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Quantitative Parameter Conditions for Stability and Coupling in GFM-GFL Converter Hybrid Systems from a Small-Signal Synchronous Perspective

Kehao Zhuang, Huanhai Xin, Hangyu Chen, Linbin Huang

TL;DR

This work addresses the stability of power systems containing both grid-forming (GFM) and grid-following (GFL) converters by developing a small-signal synchronous model that includes network line dynamics and yields a generalized second-order form with inertia, damping, and a synchronizing matrix. Using Davis–Kahan subspace perturbation theory, it derives quantitative decoupling conditions that provide analytical lower bounds on GFM inertia/damping and GFL damping to suppress coupling, clarifying when the GFM and GFL subsystems can be analyzed separately. When decoupling is not achievable, the paper applies the Small Phase Theorem to obtain decentralized stability criteria that incorporate network dynamics while keeping converter parameters separable, supplemented by a simpler static condition for practical design. The theoretical results are validated through MATLAB/Simulink simulations on the IEEE 39-bus system, showing accurate prediction of stability boundaries and highlighting the method’s usefulness for guiding parameter selection and controller tuning in converter-dominated grids.

Abstract

With the development of renewable energy sources, power systems are gradually evolving into a system comprising both grid-forming (GFM) and grid-following (GFL) converters. However, the dynamic interaction between the two types of converters, especially low-inertia GFM converters and GFL converters, remains unclear due to the substantial differences in their synchronization mechanisms. To address this gap, this paper develops a small-signal synchronous stability model for power systems containing GFM and GFL converters, which considers network line dynamics. Based on subspace perturbation theory, we reveal that GFM and GFL subsystems can be effectively decoupled when GFL converters operate near unity power factor or when GFM converters possess sufficiently large inertia or damping, and provide lower bound of control parameters ensuring decoupling. Under the decoupling condition, we propose decentralized and analytical parameter-based stability criteria which have clear physical interpretations: the positive damping of converters compensates for the negative damping of the network. In the case of coupling, we also propose decentralized stability criteria based on the small phase theorem. The effectiveness of the theoretical analysis is validated through simulations in MATLAB/Simulink.

Quantitative Parameter Conditions for Stability and Coupling in GFM-GFL Converter Hybrid Systems from a Small-Signal Synchronous Perspective

TL;DR

This work addresses the stability of power systems containing both grid-forming (GFM) and grid-following (GFL) converters by developing a small-signal synchronous model that includes network line dynamics and yields a generalized second-order form with inertia, damping, and a synchronizing matrix. Using Davis–Kahan subspace perturbation theory, it derives quantitative decoupling conditions that provide analytical lower bounds on GFM inertia/damping and GFL damping to suppress coupling, clarifying when the GFM and GFL subsystems can be analyzed separately. When decoupling is not achievable, the paper applies the Small Phase Theorem to obtain decentralized stability criteria that incorporate network dynamics while keeping converter parameters separable, supplemented by a simpler static condition for practical design. The theoretical results are validated through MATLAB/Simulink simulations on the IEEE 39-bus system, showing accurate prediction of stability boundaries and highlighting the method’s usefulness for guiding parameter selection and controller tuning in converter-dominated grids.

Abstract

With the development of renewable energy sources, power systems are gradually evolving into a system comprising both grid-forming (GFM) and grid-following (GFL) converters. However, the dynamic interaction between the two types of converters, especially low-inertia GFM converters and GFL converters, remains unclear due to the substantial differences in their synchronization mechanisms. To address this gap, this paper develops a small-signal synchronous stability model for power systems containing GFM and GFL converters, which considers network line dynamics. Based on subspace perturbation theory, we reveal that GFM and GFL subsystems can be effectively decoupled when GFL converters operate near unity power factor or when GFM converters possess sufficiently large inertia or damping, and provide lower bound of control parameters ensuring decoupling. Under the decoupling condition, we propose decentralized and analytical parameter-based stability criteria which have clear physical interpretations: the positive damping of converters compensates for the negative damping of the network. In the case of coupling, we also propose decentralized stability criteria based on the small phase theorem. The effectiveness of the theoretical analysis is validated through simulations in MATLAB/Simulink.

Paper Structure

This paper contains 19 sections, 4 theorems, 38 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. Small-Signal Modeling of Multi-Converter Power Systems
  3. Converters model
  4. Network model
  5. Closed loop system model
  6. Decoupling Analysis of GFM and GFL Converters
  7. The decoupling conditions of GFM and GFL converters
  8. The small-signal synchronous stability of decoupled systems
  9. The Decentralized Stability Conditions of the Coupled System Based on Small Phase Theorem
  10. The definition of complex matrix phase
  11. The small phase theorem
  12. The decentralized stability condition of coupled system
  13. Time-Domain Simulation
  14. Conclusions
  15. The derivation of network model
  16. ...and 4 more sections

Key Result

Proposition 3.1

The eq:2citezheng can be decoupled as two subsystem if eq:decoupled condition holds, where, $\epsilon$ is a small constant, $\left[\omega_{\min},\omega_{\max} \right]$ represents the PLL bandwidth range of the GFL converters. For ideal decoupling, we should set $\epsilon=0$. However, in practice, setting $\epsilon=0.05$ is sufficient to ensure a decoupling accuracy. If a simpler pa where $\omeg

Figures (20)

  • Figure 1: GFM and GFL control schemes of a converter connected to the grid
  • Figure 2: Vector diagram of a converter connected to the grid
  • Figure 3: GFM and GFL hybrid power system
  • Figure 4: The equivalent control block diagram of the proposed model
  • Figure 5: IEEE 39 buses system
  • ...and 15 more figures

Theorems & Definitions (13)

  • Example 1
  • Proposition 3.1: The decoupling conditions
  • proof
  • Example 2
  • Proposition 3.2: The small-signal stability conditions of decoupled systems
  • proof
  • remark 1
  • Example 3
  • Definition 4.1: Phase of a matrix
  • Lemma 4.2
  • ...and 3 more