PaperlayerView exact revision on arXiv

Source-linked AI summary

Local Energy Function-Based Power System Transient Stability Analysis with Grid Forming and Grid Following Inverters

Yingyi Tang, Junbo Zhao, Baosen Zhang

arXiv:2609.06714v1eess.SYcs.SY

TL;DR

Mixed GFM–GFL inverter systems have nonlinear interactions that complicate transient stability assessment after large disturbances. The paper models these interactions and derives a local energy function with a critical-energy threshold and stability index. Case studies validate the method on a modified IEEE 14-bus system, while analysis identifies a destabilizing effect of insufficient GFM damping.

  • Problem

    Mixed GFM–GFL systems have nonlinear interactions, and small-signal analysis cannot fully capture their behavior under large disturbances.

    Source: §I · p. 1

  • Method

    The paper models GFL–GFM controller coupling and derives a local energy function that jointly uses their damping effects to assess transient stability.

    Source: §I · p. 2

  • Results

    Case studies validate the method on a modified IEEE 14-bus system, and insufficient GFM damping can degrade GFL damping through coupling and harm overall stability.

    Source: §I · p. 2, §V · p. 14

  • Takeaways & Limitations

    The critical energy and stability index provide a fast quantitative stability assessment after large disturbances.

    Source: §I · p. 2

    The stability index does not establish classical asymptotic stability from energy decay alone; the framework relies on a separate state-dependent residual bound that vanishes near the equilibrium.

    Source: B. Critical Energy and Stability Index · p. 9

Abstract

from arXiv · show

The widespread deployment of inverter-based resources (IBRs) is reshaping power system dynamics by their control strategies. Grid-following (GFL) and grid-forming (GFM) inverters are expected to co-exist from now on. However, ensuring the transient stability of such systems remains a critical and underexplored challenge. In this paper, a comprehensive transient stability analysis of the mixed-GFM-GFL inverter system is conducted using a newly derived local energy function. The transient stability analysis model of the system is first derived to reveal the dynamic interactions between GFL and GFM inverters during transient responses. A local energy function is then proposed for analytical transient stability assessment. Instead of ignoring or making harsh assumptions on inverters' coupled damping effect, the proposed local energy function carefully addresses its impact on system transient stability by locally bounding the region where the system's energy dissipation property holds. Such a property fully considers the combined damping effect of GFL and GFM inverters. The derived critical energy and stability index provide a fast and quantitative transient stability assessment. Furthermore, the adverse impact of insufficient damping of GFM inverters on system transient stability is thoroughly investigated. The effectiveness of the proposed method is validated on the modified IEEE 14-bus system with mixed GFL and GFM inverters, and the numerical simulation results further demonstrate the small conservativeness of the proposed method.

I. INTRODUCTION

Mixed GFM–GFL systems pose a transient-stability challenge because their nonlinear interactions matter during large disturbances, while existing analytical energy methods face restrictive global-dissipation requirements. The paper develops a coupled system model and a local energy framework to assess stability after disturbances.

I. INTRODUCTION

  • Small-signal analysis relies on linearized operating-point models and cannot fully capture behavior under large disturbances, motivating transient-stability analysis of mixed-inverter systems.Source: §I · p. 1
  • The local framework requires energy dissipation outside a bounded region around the post-fault equilibrium, relaxing the global condition and supporting transient stability.Source: §I · p. 2
  • The system model permits GFL and GFM inverters at different connection points and derives coupling between their PLL and VSG controllers.Source: §I · p. 2
  • The proposed local energy function jointly accounts for GFL and GFM damping and yields critical energy and a quantitative stability index for fast post-disturbance assessment.Source: §I · p. 2
  • Poorly damped GFM inverters can degrade GFL damping through inter-inverter coupling, negatively affecting overall system stability.Source: §I · p. 2

A. AC Network Modeling with GFL and GFM Inverters

B. Dynamic Model of Coupled GFM and GFL Inverters for Transient Stability Analysis

The dynamic model expresses inverter synchronization in a common reference frame and each inverter’s local rotating frame. It models GFL synchronization through PLL dynamics and GFM synchronization through VSG control, with faster inner loops simplified.

