromagnetic transient simulations support the theoretical results.

I. INTRODUCTION

Prior network-level energy analyses either exclude trajectories entering current saturation or lack a guaranteed dissipation property there. This paper constructs a common network energy from inductive GFM port characteristics, retaining dissipation through limiting and recovery.

I. INTRODUCTION

  • Network-level methods either restrict trajectories from entering current saturation or lack guaranteed dissipation during current-limited operation.Earlier network energy functions characterized operation outside the limited region, while another construction was not guaranteed dissipative there.Source: §I · p. 2
  • The proposed construction addresses this gap with a common network energy whose derivative is nonpositive across normal operation, limiter activation, limited operation, and recovery.The construction derives a port potential by integrating each admissible GFM port’s current–voltage characteristic.Source: §I · p. 2

II. ENERGY STRUCTURE OF CURRENT-LIMITED GFM NETWORKS

  • The analysis focuses on transient angle stability and represents the network and GFM ports through algebraic phasor relations, neglecting electromagnetic transients.Source: A. GFM Port Characteristics · p. 2

A. GFM Port Characteristics

  • The port model separates current phase lag from magnitude through a scalar function of voltage-drop magnitude, with continuity, monotonicity, and zero current at zero voltage drop.The analysis adopts the inductive phase relation ϕk = π/2, preserving that relation as current magnitude is limited.Source: A. GFM Port Characteristics · p. 2
  • A reactive circular limiter increases equivalent reactance above its threshold to maintain the current limit, whereas impedance insertion can yield a single-valued current characteristic without a strict ceiling.The inductive phase structure allows port and network potentials to be combined across normal operation, limiting, and recovery.Source: A. GFM Port Characteristics · p. 3

B. Network Potential and Active Power Gradient Structure

The paper builds a network potential by combining port and line contributions, then eliminating terminal voltages through the network equations. Its angle gradient equals electrical active power, enabling transient-energy construction.

B. Network Potential and Active Power Gradient Structure

  • Integrating the inductive port current–voltage characteristic gives a port potential, and analogous line potentials combine with it into a network potential.The same port potential applies across current-limit activation and deactivation.Source: B. Network Potential and Active Power… · p. 3
  • Substituting a KCL solution for terminal voltages into the combined potential defines an effective network potential as a function of internal angles.Its angle dependence includes both internal-voltage and terminal-voltage changes.Source: B. Network Potential and Active Power… · p. 3
  • If the terminal-voltage Jacobian is singular, voltages may be nonunique; when all GFMs are limited and an unregulated voltage degree of freedom remains, additional voltage regulation is required.The effective potential and active powers remain consistent across KCL solutions for the nondecreasing characteristics considered.Source: B. Network Potential and Active Power… · p. 3
  • The effective potential’s angle gradient equals the vector of electrical active powers, enabling construction of a transient energy function with provable dissipation.The effective potential can be evaluated by solving KCL and summing port and line potentials.Source: B. Network Potential and Active Power… · pp. 3–4

C. Transient Energy Function

The transient energy combines synchronization kinetic energy with an angle potential referenced to equilibrium inputs. Damping makes its derivative nonpositive, supporting a conservative region-of-attraction estimate.

C. Transient Energy Function

  • The potential depends only on relative angles, and balanced set points make the effective inputs equal to the active-power set points at nominal frequency.Source: C. Transient Energy Function · p. 4
  • The transient energy combines frequency-deviation energy weighted by effective inertia with an angle potential whose gradient is electrical power minus effective inputs.The energy derivative cancels power-transfer terms, leaving the nonpositive damping term.Source: C. Transient Energy Function · p. 4
  • The lowest potential barrier at an unstable equilibrium defines critical energy; the bounded energy sublevel component containing the target equilibrium estimates its region of attraction conservatively.Convergence follows when that component lies within the model domain, contains no other equilibrium, and damping is positive.Source: C. Transient Energy Function · p. 4

III. TRANSIENT STABILITY ASSESSMENT VIA A TWO-GFM EXAMPLE

The two-GFM example uses the energy structure to examine how current-limited ports jointly shape power transfer and transient stability.

III. TRANSIENT STABILITY ASSESSMENT VIA A TWO-GFM EXAMPLE

  • The two-GFM assessment examines how current-limited ports jointly shape power transfer and how current utilization and dissipation affect transient stability.Source: §III · p. 4

A. Network Potential and Power Transfer

For the two-GFM network, the effective potential can be evaluated by integrating the power–angle characteristic relative to the target equilibrium. Current limiting changes that characteristic and reshapes the associated potential barrier.

