⚡ electrical engineering

A Dynamic Phasor Framework for Analysis of IBR-Induced SSOs in Multi-Machine Systems

This paper proposes a generalized dynamic phasor framework that linearizes inverter-based resources and synchronous generators in multi-machine systems to enable eigenvalue-based root-cause analysis and robust damping controller design for subsynchronous oscillations under both balanced and unbalanced conditions, including scenarios involving data center loads.

Original authors: Fiaz Hossain, Nilanjan Ray Chaudhuri, Constantino M. Lagoa

Published 2026-04-24

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CC BY 4.0

Original authors: Fiaz Hossain, Nilanjan Ray Chaudhuri, Constantino M. Lagoa

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, high-speed orchestra. For decades, the musicians (traditional power plants like coal or gas turbines) have played in perfect sync, creating a steady, reliable rhythm.

But recently, new musicians have joined the band: Inverter-Based Resources (IBRs). Think of these as solar panels and wind turbines. They are great, but they play differently. They use digital "brains" (inverters) to mimic the rhythm of the old-school musicians. Sometimes, these new musicians get a little out of sync, causing a weird, low-frequency wobble in the music. In engineering terms, this is called a Subsynchronous Oscillation (SSO). It's like the orchestra starting to sway back and forth uncontrollably, which can be dangerous if it gets too strong.

To make things even more complicated, we have Data Centers (like massive server farms for AI and cloud storage). These are like giant, hungry fans in the audience that suddenly start breathing in rhythm with the music. If they breathe at the wrong time, they can accidentally make the orchestra's wobble worse.

The Problem: The "Slow-Motion" Camera

Engineers have always used a tool called EMT (Electromagnetic Transient) simulation to study these wobbles. Think of EMT as a high-speed camera that takes a photo of the electricity every single microsecond.

The Good News: It's incredibly accurate.
The Bad News: It's painfully slow. Simulating just a few seconds of a real-world grid event can take hours on a supercomputer. It's like trying to study a car crash by watching a slow-motion video frame-by-frame for days. You can't easily use this to design a fix because the math is too messy and complex to analyze quickly.

The Solution: The "Dynamic Phasor" Framework

The authors of this paper propose a new tool called the Dynamic Phasor (DP) framework.

The Analogy:
Imagine you are trying to understand the movement of a spinning fan.

The Old Way (EMT): You take a photo of the fan blades every millisecond. You get a perfect picture, but you have millions of photos to sort through.
The New Way (Dynamic Phasor): Instead of tracking every single blade, you track the average position and speed of the fan, but you allow that average to wiggle slightly to capture the "wobble." You are essentially creating a smart, simplified sketch of the fan that still captures the dangerous shaking but ignores the tiny, irrelevant details.

Why is this cool?

It's Fast: Because it's a simplified sketch, it runs 2 times faster than the old method (and could be even faster for bigger systems).
It's Linear: The old method is like a tangled ball of yarn; you can't easily pull a single thread to see how it works. The new method is like a straight, smooth string. Because it's "straight" (mathematically linear), engineers can use powerful tools to find the exact root cause of the wobble and design a controller to stop it.

What Did They Do?

The researchers tested this new framework on a standard "test orchestra" (a modified IEEE benchmark system) where they replaced two big traditional power plants with the new digital IBRs.

They Caught the Wobble: They successfully used their new "smart sketch" to spot the dangerous subsynchronous oscillations caused by the new digital power plants.
They Built a Stabilizer: Because their model was so clean and simple, they could design a robust controller (a digital autopilot). This controller acts like a conductor who gently taps the new musicians on the shoulder to keep them in sync, damping out the wobble even when a fault (like a lightning strike) happens.
They Found the Data Center Danger: They showed that even if the orchestra is playing nicely, a Data Center (the hungry fan) sitting in the wrong spot can start breathing in a way that triggers the wobble again. Their math can predict exactly where to put the Data Center so it doesn't cause trouble.

The Bottom Line

This paper is a big step forward because it gives engineers a fast, accurate, and easy-to-analyze way to understand how the modern grid (full of solar, wind, and data centers) behaves.

Instead of waiting hours to simulate a problem, they can now run the simulation in minutes, figure out exactly what's wrong, and design a fix instantly. It's the difference between trying to fix a car engine by taking it apart piece by piece in the dark, versus having a clear, real-time diagnostic screen that tells you exactly which bolt is loose.

1. Problem Statement

The integration of Inverter-Based Resources (IBRs), particularly Grid-Following (GFL) types, into power grids has led to the prevalence of Subsynchronous Oscillations (SSOs). These oscillations pose significant reliability risks, especially with the growing integration of Data Center (DC) loads, which can potentially excite these modes.

Current analysis relies heavily on Electromagnetic Transient (EMT) simulation platforms (e.g., PSCAD/EMTDC). However, EMT simulations face two major limitations:

Scalability: They suffer from heavy computational burdens, making them slow for large multi-machine systems.
Analysis Limitations: They are unsuitable for deriving Linear Time-Invariant (LTI) models, which are essential for frequency-domain analysis, eigenvalue decomposition, and systematic controller design.