B. Dynamic Model of Coupled GFM and GFL Inverters for Transient Stability Analysis

L YLg

The model’s admittance terms encode interactions among GFL inverters, GFM inverters, and the grid source. Its coupled dynamics distinguish fixed GFM damping from GFL damping that varies with inverter angles.

L YLg

  • The reduced admittance model separates GFL, GFM, and source-bus terms, with matrix notation specifying real and imaginary components of the coupling matrices.Source: L YLg · p. 4

KIm

  • GFL dynamics include self and cross-inverter restoring and damping terms, including coupling from GFM inverters into GFL dynamics.Source: KIm · p. 5
  • GFM dynamics include self-restoring and cross-inverter restoring effects; GFM damping is fixed, whereas equivalent GFL damping varies dynamically with inverter angles.Source: KIm · p. 5
  • At the post-fault stable equilibrium, the Jacobian of the modeled dynamics is negative definite.Source: KIm · p. 5

A. Local Energy Function Candidate

The proposed local energy function combines GFL and GFM kinetic and potential energy while addressing coupling terms that complicate a systemwide energy function. Its construction makes bounded total energy imply bounded inverter frequency and angle states.

A. Local Energy Function Candidate

  • The energy candidate is decreasing within a region Ω, despite the difficulty of constructing a systemwide function in the presence of GFL lossy coupling.The GFL–GFL term Aij cos δLLij complicates a systemwide energy function.Source: A. Local Energy Function Candidate · p. 5

KIm

  • GFL potential energy includes a term designed to address the non-negligible loss term introduced by GFL–GFL coupling.The construction uses a shifted angle variable for this coupling term.Source: KIm · p. 6
  • GFM potential-energy terms are formed from the first integrals of the GFM self and coupling dynamics.The construction also applies a negative factor to the first term, consistent with the associated expression.Source: KIm · p. 6
  • The candidate energy sums kinetic and potential components for both GFL and GFM inverters.The GFL and GFM kinetic and potential terms are combined into one total energy function.Source: A. Local Energy Function Candidate · p. 5, KIm · p. 6
  • Finite total energy ensures bounded frequency and angle states because kinetic energy grows with frequency deviations and linear potential terms grow with angle deviations.The trigonometric potential terms are bounded, while the linear terms dominate as angle deviations grow.Source: KIm · p. 6

B. Critical Energy and Stability Index

The analysis uses a locally valid energy-decay condition that accounts for combined inverter damping, then defines critical energy and a stability index for post-disturbance assessment. The resulting critical-energy estimate is conservative by construction, but simulations report predicted critical clearing times close to nonlinear-simulation values.

B. Critical Energy and Stability Index

  • Energy decay is guaranteed locally where the combined GFL and GFM equivalent damping remains positive, rather than throughout the state space.The decay region also requires the GFL frequency deviation to exceed a bounded residual radius.Source: B. Critical Energy and Stability Index · pp. 6, 8
  • GFL network-loss terms persist in the energy derivative, while the GFM transfer-conductance loss is treated as negligible under the stated high X-to-R network condition.The GFL loss term arises from current-source behavior interacting with network transfer impedance.Source: B. Critical Energy and Stability Index · p. 7
  • The residual radius is governed by network coupling relative to the net damping margin and can become negligibly small under practical GFM damping settings.With substantially reduced GFM virtual damping, the radius grows, the decay region shrinks, and time-domain simulation is needed for accurate assessment.Source: B. Critical Energy and Stability Index · p. 8
  • The critical energy is the lower of the energy at the closest UEP inside Ω and the minimum energy on the damping-defined boundary ∂Ω.The estimated stability region is the set of states whose energy does not exceed this critical value.Source: B. Critical Energy and Stability Index · p. 8
  • The stability index µ compares critical energy with energy at disturbance clearing, with positive values indicating entry into the assessed region and negative values indicating instability under the criterion.The paper describes µ as a quantitative measure of stability following large disturbances.Source: B. Critical Energy and Stability Index · p. 9
  • Predicted critical clearing times are reported as very close to detailed nonlinear-simulation values, despite the critical-energy boundary being conservative by design.The authors present critical energy as a computationally efficient analytical indicator, not an exact global stability boundary.Source: B. Critical Energy and Stability Index · p. 9

C. Effects of GFM Inverters’ Damping on Transient Stability

The effect of adding GFM inverters on transient stability depends on inter-inverter damping interactions. In particular, GFL equivalent damping varies dynamically with coupling, so GFM addition does not necessarily improve stability.

