⚡ electrical engineering

A Dynamic Phasor Framework for Analysis of Subsynchronous Oscillations in Multi-Machine Systems with IBRs and Large Loads

This paper proposes a generalized dynamic phasor framework that combines dq-frame modeling for inverter-based resources with pnz-frame modeling for synchronous generators and loads to enable scalable, eigenvalue-based analysis and control design for subsynchronous oscillations in large-scale multi-machine systems, including those with AI data center loads.

Original authors: Fiaz Hossain, Nilanjan Ray Chaudhuri, Constantino M. Lagoa, Alok Sinha, ai Gopal Vennelaganti, Mohammed E. Nassar

Published 2026-07-03

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CC BY 4.0

Original authors: Fiaz Hossain, Nilanjan Ray Chaudhuri, Constantino M. Lagoa, Alok Sinha, ai Gopal Vennelaganti, Mohammed E. Nassar

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, complex orchestra. For a long time, the musicians (traditional power plants) played in perfect sync. But recently, new digital musicians (solar and wind farms using inverters) have joined the band. While they are efficient, they sometimes play in a slightly different rhythm, causing the whole orchestra to wobble or shake in a dangerous way called Subsynchronous Oscillation (SSO).

At the same time, a new "super-fan" has entered the concert hall: AI Data Centers. These are massive computers that eat electricity in huge, sudden bursts, much like a crowd suddenly jumping up and down in unison. This can make the orchestra's rhythm stumble and even damage the physical instruments (the turbine shafts) if they vibrate too hard.

The Problem: The "Slow Motion" Camera Issue

To study these wobbles and jumps, engineers usually use a tool called EMT (Electromagnetic Transient) simulation. Think of this like trying to film a hummingbird's wings with a camera that takes a picture every single microsecond. It captures every tiny detail perfectly, but it's so slow and computationally heavy that you can't film a whole symphony (a large power grid) without your computer crashing or taking days to finish.

The Solution: The "Dynamic Phasor" Framework

The authors of this paper invented a new way to film the orchestra, which they call a Dynamic Phasor (DP) framework.

Instead of taking a picture of every single micro-movement, this new method takes a "smart snapshot." It looks at the main rhythm and the most important wobbles, ignoring the tiny, irrelevant vibrations that don't matter.

The Analogy: Imagine watching a dance. The EMT method tries to track every muscle twitch of every dancer. The DP method tracks the dancers' overall steps and the rhythm of the music.
The Benefit: Because it ignores the tiny details, it runs 12 times faster than the old method. This means engineers can simulate the entire power grid (including the new AI data centers and the new digital musicians) in minutes instead of days.

How It Works: Speaking Different Languages

The grid has different parts that "speak" different languages:

The Inverters (Solar/Wind): They speak "dq-frame" (a specific digital language).
The Big Turbines and Power Lines: They speak "pnz-frame" (a language for physical rotation and waves).

The authors built a universal translator (the DP framework) that lets these two groups talk to each other seamlessly. They can model:

Grid-Following (GFL): Inverters that just follow the grid's rhythm.
Grid-Forming (GFM): Inverters that try to lead the rhythm.
Multi-Mass Turbines: Real-world turbines that have heavy metal shafts with different weights, which can twist and bend like a long rubber band.

What They Discovered (The Experiments)

Using their new fast camera, the team ran several tests on a model of a large power system (the IEEE 68-bus system):

Fixing the Wobble: They found a specific "wobble" (an unstable SSO mode) caused by the solar/wind inverters.
- Fix A: They built a special "damping controller" (like a shock absorber) using a smart algorithm. This controller quickly calmed the wobble.
- Fix B: They swapped one of the "follower" inverters for a "leader" (Grid-Forming) inverter. This change completely made the dangerous wobble disappear.
The AI Data Center Impact: They simulated AI data centers ramping up and down their power usage (like a computer cluster starting a massive training job).
- Result: These sudden changes didn't just affect the grid's speed; they caused the heavy metal shafts inside the turbines to twist and stress.
- The Danger: If the rhythm of the AI's power usage matched the natural "twisting frequency" of the turbine shafts, it could cause fatigue, potentially breaking the shaft over time. The new framework allowed them to calculate exactly how much stress these AI loads put on the turbines.