A. Network Potential and Power Transfer

  • The two-GFM effective potential is obtained by integrating the power–angle characteristic, referenced to the target SEP.The signed area between Pe(δ) and P0 measures total potential relative to equilibrium.Source: A. Network Potential and Power Transfer · p. 4
  • In Fig. 2, power–angle curve areas relative to P0 represent potential changes, and the stable equilibrium and boundary unstable equilibrium are marked.The limiting curves use the reactive circular current limiter at both GFMs.Source: Figure 2 · p. 5
  • The common current is determined by port current–voltage characteristics and network voltage drops, with continuity, monotonicity, and X12 > 0 ensuring uniqueness.The network equations equate both port currents to the common current and sum the port drops and line drop to ∆E.Source: A. Network Potential and Power Transfer · p. 5
  • Without current limiting, the familiar sinusoidal power–angle relation returns; with limiting, F changes the curve while preserving its relation to network potential.The resulting curve reshapes the associated potential barrier.Source: A. Network Potential and Power Transfer · p. 5

B. Energy-Based Transient Stability Assessment

The two-GFM stability assessment separates center-of-inertia motion from relative motion under equal damping-to-inertia ratios. It evaluates the critical energy barrier and estimates the region of attraction from the relative-motion energy.

B. Energy-Based Transient Stability Assessment

  • With equal damping-to-inertia ratios, center-of-inertia motion decays independently and does not feed relative motion, so stability can be assessed using (δ, ωδ), even when ωc(0) ≠ 0.The relative-motion energy combines kinetic energy and potential, with dissipation specified by the model.Source: B. Energy-Based Transient Stability… · p. 5
  • For P0 > 0, the UEP immediately right of the target SEP gives the lower barrier, measured by the area between Pe(δ) and P0 from δ0 to δu.The power–angle curves considered have one UEP per 2π period.Source: B. Energy-Based Transient Stability… · pp. 5–6
  • Under the stated Section II-C conditions, the energy-based region-of-attraction estimate is the connected component containing (δ0, 0) of the specified energy sublevel set.The estimate is obtained by applying the stability criterion to the relative-motion energy.Source: B. Energy-Based Transient Stability… · pp. 5–6

C. Physical Insights into Transient Stability

The section shows how limiter characteristics shape transient stability margins and voltage regulation, while the energy-based ROA estimate remains conservative. Simultaneous current limiting can preserve angle stability under the stated conditions, but may allow terminal-voltage drift.

Reactive circular current limiter

  • The smaller current limit determines saturation of F; increasing only the larger limit leaves the power–angle curve, potential barrier, and ROA estimates unchanged.This holds with baseline reactances fixed and other system parameters unchanged.Source: C. Physical Insights into Transient… · p. 6
  • With equal current limits, simultaneous saturation need not eliminate the target SEP or its ROA; the same dissipative energy function applies through limiter activation and deactivation.The sublevel component can extend into the region where both GFMs are current-limited, and states there return to the SEP under the conditions in Section II-C.Source: C. Physical Insights into Transient… · p. 6
  • In the equal-limit case, simultaneous saturation can let terminal voltages drift during a transient despite relative-angle recovery when no additional voltage regulation is present.The terminal-voltage Jacobian is singular because the series relation fixes the sum of port voltage drops, not their allocation.Source: C. Physical Insights into Transient… · p. 6

Current utilization with impedance insertion

  • Impedance insertion’s lower current at intermediate voltage separations lowers the power–angle curve, moves the boundary UEP closer to the SEP, and reduces the energy-based ROA estimate.Current utilization throughout the transient, as well as maximum current, determines the stability margin.Source: C. Physical Insights into Transient… · p. 6
  • Impedance insertion retains voltage regulation during limiting but yields lower current utilization and a smaller transient angle-stability margin.Its current continues to rise with voltage drop after activation; at intermediate voltage separations it supplies less current, lowering the power–angle curve and potential barrier.Source: C. Physical Insights into Transient… · p. 6

Conservatism of the energy estimate

  • Energy sublevel estimates remain inside the exact ROAs under current limiting, and their gap narrows when damping decreases or inertia increases.The barrier-based estimate does not credit energy dissipated along the subsequent trajectory, so exclusion from it does not establish instability.Source: C. Physical Insights into Transient… · p. 6

IV. SIMULATION VALIDATION

EMT simulations evaluate the energy-based transient-stability approach under phase jumps and three-phase short-circuit faults. Fig. 4 compares energy-based ROA estimates with exact boundaries across limiter, damping, and inertia cases.