C. Effects of GFM Inverters’ Damping on Transient Stability

  • Adding GFM inverters does not necessarily enhance transient stability because GFL damping is dynamically affected by inter-inverter coupling.The analysis makes the effect dependent on whether the associated matrix term is positive definite.Source: C. Effects of GFM Inverters’ Damping… · p. 9
  • The stability effect depends on a positive-definiteness condition involving the inter-inverter damping interactions.The passage identifies this condition as the factor governing the nontrivial effect of GFM addition.Source: C. Effects of GFM Inverters’ Damping… · p. 9

L · YIm LM)+

The damping analysis distinguishes normal operating assumptions from weak-coupling conditions and inadequate GFM damping. The resulting effect of GFM inverters on energy dissipation can be beneficial or adverse depending on these conditions.

YRe

  • Under normal operation, angle differences within (−π/2, π/2) make each stated GFL–GFM damping term positive.The passage ties this sign to the positivity of the relevant cosine term.Source: KRe · p. 9

KRe

  • When the coupling contribution is small and GFM damping is adequate, GFM inverters can accelerate energy dissipation after large disturbances.The passage attributes the small coupling contribution to weak coupling terms and bounded coefficients.Source: KRe · p. 10
  • Poorly damped GFM inverters can reduce energy dissipation and may increase system energy, so damping settings matter to whether GFM integration supports stability.The passage describes this adverse effect as arising through inter-inverter coupling.Source: KRe · p. 10

D. Effects of PLL Parameters on Transient Stability

The PLL parameters enter the analysis through ΛL. Under a shared-ratio simplification, the paper examines how PLL tuning affects the sufficient damping condition.

D. Effects of PLL Parameters on Transient Stability

2 DLMDM −1D⊤

The passage bounds the PLL ratio using system damping and identifies gain extremes that compromise transient stability.

2 DLMDM −1D⊤

  • λmax denotes the maximum eigenvalue of the matrix.Source: §2 · p. 10
  • The ratio KPLL/I_i has an upper bound determined by system dynamic damping to ensure energy dissipation.Source: §2 · p. 10
  • An overly small KPLL^I or overly large KPLL^P would compromise transient stability of the mixed GFM-GFL inverter system.Source: §2 · p. 10

IV. NUMERICAL RESULTS

On the modified IEEE 14-bus test system, the energy threshold produced CCT estimates close to simulated CCTs. The varied-damping cases also show changes in stability indices and transient trajectories.

A. Validation of System Critical Energy Value

B. System Dynamics with Varied Di under Different Fault Types

C. Adverse Effect of Insufficient GFM Inverter Damping on System Transient Stability

The section compares transient responses under low and higher GFM damping, then examines how PLL parameter ratio relates to stability in the mixed-inverter system.

C. Adverse Effect of Insufficient GFM Inverter Damping on System Transient Stability

D. Impacts of PLL Parameters on Transient Stability

  • At tc = 0.075 s, λ = 0.01 trajectories converge while λ = 1 trajectories diverge, with system energy exceeding the critical threshold and growing unbounded.At λ = 0.01, energy peaks near the critical value at fault clearance and then decays to zero.Source: Figure 11 · p. 14, Figure IV · p. 14, Figure 12 · p. 14

V. CONCLUSIONS

  • The conclusion reports IEEE 14-bus case studies validating the assessment tool, while leaving inner-loop, reactive-power, fault-ride-through, and current-limiter dynamics for future work.It also identifies repeated current-limiter saturation and limit-cycle phenomena as topics for further study.Source: §V · p. 14
Loading 2609.06714v1…