The Bottom Line

This paper doesn't just say "we have a faster computer." It says, "We have a fast, accurate, and flexible tool that lets us understand how new digital power sources and giant AI computers interact with old-school power plants."

Because the tool is so fast, engineers can now test "what-if" scenarios (like "What if an AI data center suddenly turns on?" or "What if a power line breaks?") and design better controllers to keep the lights on and the turbines from breaking, all without waiting days for the computer to finish the math.

Technical Summary: A Dynamic Phasor Framework for Analysis of Subsynchronous Oscillations in Multi-Machine Systems with IBRs and Large Loads

1. Problem Statement

The modern power grid faces unprecedented challenges regarding scalability in time- and frequency-domain analysis. Specifically, the increasing integration of Grid-Following (GFL) Inverter-Based Resources (IBRs) has led to a rise in Subsynchronous Oscillations (SSOs). Simultaneously, the rapid expansion of Artificial Intelligence (AI) Data Center (DC) loads introduces new reliability concerns, including large, steep power ramps and synchronized training cycles that can induce high-amplitude fluctuations. These fluctuations threaten the fatigue life of turbine-generator shafts and affect primary frequency response.

While Electromagnetic Transient (EMT) simulation tools can model these phenomena, they suffer from poor scalability due to high computational demands. Furthermore, EMT models are generally unsuitable for developing the Linear Time-Invariant (LTI) models required for eigenvalue analysis, root-cause identification, and systematic controller design. Existing Dynamic Phasor (DP) frameworks have largely been restricted to single-machine or single-IBR systems, or they have failed to adequately address multi-machine systems with IBRs under unbalanced conditions.

2. Methodology

The authors propose a generalized Dynamic Phasor (DP) framework designed to analyze SSOs in multi-machine systems containing GFL and Grid-Forming (GFM) IBRs, multi-mass synchronous generators (SGs), and large loads (specifically AI DCs) under both balanced and unbalanced conditions.

Core Modeling Strategy

The framework utilizes a hybrid coordinate system to leverage the strengths of different reference frames:

IBRs (GFL and GFM): Modeled in the dq-frame using dynamic phasors. This aligns with standard vector control implementations. The models retain DP coefficients for $k=0$ $k = 0$ (positive sequence) and $k=\pm2$ $k = \pm 2$ (negative sequence) to capture unbalanced dynamics.
- GFM IBRs: Decomposed into droop mechanisms, outer voltage loops, inner voltage/current loops, and output filter dynamics. Current limiting strategies (constant-angle and $i_q$ -priority) are explicitly modeled.
- GFL IBRs: Modeled with Phase-Locked Loop (PLL) dynamics, outer active/reactive power control loops, inner current control loops, and the plant (series R-L filter).
Network and SGs: Modeled in the pnz-frame (positive, negative, zero sequence).
- Transmission Lines: Represented using lumped $\pi$ -section models.
- Synchronous Generators: Detailed multi-mass turbine-generator models (including HP, IP, LPA, LPB turbines, generator, and exciter) are represented. Mechanical dynamics are captured using the $k=0$ component, while electrical transients utilize $k=0, \pm2$ components.
- Loads: AI DC loads are modeled as constant power demands with unity power factor (positive sequence only).
Coupling: Interface variables (terminal voltages and currents) are transformed between the dq-frame (IBRs) and pnz-frame (network/SGs) using specific transformation equations derived in the paper.

Key Properties

Time-Invariance: The formulation results in a time-invariant system, enabling eigenvalue decomposition.
Linearizability: The model can be linearized to identify SSO modes, perform root-cause analysis, and design damping controllers.
Scalability: By truncating the Fourier series to dominant coefficients, the framework supports faster time-domain simulations compared to EMT.

