⚡ electrical engineering

Impacts of Heterogeneous Grid-Forming Devices on Power System Dynamics Quantified by DW Shells

This paper proposes a decentralized and scalable stability analysis method for power systems with heterogeneous grid-forming and grid-following converters by utilizing Davis-Wielandt shells and two easily obtainable indices (local passivity and imaginary-axis) to quantify system dynamics without requiring detailed converter parameters.

Original authors: Liangxiao Luo, Linbin Huang, Hangyu Chen, Ruohan Leng, Zhixian Hou, Kehao Zhuang, Huanhai Xin

Published 2026-08-25

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CC BY 4.0

Original authors: Liangxiao Luo, Linbin Huang, Hangyu Chen, Ruohan Leng, Zhixian Hou, Kehao Zhuang, Huanhai Xin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Modern electricity grids are undergoing a quiet revolution. For decades, the system relied on massive, spinning turbines that naturally kept the lights on and the frequency steady. Today, those heavy machines are being replaced by electronic converters that connect solar panels, wind farms, and battery storage to the grid. These new devices are incredibly efficient, but they behave differently. Some simply follow the rhythm set by the grid, like a dancer matching a partner's steps. Others are designed to create their own rhythm, acting as the leader that sets the tempo for the whole group. As the grid fills up with a mix of these different devices, engineers face a difficult puzzle: how to predict if this complex mix will remain stable or if it will start to wobble and fail. The challenge is that every type of device reacts to the grid in its own unique way, and traditional methods of checking stability often require knowing the exact internal secrets of every single machine, which is often impossible when dealing with proprietary technology from different manufacturers.

A team of researchers at Zhejiang University has developed a new way to solve this problem without needing those internal secrets. Instead of trying to model every intricate detail of how a battery or a wind turbine is controlled, they created a geometric method to measure how these devices interact with the grid as a whole. They focused on two specific, measurable traits of the devices: how well they resist changing direction (a concept they call a passivity index) and how they behave on a specific imaginary axis of frequency (an imaginary-axis index). By testing these two traits, the researchers found they could compress the complex behavior of a device into a simple, compact description. This allowed them to treat the grid not as a chaotic collection of individual parts, but as a unified system where the "leaders" (the grid-forming devices) effectively strengthen the foundation for the "followers" (the grid-following devices).

The researchers tested their approach on two different scenarios. First, they looked at a small system with just three converters to prove the concept. They showed that when they ignored the influence of the grid-forming device, the system appeared unstable and risky. However, once they used their new method to fuse the behavior of that device into the grid itself, the picture changed completely. The analysis revealed that the grid-forming device was actually acting as a stabilizer, increasing the strength of the grid and allowing the other devices to operate safely. This was confirmed by computer simulations that showed the system remained steady even when a small disturbance occurred.

To prove the method works on a larger scale, the team applied it to a modified version of a standard 68-bus power system, a model that represents a complex regional grid. In this scenario, they placed four different types of grid-forming devices alongside eleven grid-following devices. They compared two situations: one where the grid-forming devices used specific electronic control strategies, and another where one of those devices was replaced by a traditional synchronous generator, a heavy machine with a spinning rotor. The results were precise and revealing. In the first case, the analysis predicted that the system would become unstable at certain frequencies, a prediction that was confirmed when the computer simulation showed the system actually oscillating and failing. In the second case, where the traditional generator was used, the new method showed that the system was stable. The difference came down to the specific "passivity" of the devices; the traditional generator offered a stronger stabilizing effect in the critical frequency range than the electronic controls in the first scenario.

The significance of this work lies in its ability to handle diversity without getting bogged down in complexity. The researchers demonstrated that you do not need to know the detailed control schemes or internal parameters of every device to ensure the grid is safe. Instead, by measuring just two local indices, engineers can determine whether a specific mix of heterogeneous devices will hold together. This approach provides a scalable way to analyze massive power systems, offering a clear path forward as the world integrates more renewable energy and storage. The study confirms that while the grid is becoming more complex, it is possible to understand and secure it by looking at the right geometric properties of the devices, ensuring that the lights stay on even as the technology behind them changes.

