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arxiv: 2603.26335 · v2 · submitted 2026-03-27 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Transient Stability of GFL Converters Subjected to Switching of Droop-Controlled GFM Converters

Bingfang Li , Songhao Yang , Pu Cheng , Zhiguo Hao
Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:47 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords transient stabilitygrid-forming convertergrid-following converterswitched systemdroop controlcurrent limit controlvirtual fixed d-axis control
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The pith

The stability boundary of the switched GFL-GFM system coincides with the current-limit control subsystem, and a local virtual fixed d-axis control strategy restores convergence to constant-voltage mode.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper models the switched dynamics that arise when droop-controlled grid-forming converters toggle between constant-voltage and current-limit modes inside a grid-following-converter-dominated network. It derives angle-dependent switching conditions and shows that the overall stability limit of the grid-following converter is identical to the limit of the current-limit subsystem alone. The authors then introduce a virtual fixed d-axis control method that decouples the two converter types and drives the grid-forming unit back to voltage control using only local measurements.

Core claim

The switched system's stability boundary coincides with that of the CLC subsystem. To enhance GFLC's transient stability and ensure GFMC converges to the CVC mode, this paper introduces a virtual fixed d-axis control (VFDC) strategy that achieves decoupling and self-stabilization using only local state variables from individual converters.

What carries the argument

Virtual fixed d-axis control (VFDC) strategy that fixes the d-axis reference from local converter states to decouple GFL and GFM dynamics during mode switching.

If this is right

  • Stability margins for GFL converters can be designed from the CLC boundary alone even when GFMs switch modes.
  • The VFDC method removes the need for communication links between converters to achieve stabilization.
  • GFM units are driven out of current-limit mode during transients, reducing sustained stress on the network.
  • Subsystem analysis of the CLC mode supplies a conservative bound for the entire switched system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If switching is strictly angle-dependent, then GFL angle oscillations directly tighten the effective stability margin by triggering GFM mode changes.
  • The local-only VFDC approach may scale to large inverter fleets without centralized coordination.
  • Similar local decoupling could be tested on other switching events such as fault ride-through or protection actions.

Load-bearing premise

The switched system can be modeled with angle-dependent triggers so that stability of the full system reduces to stability of the current-limit subsystem without major unmodeled interactions.

What would settle it

A test case in which the full switched system becomes unstable at an operating point inside the CLC stability boundary, or in which VFDC fails to drive the GFM converter back to CVC using only local variables.

Figures

Figures reproduced from arXiv: 2603.26335 by Bingfang Li, Pu Cheng, Songhao Yang, Zhiguo Hao.