Existing Dynamic Phasor (DP) frameworks have largely been limited to single-machine or single-IBR systems and often fail to handle unbalanced conditions or detailed multi-machine interactions.

2. Methodology

The authors propose a Generalized Dynamic Phasor (DP) Framework capable of modeling multi-machine systems with IBRs under both balanced and unbalanced conditions.

A. Core Modeling Approach

The framework utilizes the Generalized Averaging Theory, representing time-domain waveforms as a sum of Fourier coefficients (Dynamic Phasors) that vary slowly over time.

GFL IBRs: Modeled in the dq-frame using DP coefficients $k=0$ (positive sequence) and $k=\pm 2$ (negative sequence). This aligns with the vector control structure of IBRs. The model includes the Phase Locked Loop (PLL), outer power/voltage loops, inner current loops, and the R-L filter plant.
Synchronous Generators (SGs) & Network: Modeled in the pnz-frame (positive, negative, zero sequence).
- SG stator transients use $k=\pm 1$ DPs.
- SG mechanical dynamics use $k=0$ DP.
- Transmission lines and loads use $k=\pm 1$ DPs to capture dynamics around the synchronous frequency (60 Hz).
Interfacing: A transformation matrix converts dq-frame DP quantities (from IBRs) to pnz-frame quantities (for the network) and vice versa, ensuring seamless integration.
DC Loads: Modeled as constant power loads with unity power factor. Crucially, the framework analyzes how DC load injections ( $k=\pm 1$ ) interact with SSO modes.

B. Controller Design

The framework's time-invariance and linearizability allow for eigenvalue decomposition. The authors design a Robust Decentralized $H_\infty$ Damping Controller:

Linearization: The system is linearized around an operating point.
Sequential Design: Controllers are designed iteratively for each IBR. The first controller is designed to meet a settling time ( $T_s$ ), then the closed-loop system is reduced (via Schur balanced truncation) and treated as the new plant for the next IBR.
Inputs/Outputs: Local feedback signals are the Point of Interconnection (POI) voltage magnitude; control inputs modulate the current references ( $i^*_{td}, i^*_{tq}$ ).

C. Case Study

The framework is validated on a modified IEEE two-area benchmark system where 50% of the generation is replaced by GFL IBRs.

Validation: Results are compared against high-fidelity EMTDC/PSCAD simulations.
Scenarios: Balanced/unbalanced faults, SSO damping, and DC load excitation.

3. Key Contributions

Generalized Multi-Machine DP Framework: The first work to demonstrate a scalable DP framework for multi-machine systems with IBRs that handles unbalanced conditions (e.g., single-line-to-ground faults).
SSO Capture and Validation: The model accurately captures GFL-induced SSOs and shows reasonable agreement with EMT simulations while offering significant computational speedups.
Systematic Controller Design: Leverages the LTI nature of the DP model to design a robust decentralized $H_\infty$ controller, effectively damping poorly damped SSO modes using local IBR signals.
DC Load Impact Analysis: Demonstrates that even well-damped SSO modes can be excited by DC loads. The paper provides an analytical method to quantify the locational impact of DC loads on mode excitation using frequency-dependent gains.

4. Results

Accuracy: The DP model's eigenvalues for SSO modes closely match EMT results. For a 20 Hz PLL bandwidth, the SSO frequency was ~6.36 Hz (DP) vs. ~6.46 Hz (EMT), with damping ratios of 0.3% vs. 0.8%.
Fault Response: In a single-line-to-ground (L-G) fault simulation, the DP framework successfully captured transients and the persistence of poorly damped SSOs after fault clearance, matching EMT behavior.
Computational Efficiency: In a statistical analysis of 50 simulations (5-second duration post-fault), the DP model ran 2x faster on average than the EMT model (Mean time: ~14.8s for DP vs. ~30.3s for EMT).
Controller Performance: The decentralized $H_\infty$ controller successfully damped the SSO mode following an L-G fault, reducing oscillation amplitudes significantly compared to the uncontrolled case.
DC Load Excitation: When a DC load was introduced, it excited the SSO mode. The ratio of excitation amplitude between a load at Bus 7 vs. Bus 9 was observed to be ~6.7 in time-domain simulations, which was closely predicted by the frequency-domain analysis (ratio ~8.7), validating the proposed locational impact index.

5. Significance

This paper bridges a critical gap in power system analysis by extending Dynamic Phasor theory to complex, multi-machine systems with IBRs. Its significance lies in:

Enabling Frequency-Domain Analysis: It allows for the use of powerful linear control theory tools (eigenvalue analysis, $H_\infty$ synthesis) on systems that were previously only analyzable via time-consuming non-linear EMT simulations.
Handling Unbalance: Unlike previous DP works, this framework handles unbalanced faults, making it suitable for realistic grid studies.
Data Center Integration: It specifically addresses the emerging reliability risk posed by data center loads interacting with inverter-dominated grids.
Scalability: The demonstrated 2x speedup suggests the framework is viable for larger, more complex grids where EMT simulations are currently prohibitive.

The authors conclude that this framework is a foundational step toward analyzing and controlling SSOs in future high-IBR penetration grids, with future work planned for larger systems and more systematic controller design approaches.