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Quantitative Stability Conditions for Grid-Forming Converters With Complex Droop Control

Xiuqiang He, Linbin Huang, Irina Subotić, Verena Häberle, Florian Dörfler

TL;DR

This paper addresses the transient stability of grid-forming converters under grid disturbances when using complex droop control (dVOC). It adopts a multi-time-scale modeling approach, deriving reduced-second-order and full-order models, and proves analytical, quantitative stability conditions via Lyapunov-based and nested singular perturbation techniques. The main contributions are: (i) existence of equilibria under complex droop control where classical droop may fail, (ii) global asymptotic stability guarantees under explicit parametric conditions, (iii) characterization of transient instability as bounded limit cycles with an upper voltage bound, and (iv) extension to full-order dynamics with practical tuning guidelines, validated by simulations and experiments. These results provide rigorous stability certificates and actionable guidance for deploying dVOC in grid-connected converters with robust transient performance.

Abstract

In this paper, we analytically study the transient stability of grid-connected converters with grid-forming complex droop control, also known as dispatchable virtual oscillator control. We prove theoretically that complex droop control, as a state-of-the-art grid-forming control, always possesses steady-state equilibria whereas classical droop control does not. We provide quantitative conditions for complex droop control maintaining transient stability (global asymptotic stability) under grid disturbances, which is beyond the well-established local (non-global) stability for classical droop control. For the transient instability of complex droop control, we reveal that the unstable trajectories are bounded, manifesting as limit cycle oscillations. Moreover, we extend our stability results from second-order grid-forming control dynamics to full-order system dynamics that additionally encompass both circuit electromagnetic transients and inner-loop dynamics. Our theoretical results contribute an insightful understanding of the transient stability and instability of complex droop control and offer practical guidelines for parameter tuning and stability guarantees.

Quantitative Stability Conditions for Grid-Forming Converters With Complex Droop Control

TL;DR

This paper addresses the transient stability of grid-forming converters under grid disturbances when using complex droop control (dVOC). It adopts a multi-time-scale modeling approach, deriving reduced-second-order and full-order models, and proves analytical, quantitative stability conditions via Lyapunov-based and nested singular perturbation techniques. The main contributions are: (i) existence of equilibria under complex droop control where classical droop may fail, (ii) global asymptotic stability guarantees under explicit parametric conditions, (iii) characterization of transient instability as bounded limit cycles with an upper voltage bound, and (iv) extension to full-order dynamics with practical tuning guidelines, validated by simulations and experiments. These results provide rigorous stability certificates and actionable guidance for deploying dVOC in grid-connected converters with robust transient performance.

Abstract

In this paper, we analytically study the transient stability of grid-connected converters with grid-forming complex droop control, also known as dispatchable virtual oscillator control. We prove theoretically that complex droop control, as a state-of-the-art grid-forming control, always possesses steady-state equilibria whereas classical droop control does not. We provide quantitative conditions for complex droop control maintaining transient stability (global asymptotic stability) under grid disturbances, which is beyond the well-established local (non-global) stability for classical droop control. For the transient instability of complex droop control, we reveal that the unstable trajectories are bounded, manifesting as limit cycle oscillations. Moreover, we extend our stability results from second-order grid-forming control dynamics to full-order system dynamics that additionally encompass both circuit electromagnetic transients and inner-loop dynamics. Our theoretical results contribute an insightful understanding of the transient stability and instability of complex droop control and offer practical guidelines for parameter tuning and stability guarantees.
Paper Structure (25 sections, 13 theorems, 76 equations, 15 figures, 4 tables)

This paper contains 25 sections, 13 theorems, 76 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions of Relevant Complex Variables
  4. From Classical Droop Control to Complex Droop Control
  5. Modeling and Stability Problem Statement
  6. Full-Order System With Multi-Time-Scale Dynamics
  7. Reduced-Order Systems
  8. Stability Problem Statement
  9. Stability of Second-Order GFM Dynamics
  10. Existence of Equilibrium Points
  11. Transient Stability of Complex Droop Control
  12. Transient Instability as Limit Cycle Oscillations
  13. Stability of Full-Order System Dynamics
  14. Simulation and Experimental Validations
  15. Case Study I: Transient Response for Different Orders
  16. ...and 10 more sections

Key Result

Proposition 1

Consider the linear system in eq:model-grid-following (i.e., $\alpha = 0$). The system is globally asymptotically stable if and only if

Figures (15)

  • Figure 1: Complex droop control in complex-angle coordinates and equivalent dVOC in complex-voltage rectangular coordinates.
  • Figure 2: The block diagram of a typical grid-connected converter system, where dVOC (i.e., complex droop control) serves as a GFM control, and the inner voltage and current controls are implemented in the $\alpha\beta$ coordinate frame (also possible in the $dq$ coordinate frame).
  • Figure 3: (a) The full-order model in \ref{['eq:full-order-system-0']} in a multi-time-scale nested form. (b) The reduced second-order model in \ref{['eq:reduced-order-system']} with GFM dynamics solely.
  • Figure 4: Transient response of systems of different orders. (a) $\eta = 0.02\omega_0$ rad/s, (b) $\eta = 0.06\omega_0$ rad/s. The decay rate of different time-scales of dynamics in the full-order system is represented by the error coordinates (normalized by their maximums): $\norm{\hat{\boldsymbol{v}} - \boldsymbol{v}_{\rm s}}/\norm{\hat{\boldsymbol{v}} - \boldsymbol{v}_{\rm s}}_{\max}$, $\norm{\boldsymbol{y}_2}/\norm{\boldsymbol{y}_2}_{\max}$, $\norm{\boldsymbol{y}_3}/\norm{\boldsymbol{y}_3}_{\max}$, and $\norm{\boldsymbol{y}_4}/\norm{\boldsymbol{y}_4}_{\max}$, see \ref{['eq:error-coodinates']} in Appendix \ref{['sec:appendix-b']} for the definition of the error coordinates $\boldsymbol{y}_i$, $i \in \{2,3,4\}$.
  • Figure 5: Comparison of the parametric stability ranges between the analytical result in \ref{['eq:condition-a']} and \ref{['eq:condition-b']} and the precise result by simulations traversing the parameter space. (a) $r_{\rm g} = 0.08$ pu and (b) $r_{\rm g} = 0.20$ pu.
  • ...and 10 more figures

Theorems & Definitions (38)

  • Remark 1: Comparison between classical and complex droop controls
  • Remark 2: Equivalence to dVOC he2022complex
  • Remark 3: Comparison of modeling coordinates and analysis methods
  • Remark 4: Current limiting issue
  • Remark 5: Interpretation of the stability conditions
  • Remark 6: Grid strength of inductive-resistive networks
  • Remark 7: Upper bound of the voltage profile
  • Remark 8: Conservatism and practicality of the stability conditions
  • Remark 9: Region of attraction
  • Example 1: Voltage-amplitude-following mode
  • ...and 28 more