Figure 2
Figure 2. Figure 2: Relation of reference axes and system vectors B. Dynamic Model of the Switched System To facilitate analysis and focus on the primary issue of stability determination, the following reasonable assumptions are made: 1) In co-located systems, the GFMC is expected to provide a stable voltage and angle reference to facilitate robust phase locking by the GFLC [6]. Therefore, with proper parameter design, the ba… view at source ↗
Figure 1
Figure 1. Figure 1: Topology and control structure of the co-located system A. Co-located System Overview [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: GFMC angle-dependent Switching conditions on the Pc1-δ plane Eq.(8) shows that the PLL angle  influences the GFMC’s switching conditions through c2. If the PLL is converged,  equals 0. Here, 0 is not an independent variable but rather a dependent variable that changes with δ. Thus, δ L k and δ R k remain constant. However, if the PLL has not converged, δ L k and δ R k will fluctuate with . 2) GFLC an… view at source ↗
Figure 4
Figure 4. Figure 4: Phasor diagrams in CVC and CLC subsystems. Ug 0° β1Ic1 max ( c1+90°) δ0=δp O x CL2 2+ U γIc2 (+) c1 δp ΓL1 ΓL2 CV ΓV δp (a) CVC to CLC transition point δ0=δp O x CL1 + ΓL1 ΓL2 CV ΓV δp (b) CLC to CVC transition point [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Phasor diagram of GFLC angle-dependent Switching conditions. Given the dynamic decoupling of slow variable δ and fast variable , we fix δ=δp to identify the critical  values at the switching instant. After fault clearance, all the phasors change continuously until the current saturation angle undergoes an abrupt change (typically following current magnitude saturation). Thus, the phasor diagrams are iden… view at source ↗
Figure 6
Figure 6. Figure 6: Critical energy comparison between CVC and CLC subsystem. The PLL’s Lyapunov function can be chosen as[25]: ( )       {V,L} ( ) s V,L V,L V,L Ec2 Mc2 2 c2 1 ,d 2 V PP T       = + −  , (13) where the superscript {V, L} denotes variables either in the CVC or CLC subsystems. Ignoring the second-order damping in (6), the time derivation of (13) is dV/dt=-D {V,L} C2 ϖ2 . The equilibrium point is st… view at source ↗
Figure 7
Figure 7. Figure 7: Switching dynamics on the P {V,L} Ec2 - plane Let ∆Vo denote the overall periodic energy increment for the full switching sequence from CLC to CVC and then back to CLC. The potential energy discontinuities at points E and F are represented by ∆V E p and ∆V F p , respectively. The periodic energy change attributable to damping is denoted by ∆VD. Thus, ( ) ( ) ( ) ( ) 11 VL ss p 22 LV ss p V V L L VL Ec2 Mc… view at source ↗
Figure 8
Figure 8. Figure 8: Control block diagram of VFDC. A. Virtual d-Axis Fixed Control Strategy Under traditional grid-following control of GFLC or the CLC control of GFMC, the converter’s d-axis is typically aligned with θ or δ. This alignment keeps the d-axis component of the GFLC or GFMC current constant despite variations in θ or δ. Consequently, as θ increases, this constant current component can provide sustained “accelerat… view at source ↗
Figure 9
Figure 9. Figure 9: System vectors after applying the proposed VFDC B. Demonstration of Stability Enhancement Subsequent analysis will show that for co-located GFLCs and droop-controlled GFMCs systems, VFDC achieves superior transient stability over conventional current control of GFLCs and GFMCs. 1) Demonstration of enhanced PLL stability In engineering practice, renewable energy stations typically incorporate multiple GFLCs… view at source ↗
Figure 11
Figure 11. Figure 11: Simulation Results of Cases 1, 2, and 3. (a) PLL angle versus time curves; (b) CVC and CLC state indicator. time (s) δ,  −/ 2(rad) S δ  1a 1b 2a 2b 3a 4a 5a 6a 3b 4b 5b 6b s V -/ 2 (a) (b) Indicator (S=1, CVC; S=0, CLC) [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Simulation Results of Case 2. (a) PSL and PLL angle versus time curves; (b) CVC and CLC state indicator. The corresponding results are presented in [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: GFMC active power versus angle curves with and without VFDC in Cases 4-7. δ (rad) time (s) Case 7 Case 7 (with VFDC) SEP in CVC CVC CLC CVC Pc1 (MW)  (rad) E E (a) (b) ref Pc1 [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of results with and without VFDC in Case 1. (a) PLL angle versus time curves; (b) GFMC active power versus angle curves. Regarding the recovery of PSL-based GFMC from CLC to CVC, [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparative simulation results of the proposed VFDC strategy and existing methods. Host Computer Dspace RTDS Circuit Model Real-time Data Real-time Data Voltage and Current Sampling PWM signal [PITH_FULL_IMAGE:figures/full_fig_p011_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Configuration of the CHIL platform. B. Test System 2: Controller Hardware in the Loop Test To further validate the proposed VFDC strategy, a controller hardware-in-the-loop (CHIL) platform based on Real-Time Digital Simulator (RTDS) and dSPACE rapid control prototyping system is constructed ( [PITH_FULL_IMAGE:figures/full_fig_p011_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Experiment results of Case 10. VII. CONCLUSION Upon integration of GFMCs, their current saturation characteristics transform the GFLC-dominated power system into a state-dependent switching system. There are two distinct switching modes exist: “GFMC angle-dependent switching” at the PSL dynamic timescale and “GFLC angle-dependent switching” at the PLL dynamic timescale. Under these dynamics, the system fa… view at source ↗
Figure 19
Figure 19. Figure 19: Vector relationships in the CVC subsystem at switching instant. For a conservative analysis, i d2 c1 can be approximated by its maximum value, I max c1 , which represents the most critical operating condition for PLL stability. In (11) and (12), given that Lc1 is negligible compared to Lg, θ V s -θ L s is expressed as: 2 max d VL 1 c1 c2 s s p p g arcsin Ii U     s  + −  − = − . (34) From [PITH_FU… view at source ↗
read the original abstract

Integrating grid-forming converters (GFMCs) into grid-following converter (GFLC)-dominated power systems enhances the grid strength, but GFMCs' current-limiting characteristic triggers dynamic switching between constant voltage control (CVC) and current limit control (CLC). This switching feature poses critical transient stability risks to GFLCs, requiring urgent investigation. This paper first develops a mathematical model for this switched system. Then, it derives switching conditions for droop-controlled GFMCs, which are separately GFMC angle-dependent and GFLC angle-dependent. On this basis, the stability boundaries of GFLC within each subsystem are analyzed, and the impact of GFMC switching arising from GFLC angle oscillation is investigated. The findings reveal that the switched system's stability boundary coincides with that of the CLC subsystem. To enhance GFLC's transient stability and ensure GFMC converges to the CVC mode, this paper introduces a virtual fixed d-axis control (VFDC) strategy. Compared with existing methods, this method achieves decoupling and self-stabilization using only local state variables from individual converters. The conclusions are validated through simulations and Controller Hardware-in-the-Loop tests.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper develops a mathematical model for the switched system of grid-following converters (GFLCs) interacting with droop-controlled grid-forming converters (GFMCs) that dynamically switch between constant voltage control (CVC) and current limit control (CLC) modes. It derives GFMC-angle-dependent and GFLC-angle-dependent switching conditions, analyzes GFLC stability boundaries in each subsystem, finds that the overall switched-system stability boundary coincides with the CLC subsystem, and proposes a virtual fixed d-axis control (VFDC) strategy that achieves decoupling and self-stabilization using only local converter states. Results are validated by simulations and Controller Hardware-in-the-Loop tests.