Simulation validation

  • EMT simulations in MATLAB/Simulink evaluate phase jumps and three-phase short-circuit faults using the listed baseline parameters.The EMT model, including limiter implementations, is stated to be made publicly available.Source: §IV · p. 6, Table I · p. 7
  • Fig. 4 compares exact ROA boundaries with energy-sublevel estimates for limiter, damping, and inertia variations.The exact boundaries are computed by backward integration of stable manifolds of boundary UEPs; shaded regions are estimates, dots mark the target SEP, and crosses mark UEPs.Source: Figure 4 · p. 7
  • The baseline EMT port reactance combines 0.0618 p.u. physical reactance and 0.1 p.u. virtual reactance.The balanced set points give nominal synchronous frequency; each converter uses M_i = 2H/ω0 and D_i = Dp/ω0.Source: §IV · pp. 6–7

A. ROA Estimation and Interpretation

The section estimates the ROA using the target-SEP-containing component of an energy sublevel set and compares it with numerical boundaries. Its convergence guarantee applies to the reduced angle model under stated conditions; EMT simulations provide numerical support for predictions with implemented controls.

A. ROA Estimation and Interpretation

  • The energy-based ROA estimate is the connected component containing the target SEP of the energy sublevel set in (31).Numerical ROA boundaries are reconstructed using the stable-manifold method under the applicability conditions of.Source: A. ROA Estimation and Interpretation · p. 7
  • Numerical ROA boundaries provide a reference for interpreting EMT post-jump and fault-clearing states.Source: A. ROA Estimation and Interpretation · p. 7
  • Convergence to the target SEP is guaranteed for the reduced angle model only under Section II-C conditions; EMT simulations numerically support predictions for implemented controls.Source: A. ROA Estimation and Interpretation · p. 7

B. Transient Stability Under Phase Jumps and Short-Circuit Faults

The energy criterion and reconstructed ROA are tested against EMT for phase jumps and faults, while energy dissipation and transient-stability margins are also assessed.

B. Transient Stability Under Phase Jumps and Short-Circuit Faults

  • Five certified disturbance states return to the target SEP in EMT; the circular-limiter B2 state also returns despite Eδ/Ecr = 1.203, outside the energy estimate but inside the reconstructed ROA.The energy criterion is sufficient, while the reconstructed ROA includes additional convergent initial states.Source: B. Transient Stability Under Phase… · p. 7
  • At nearly identical 55°-jump states, circular limiting returns to the target SEP while impedance insertion loses synchronism; the 40-ms fault cases show the same contrast, though neither is energy-certified.For the tested characteristics, impedance insertion lowers the critical energy and corresponding energy estimate.Source: B. Transient Stability Under Phase… · pp. 7–8

C. Energy Balance and Transient Stability Margins

  • Over the 8-s records, the maximum normalized energy-balance residual is 3.50% for circular limiting and 2.04% for impedance insertion.The comparison excludes the imposed jump and tests energy decrease against the accumulated damping integral.Source: C. Energy Balance and Transient… · p. 8
  • Energy-derived jump thresholds agree with EMT transition intervals to within 0.85% for circular limiting and 0.55% for impedance insertion.The table reports energy-based assessments alongside EMT results.Source: C. Energy Balance and Transient… · pp. 8–9, Table II · p. 9

D. Effect of Unequal Current Limits

In the two-GFM case, the lower current limit governs stability margins, while the proposed energy construction supports analysis across limiter operating modes and is formulated for arbitrary-size GFM networks.

D. Effect of Unequal Current Limits

  • With limits of (1.2, 1.4) or (1.2, 1.2) p.u., the shared lower limit yields the same theoretical potential, energy estimate, and ROA, and both EMT responses lose synchronism.Raising both limits to 1.4 p.u. places the clearing state inside the energy estimate at approximately 0.927Ecr; the reduced-model criterion and EMT both predict return to the target SEP.Source: D. Effect of Unequal Current Limits · p. 9

V. Conclusion

  • The common energy function covers normal operation, limiter activation, current-limited operation, and deactivation without switching mode-dependent energy functions.For the inductive port and network model, current limiting reshapes the effective potential and synchronization damping dissipates transient energy.Source: §V · p. 9
  • The two-GFM analysis shows that current limiting shapes power transfer and transient stability, with EMT simulations supporting the predicted behavior and energy dissipation.The comparisons also show that current utilization shapes the potential barrier and stability margin.Source: §V · p. 9
  • Although illustrated with two GFMs, the construction is formulated for arbitrary numbers of GFMs under the stated modeling assumptions.The paper presents this formulation as a theoretical foundation for scaling energy-based transient-stability assessment to large GFM networks under current limiting.Source: §V · p. 9
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