⚡ electrical engineering

On the Convergence of the Current-Constrained Power Angle Curve of Virtual Admittance-Based Grid Forming Converters

This paper investigates the non-uniform convergence of the current-constrained power-angle curve in virtual admittance-based grid-forming converters by employing eigen-sweep analysis to theoretically confirm the phenomenon and establish the necessary open-loop stability and response-rate conditions for ensuring convergence, thereby addressing critical gaps in existing transient synchronization stability assessments.

Original authors: Haoxiang Zong, Chen Zhang, Yu Zhang, Xu Cai, Marta Molinas

Published 2026-07-10

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CC BY 4.0

Original authors: Haoxiang Zong, Chen Zhang, Yu Zhang, Xu Cai, Marta Molinas

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 you're trying to predict how a high-tech robot dancer (a "grid-forming converter") will move when the music (the power grid) gets intense. Engineers have a famous "dance map" called the Power-Angle Curve (PAC) that tells them exactly how much power the robot can handle at every step of its spin. For years, everyone assumed this map was always perfect and instant, like a GPS that updates the moment you turn a corner. They believed that no matter how fast the robot spun, it would instantly settle into a smooth, steady rhythm.

But this paper says: Hold on, that map might be lying to you.

The authors, a team of curious engineers, discovered a weird glitch where the robot's dance map doesn't update evenly. They call this the "non-uniform convergent issue." Here's what they found:

The "Too Fast to See" Problem

Think of the robot's internal brain (its control loops) as a camera trying to take a photo of its own movement. If the camera's shutter speed (the current loop bandwidth) is too slow, the photo comes out blurry. The paper shows that when the robot enters a "danger zone" where it has to limit its current (like a runner hitting a speed limit), this blur gets worse.

In their simulations, when they slowed down the camera's shutter speed to 110 Hz, the robot's actual movement started to wobble and oscillate wildly instead of following the smooth, steady line the old map predicted. It's like trying to follow a recipe that says "stir instantly," but your hand is moving so slowly that the batter starts splashing everywhere. The map said, "You're stable!" but the robot was actually spinning out of control.

The Two Rules for a Good Map

To fix this, the team built a super-detailed, full-order model (a massive, complex simulation of the whole robot, not just a sketch). They used a technique called "eigen-sweeping" to look at the robot's stability across the entire dance floor. They found that for the old, simple map to be trusted, two strict rules must be met:

The Stability Rule: The robot's internal brain must be stable everywhere. In their tests, when the robot was in the "unconstrained" zone (moving freely), all its stability numbers were negative (safe). But as soon as it hit the "constrained" zone (the speed limit), one of its stability numbers flipped to positive (1.044), meaning it became unstable and started to diverge.
The Speed Rule: Even if the robot is stable, it has to react fast enough. Imagine the robot is trying to keep up with a spinning turntable. If the turntable spins at a certain speed (represented by a drive frequency of 0.06 p.u.), the robot's internal reaction must be at least four times faster to keep up. If the robot's reaction is too sluggish, the map fails to capture the real behavior, even if the robot isn't technically "falling over."

What They Didn't Find (And What They Ruled Out)

The paper is very careful not to overpromise. They didn't say this happens all the time or that the old maps are useless forever. They specifically ruled out the idea that the "quasi-steady-state assumption" (the idea that the robot instantly settles) is always true. They showed that this assumption breaks down specifically when the robot is in the current-limited area and the control loops aren't fast enough.

They didn't test every single type of robot control in the world; they focused on a specific setup using "virtual admittance" and "circular-based current limitation." They noted that other controls might act similarly, but they didn't prove it for those yet.

The Bottom Line

This isn't a "we fixed everything" story; it's a "we found a hidden trap" story. The authors suggest that many existing safety checks for these power robots might be based on a map that doesn't work in the danger zones. They proved through simulations and mathematical analysis that the map can oscillate and diverge if the robot's internal speed isn't tuned correctly.

So, the next time an engineer looks at a power-angle curve, they should ask: "Is this map updating fast enough, or is it just a blurry photo of a spinning robot?" The paper doesn't give the final answer for every robot, but it definitely opens the door to re-checking the rules of the dance.

Technical Summary: On the Convergence of the Current-Constrained Power Angle Curve of Virtual Admittance-Based Grid Forming Converters

Problem Statement
Transient synchronization stability (TSS) analysis for virtual admittance-based (VA) grid-forming (GFM) converters relies heavily on the current-constrained power-angle curve (PAC). Current formulations of this PAC depend critically on a quasi-steady-state assumption: that active power can converge to its steady-state value across the entire angle space in a stable and fast manner. While this assumption is intuitively perceived, the paper notes a lack of formal clarification regarding the specific conditions under which the actual PAC converges to its steady-state. Consequently, the applicability and limitations of existing TSS analyses, which utilize these PACs, remain unclear. Specifically, there is insufficient understanding of the underlying behaviors when this assumption is violated, particularly within the current-constrained operating region.

Methodology
To address this gap, the authors develop a full-order VA-PAC model that incorporates detailed inner-loop current control, outer-loop voltage control, and circuit dynamics, moving beyond the idealized voltage control assumptions of previous models.

Modeling: A full-order model, denoted as $H_{sync}(s)$ , is established by linearizing the system dynamics. This model accounts for the piecewise nature of the system, specifically the switching behavior of the circular-based current limiter. The model structure integrates the virtual admittance loop, current loop, voltage feedforward, and AC grid/filter inductances.
Analysis: The authors employ an eigen-sweep-based analysis to track the dominant eigenvalues of $H_{sync}(s)$ across the entire angle space. This approach allows for the theoretical confirmation of convergence behaviors by examining pole distributions (stability) and real parts of eigenvalues (response rates).
Validation: The theoretical findings are validated against time-domain simulations, comparing the measured PAC from the full-order model against the ideal analytical PAC derived under quasi-steady-state assumptions.

Key Findings and Results
The analysis uncovers a phenomenon termed "non-uniform convergence" of the VA-PAC. The results indicate that the quasi-steady-state assumption does not hold uniformly across all operating conditions:

Stability Violation: In the unconstrained region, the system poles remain in the left-hand plane, ensuring convergence. However, as the power angle enters the current-constrained region, the analysis reveals the emergence of right-hand plane (RHP) poles. This indicates a loss of stability in the power-angle dynamics, causing the PAC to oscillate and diverge from its steady-state value rather than converging.
Response Rate Limitation: Even when stability conditions are met (all poles in the left-hand plane), the paper identifies that the response rate of the full-order model is critical. According to singular perturbation theory, the PAC only approximates a steady-state if the dynamics of $H_{sync}(s)$ are sufficiently fast compared to the timescale of the angle movement (synchronization control). If the response rate is too slow relative to the angle dynamics, non-convergent or poorly convergent behaviors occur despite the system being technically stable.

Significance and Claims
The paper claims to provide the first formal theoretical justification and demonstration of the non-uniform convergence issue in VA-based GFM converters. By clarifying the open-loop stability and response-rate conditions required for the PAC to converge, this work offers deeper insights into the validity boundaries of existing TSS analyses. The authors posit that these findings necessitate a re-examination of the applicability of current PAC-based TSS methods for GFM converters. Rather than proposing immediate new control strategies, the paper aims to provoke new analyses regarding the limitations of current stability assessment tools and to establish a more rigorous theoretical foundation for future TSS studies in renewable-dominated power systems.

The "Too Fast to See" Problem

The Two Rules for a Good Map

What They Didn't Find (And What They Ruled Out)

The Bottom Line

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