⚡ electrical engineering

Large-Signal Stability of Power Systems with Mixtures of GFL, GFM and GSP Inverters

This paper elucidates the large-signal stability mechanisms of power systems with mixed grid-following, grid-forming, and grid-supporting inverters by deriving a generalized model, employing a manifold method with reduced-order models to determine regions of attraction, and demonstrating through simulations and experiments that grid-forming and grid-supporting inverters significantly enhance stability margins via voltage support, particularly when positioned at voltage minima.

Original authors: Yifan Zhang, Yaoxin Wang, Yunjie Gu, Yitong Li, Sijia Geng, Yue Zhu, Hsiao-Dong Chiang, Timothy C. Green

Published 2026-04-01

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CC BY 4.0

Original authors: Yifan Zhang, Yaoxin Wang, Yunjie Gu, Yitong Li, Sijia Geng, Yue Zhu, Hsiao-Dong Chiang, Timothy C. Green

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

Imagine the electrical grid as a massive, delicate dance floor. For decades, the main dancers were Synchronous Generators (huge spinning turbines in power plants). They were heavy, predictable, and kept the rhythm naturally. If someone stumbled (a fault), the whole group would sway but usually stay in sync.

Now, we are replacing these heavy dancers with Inverters (electronic devices connecting solar panels and wind turbines). These new dancers are light, fast, and follow different rules. The paper you provided is a study on how to keep this new dance floor from collapsing when things go wrong.

Here is the breakdown of the paper in simple terms:

1. The Three Types of Dancers

The paper studies three specific types of inverter "dancers":

GFL (Grid-Following): These are the "followers." They need a leader to tell them the rhythm. They use a "Phase-Locked Loop" (PLL) to listen to the grid's voltage and copy it. If the grid gets noisy or the leader stumbles, the follower gets confused and might lose the beat entirely.
GFM (Grid-Forming): These are the "leaders." They don't just listen; they create their own rhythm and voltage. They act like the old heavy turbines, providing stability to the whole group.
GSP (Grid-Supporting): These are the "helpers." They are mostly followers, but they have a special trick: they can inject extra energy (voltage support) when they see the grid getting weak, helping to stabilize the neighbors.

2. The Problem: The "Energy Map" is Broken

In the old days, engineers had a magical map called a "Global Energy Function." Think of this as a topographic map of a valley. If a ball (the system) is in the valley, you can easily predict if it will roll back to the bottom (stable) or roll over the hill and fall off (unstable).

The paper's big discovery: When you mix these new electronic dancers (especially the "followers"), that magical map disappears. The landscape becomes a jagged, chaotic maze with no clear "valley." You can't use the old rules to predict if the system will crash. It's like trying to navigate a foggy mountain with a map that only works for flat plains.

3. The Solution: The "Manifold Method" and the "Safety Bubble"

Since the old map is gone, the authors invented a new way to navigate.

The Manifold Method: Instead of looking for a smooth valley, they trace the "edges" of the safe zone. Imagine a trapeze artist. They don't need to know the whole circus; they just need to know exactly where the edge of the net is. This method calculates the precise boundary of safety.
The Stability Radius (SR): To make this easy to understand, they created a "Safety Bubble." Imagine a circle drawn around the stable dancing spot. The Stability Radius is the distance from the center of that circle to the edge of the net.
- If a disturbance (like a power line fault) pushes the dancers outside this bubble, they crash.
- If they stay inside, they are safe.
- This bubble gives a "conservative" safety estimate: if you are inside the bubble, you are definitely safe.

4. The Experiments: Hardware and Simulation

The authors didn't just do math on a computer. They built a Power Hardware-in-the-Loop (PHIL) setup.

The Setup: They built a real, small-scale power system in a lab with actual inverters and wires.
The Test: They simulated "earthquakes" (short-circuit faults) on the grid to see how long the system could survive before the dancers lost sync.
The Result: Their new "Safety Bubble" method predicted the crash point almost perfectly, matching both the computer simulations and the real-world hardware.

5. Key Findings: Who Saves the Day?

The paper tested different combinations of dancers to see who makes the system most stable:

Two Followers (GFL + GFL): This is the most dangerous. If one stumbles, the other gets confused, and they both crash quickly.
One Follower + One Leader (GFL + GFM): The "Leader" (GFM) saves the "Follower." The system becomes much more stable.
One Follower + One Helper (GFL + GSP): The "Helper" (GSP) also saves the day, almost as well as the Leader, but only if it is tuned correctly. If the helper is too weak, it doesn't help much. If it's strong, it acts just like a Leader.

The Golden Rule of Location:
Where you place the "Helper" or "Leader" matters immensely.

Bad Spot: If you put the helper right next to the infinite power source (the grid), it's useless because the grid is already strong there.
Bad Spot: If you put the helper right next to the weak inverter, the distance to the grid is too long, and the helper can't push enough power through the resistance.
Best Spot: The Middle of the Road. Placing the GFM or GSP inverter in the middle of the transmission line provides the maximum stability boost. It's like a lifeguard standing in the middle of a pool rather than at the edge; they can help swimmers on both sides.

Summary

This paper tells us that the old rules for keeping power grids stable don't work anymore with our new solar and wind technology. We need a new way to measure safety. By using a "Safety Bubble" (Stability Radius), we can predict exactly when a system will crash. Furthermore, we can make our grids much safer by adding "Grid-Supporting" inverters, especially if we place them in the middle of the transmission lines to act as stabilizers for the whole network.

1. Problem Statement

The transition to renewable energy has led to a power grid dominated by Inverter-Based Resources (IBRs), specifically Grid-Following (GFL) inverters. Unlike traditional Synchronous Generators (SGs), GFL inverters rely on Phase-Locked Loops (PLLs) for synchronization, exhibiting fundamentally different large-signal stability characteristics.