3. Key Contributions

The paper claims the following specific contributions:

Generalized Framework: Presentation of a DP-based modeling framework capable of analyzing subsynchronous phenomena in systems with multi-mass turbines, both GFL and GFM IBRs, and transmission line dynamics under balanced and unbalanced conditions.
Benchmarking: Demonstration of the framework's speedup over EMT models in a multi-machine system.
Control Design Application: Leveraging the time-invariance and linearizability of the framework to design a supplementary low-order decentralized controller for damping poorly-damped SSO modes in a modified IEEE 68-bus system.
GFM vs. GFL Analysis: Demonstration via time- and frequency-domain analysis that replacing a GFL IBR with a GFM IBR eliminates specific poorly-damped SSO modes.
AI Load Impact Study: Analysis of the impact of AI DC loads on primary frequency response and the torsional oscillations (shear stress) of multi-mass turbine-generator shafts, providing a method to assess fatigue life risks.

4. Results and Validation

The framework was validated and tested using three primary systems:

A. Validation against EMT Models

IEEE First Benchmark Model (FBM) for SSR: The DP model's eigenvalues and time-domain responses (following a three-phase-to-ground fault) matched closely with EMT results and literature values.
GFM IBR with Current Limits: Validated against an EMTDC/PSCAD model. The DP model accurately replicated responses to self-clearing Line-to-Line (L-L) faults under both constant-angle and $i_q$ -priority current limiting strategies.
Modified IEEE 4-Machine System: A system with two GFL IBRs replacing synchronous generators was simulated. The DP model (using linearization) and EMT model (using Prony analysis) showed close agreement in SSO mode frequency and damping ratios for different PLL bandwidths.
Speedup: In the IEEE 4-machine system, the DP model achieved an average 2x speedup over the EMT model. In the IEEE 39-bus system context (referenced from prior work), a 10x speedup was noted, though the current paper focuses on the 68-bus system.

B. SSO Damping in Modified IEEE 68-Bus System

Two solutions were proposed to damp a poorly-damped SSO mode (4.76 Hz, 0.2% damping) induced by GFL IBRs:

Decentralized Controller: A 5th-order controller designed using Particle Swarm Optimization (PSO). It uses terminal voltage magnitude as feedback to modulate reactive current references. Simulations showed effective damping of the SSO mode following various tie-line outages and unbalanced faults.
GFM Replacement: Replacing one GFL IBR with a GFM IBR of the same rating resulted in the complete disappearance of the poorly-damped SSO mode, as confirmed by eigenvalue analysis.

C. Impact of AI DC Loads

Frequency Response: Simulations of AI DC load ramps (transitioning between compute and rest periods) showed significant frequency deviations, triggering ancillary frequency support but not load shedding.
Shaft Stress: By tuning AI DC load oscillations to coincide with torsional modes (e.g., ~7.83 Hz), the study demonstrated that shaft stresses could reach limits defined by the augmented modified Goodman diagram, potentially leading to fatigue life loss.
Sensitivity Analysis: Monte Carlo simulations (100 runs) varying amplitude, frequency, and phase of the AI load oscillations revealed that shaft stresses are highly sensitive to small frequency deviations from the torsional mode.

D. Computational Performance

In the modified IEEE 68-bus system (containing 8 multi-mass SGs, 8 single-mass SGs, 1 GFL, 1 GFM, and dynamic transmission lines):

The DP framework simulated a 20-second forced oscillation scenario from 7 AI DCs in an average of 2.5 minutes.
This establishes the scalability of the framework for large-scale systems where EMT simulations would be computationally prohibitive.

5. Significance and Claims

The authors explicitly state that no novelty is claimed regarding the design of the SSO damping control itself or the general fact that GFM control can remedy SSO problems.

The primary significance of this work lies in showcasing the capability of the DP-based framework to:

Capture these complex phenomena (multi-machine SSOs, unbalanced faults, and AI load impacts) for the first time in a unified, scalable, and linearizable environment.
Enable the transition from purely time-domain EMT simulations to frequency-domain analysis (eigenvalue decomposition) for large systems with IBRs.
Provide a practical tool for analyzing the fatigue life of turbine-generator shafts under the specific stress conditions induced by modern AI data center loads.

The paper concludes that the proposed framework successfully bridges the gap between the accuracy of EMT and the analytical utility of linear models, offering a scalable solution for the analysis and control of future power grids.