Technical Summary: Impacts of Heterogeneous Grid-Forming Devices on Power System Dynamics Quantified by DW Shells

Problem Statement
Modern power systems are increasingly characterized by the large-scale integration of heterogeneous converter-based resources, including renewable generation, energy storage, and HVDC links. While Grid-Forming (GFM) converters are critical for supporting system stability in converter-dominated grids, quantifying their impact on system dynamics remains challenging. Existing methods face significant limitations: detailed state-space or impedance models require proprietary control parameters and become computationally intractable for large-scale systems; static or quasi-static grid strength metrics (e.g., generalized short-circuit ratio) fail to distinguish devices with different frequency-domain dynamics; and existing decentralized stability certificates often treat GFM devices merely as part of the "device" side, failing to explicitly capture their contribution to grid strength when multiple heterogeneous units interact with Grid-Following (GFL) converters.

Methodology
The paper proposes a geometric, decentralized stability analysis framework that avoids the need for detailed internal models of GFM converters. The methodology proceeds through three main stages:

Equivalent Network Reformulation: The system is partitioned into GFL and GFM nodes. Using the Schur complement, the dynamics of the GFM devices are mathematically "fused" into the power network model. This results in an equivalent network admittance matrix ( $\tilde{Y}_{GridC}$ ) seen by the GFL converters, which inherently includes the stabilizing effects of the GFM devices.
Local Index Definition: To characterize the heterogeneous GFM devices without detailed models, the authors define two local indices derived from the black-box admittance models:
- Passivity Index ( $\nu_i$ ): The minimum eigenvalue of the Hermitian part of the rescaled GFM admittance.
- Imaginary-Axis Index ( $\mu_i$ ): The maximum singular value of the imaginary Hermitian part of the rescaled GFM admittance.
  These indices can be obtained via frequency scanning or black-box testing.
Geometric Stability Analysis via DW Shells: The paper utilizes Davis–Wielandt (DW) shells to characterize the dynamics of the converters and the equivalent network. The authors derive an outer envelope for the DW shell of the equivalent network ( $\tilde{Y}_{GridC}$ ) based on the GFM indices. This envelope is defined by a lower bound on the real-axis projection (related to grid strength) and an upper bound on the imaginary-axis projection (related to phase sector).

Key Contributions

Decentralized Stability Certificates: The paper derives stability conditions where the stability of the closed-loop system (GFL converters interacting with the GFM-enhanced network) is guaranteed if the DW shell of the GFL converters is separated from the DW shell envelope of the equivalent network.
Explicit Impact Quantification: Theoretical bounds (Theorem IV.2) are derived showing that the GFM passivity index directly increases the lower bound of the real-axis projection (effectively increasing the equivalent grid strength), while the imaginary-axis index influences the phase sector bounds.
Scalability: The approach allows for the analysis of large-scale systems with heterogeneous GFM devices without constructing high-order system impedance matrices or requiring detailed control parameters. The analysis relies only on the two local indices and the network topology.

Results
The proposed method was validated through two case studies:

Three-Converter System: A system with two GFL and one GFM converter demonstrated that fusing GFM dynamics into the network model significantly reduced the conservativeness of stability analysis compared to treating GFM devices as separate entities. Time-domain simulations confirmed that the system was stable, aligning with the geometric analysis which showed separation between the GFL and equivalent network DW shells.
Modified IEEE 68-Bus System: This large-scale system included 11 GFL converters and 4 heterogeneous GFM devices (using VSG, droop, and matching controls). The analysis identified specific frequency ranges where stability risks existed (zero stability distance) for certain GFL converters. By replacing one GFM converter with a Synchronous Generator (SG) having a higher passivity index in a critical frequency range, the stability distance became positive, and time-domain simulations confirmed the system became stable. The method successfully identified the specific GFL converters (GFL 5 and 6) responsible for the instability.

Significance and Claims
The paper claims to provide a scalable, low-dimensional framework for analyzing the stability of power systems with heterogeneous GFM resources. Its primary significance lies in:

Bridging the Gap: It offers a way to quantify the stabilizing support of GFM devices without needing proprietary control details, addressing a major barrier in analyzing modern, diverse power grids.
Physical Insight: It explicitly links local GFM properties (passivity and imaginary-axis indices) to global system stability metrics (equivalent grid strength and phase sectors) through geometric interpretations.
Practical Application: The method enables the identification of critical instability risks and specific devices contributing to them, facilitating better placement and capacity allocation of GFM resources.

The authors note that the passivity index defined is a "rotated" index due to the rescaling of the admittance matrix, and they position this work as a foundation for future research on GFM controller synthesis based on DW shells.

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