Limitations of Existing Methods: Traditional large-signal stability assessment relies on the Equal-Area Criterion and Global Energy Functions (Lyapunov functions). However, these methods fail in systems with GFL inverters because the interaction terms in the differential equations (specifically cosine terms introduced by PLL dynamics) prevent the existence of a global energy function.
The Gap: Existing studies often focus on single inverters or assume weak coupling between multiple inverters. There is a lack of rigorous analysis regarding the large-signal stability boundaries of systems containing heterogeneous mixtures of GFL, Grid-Forming (GFM), and Grid-Supporting (GSP) inverters, particularly under severe faults (large perturbations).

2. Methodology

The authors propose a novel framework combining model order reduction with the manifold method to assess stability without relying on global energy functions.

A. Model Order Reduction

To make the analysis computationally feasible for high-order systems, the authors derive reduced-order models based on multiscale analysis:

Assumptions: Inner control loops (current/voltage) are fast and treated as controlled sources. PLLs (for GFL) and droop controls (for GFM) are designed to be overdamped.
Result: The complex dynamics of GFL, GFM, and GSP inverters are reduced to first-order models where the internal angle ( $\delta$ $δ$ ) is the sole state variable.
- GFM: Modeled via $p-\omega$ droop control.
- GFL: Modeled via PLL dynamics (proportional dominant).
- GSP: Modeled as a GFM with an added voltage control loop ( $v_d-i_q$ ) to inject reactive current.

B. The Manifold Method & Stability Radius (SR)

Since a global energy function does not exist for heterogeneous systems, the authors employ the Manifold Method:

Region of Attraction (ROA): The ROA is defined as the region enclosed by the stable manifolds emanating from all Unstable Equilibrium Points (UEPs).
Stability Radius (SR): To quantify the stability margin, the authors introduce the Stability Radius, defined as the shortest distance from the post-fault Stable Equilibrium Point (SEP) to the boundary of the ROA.
- The SR serves as a conservative but generic metric for stability, independent of specific fault locations or types.
- It allows for the estimation of the Critical Clearing Time (CCT).

C. Validation

The theoretical findings are validated through two rigorous channels:

Electromagnetic Transient (EMT) Simulations: Using MATLAB/Simulink.
Power Hardware-in-the-Loop (PHIL): A physical testbed using real MOSFET-based converters and a real-time simulator (RTScale) to emulate the grid and the second inverter.

3. Key Contributions

Proof of Non-Existence of Global Energy Functions: The paper mathematically demonstrates that for systems containing GFL inverters, the cosine interaction terms in the differential equations preclude the existence of a global energy function, rendering traditional direct methods inapplicable.
Generalized Reduced-Order Model: A unified first-order model is derived for all combinations of GFL, GFM, and GSP inverters in a two-inverter system, facilitating the application of the manifold method.
Stability Radius (SR) Metric: Introduction of the SR as a practical, conservative metric to assess large-signal stability margins and estimate CCT in systems where traditional energy functions fail.
Identification of New Instability Modes: The analysis reveals a specific instability mode where trajectories converge to a Stable Periodic Orbit (SPO) rather than a stable equilibrium, leading to sustained oscillations and "pole slipping" (loss of synchronization).
Optimal Placement Strategy: The study identifies that the stability enhancement provided by GFM and GSP inverters is maximized when they are placed at the midpoint of the transmission line (where voltage is lowest), rather than near the grid or near the GFL inverter.

4. Key Results

GFL-GFL Interaction:
- Two GFL inverters are highly susceptible to large-signal instability.
- The manifold method accurately predicts the ROA and CCT.
- A new instability mode involving convergence to a Stable Periodic Orbit (SPO) was observed under high grid impedance, leading to continuous angle divergence and potential relay maloperation.
GFL-GFM/GSP Interaction:
- Voltage Support: Both GFM and GSP inverters significantly enhance the large-signal stability of a neighboring GFL inverter by providing voltage support.
- GFM vs. GSP: GFM inverters generally offer slightly better stability margins than GSP inverters. However, a GSP inverter with high voltage control gain ( $k_v$ ) and fast PLL bandwidth can achieve stability performance comparable to a GFM inverter.
- Location Sensitivity: Placing the GFM/GSP inverter at the midpoint of the transmission line yields the largest Stability Radius. Placing it too close to the infinite bus (grid) reduces its ability to regulate voltage, while placing it too close to the GFL inverter increases impedance to the grid, reducing transmission capacity.
GFM-GSP Interaction:
- Adding a GSP inverter to a GFM system improves stability, but high voltage control gains can compromise small-signal stability, creating a trade-off.

5. Significance

Theoretical Advancement: This work bridges a critical gap in power system theory by providing a rigorous method to analyze large-signal stability in inverter-dominated grids where traditional energy-based methods fail.
Practical Application: The Stability Radius provides system operators and planners with a concrete metric to assess the safety margin of mixed-inverter grids against severe faults.
Grid Planning: The findings on optimal inverter placement (midpoint of lines) offer actionable guidelines for integrating renewable resources and grid-forming technologies to maximize system resilience.
Validation: The use of PHIL experiments alongside simulations ensures that the theoretical models and reduced-order approximations are robust and accurate in real-world hardware scenarios, including the effects of inner-loop dynamics and switching.

In conclusion, the paper establishes that while GFL inverters introduce significant large-signal stability challenges, the strategic integration and optimal placement of GFM and GSP inverters can effectively mitigate these risks, provided that advanced analysis tools like the manifold method are used for assessment.