Significance. If the stability-boundary coincidence holds and the VFDC provides the claimed decoupling without new instability modes, the work would be significant for transient-stability analysis of mixed GFL-GFM systems and for supplying a practical, local-only control fix that avoids communication or global measurements.

major comments (1)
  1. [stability analysis section] The central claim that the switched system's stability boundary coincides with that of the CLC subsystem (abstract and stability-analysis section) rests on the unverified assumption that angle-triggered hybrid transitions introduce no new instability modes via dwell-time effects, chattering, or grid-mediated interactions. Separate subsystem analysis is insufficient; the manuscript should supply either explicit dwell-time bounds or a hybrid Lyapunov function to confirm that CVC intervals cannot destabilize trajectories already at the CLC limit.
minor comments (1)
  1. [Abstract] The abstract states that conclusions are validated through simulations and CHIL tests but provides no quantitative metrics (e.g., critical clearing times, oscillation damping ratios, or parameter ranges), making it difficult to judge the practical margin of the reported improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We agree that a rigorous treatment of the hybrid switched-system dynamics is essential to fully support the central stability-boundary claim. In the revised manuscript we will augment the stability-analysis section with an explicit hybrid Lyapunov function and dwell-time bounds derived from the angle dynamics, confirming that CVC intervals cannot destabilize trajectories already at the CLC limit.

read point-by-point responses

  1. Referee: [stability analysis section] The central claim that the switched system's stability boundary coincides with that of the CLC subsystem (abstract and stability-analysis section) rests on the unverified assumption that angle-triggered hybrid transitions introduce no new instability modes via dwell-time effects, chattering, or grid-mediated interactions. Separate subsystem analysis is insufficient; the manuscript should supply either explicit dwell-time bounds or a hybrid Lyapunov function to confirm that CVC intervals cannot destabilize trajectories already at the CLC limit.

    Authors: We appreciate this observation. The manuscript derives GFMC- and GFLC-angle-dependent switching conditions and shows, via subsystem Lyapunov functions and extensive simulation/CHIL validation, that trajectories reaching the CLC boundary remain inside the region of attraction of the CVC subsystem upon switching back; no chattering or grid-mediated instabilities appear in the tested cases. Nevertheless, we acknowledge that a complete hybrid stability argument is required. In revision we will construct a hybrid Lyapunov function by combining the individual subsystem functions with a dwell-time condition obtained from the bounded rate of angle change under the proposed VFDC. This will rigorously demonstrate that CVC intervals cannot push the state outside the CLC stability region. revision: yes

Circularity Check

0 steps flagged

Derivation chain self-contained; stability boundary result obtained from switched-system model without reduction to inputs

full rationale

The paper first constructs an explicit switched-system model incorporating angle-dependent switching conditions for both GFMC and GFLC. It then separately computes the stability boundaries of the GFLC within the CVC and CLC subsystems and compares them to the full switched trajectory. The reported coincidence is presented as an outcome of this comparison rather than a definitional identity or a fitted parameter renamed as prediction. The VFDC strategy is introduced as an additional local control law whose decoupling property is verified by simulation, not presupposed in the model. No load-bearing self-citation, uniqueness theorem imported from prior work, or ansatz smuggled via citation appears in the derivation steps. The analysis therefore remains independent of its own conclusions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The paper relies on standard domain assumptions from power converter control theory for modeling droop behavior and current limiting; the VFDC is introduced as a new control entity without independent evidence beyond the proposed simulations.

free parameters (1)
  • Switching thresholds
    Parameters determining when GFMC switches between CVC and CLC modes are required for the model but not quantified in the abstract.
axioms (2)
  • domain assumption Droop control law governs GFMC voltage and frequency response
    Invoked to derive the angle-dependent switching conditions for droop-controlled GFMCs.
  • domain assumption Current limit control activates precisely under overcurrent conditions
    Fundamental to defining the switched system and its subsystems.
invented entities (1)
  • Virtual fixed d-axis control (VFDC) no independent evidence
    purpose: To achieve decoupling and self-stabilization of the switched system using only local state variables
    New control strategy proposed to ensure convergence to CVC mode and improve GFLC stability.

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Reference graph

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