⚡ electrical engineering

Eigenvalue Tracking of Large-Scale Systems Impacted by Time Delays

This paper presents a continuation-based numerical approach for tracking eigenvalue trajectories in large-scale power systems with time delays, demonstrating its effectiveness on both a modified IEEE 39-bus system and a real-world Irish transmission network model.

Original authors: Andreas Bouterakos, Georgios Tzounas

Published 2026-02-23

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Andreas Bouterakos, Georgios Tzounas

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 a massive, complex orchestra playing a piece of music. This orchestra represents a modern power grid, with thousands of instruments (generators, solar panels, wind turbines) all trying to stay in perfect rhythm.

In the old days, the conductor (the control system) could hear the musicians instantly and adjust the tempo immediately. But today, our power grid is becoming digital. The conductor now relies on high-speed data messages to hear the musicians. The problem? Time Delays.

Just like a bad video call where your friend's voice arrives a split second late, these digital signals in the power grid take a tiny bit of time to travel. If the delay is too long, the conductor might adjust the tempo after the musicians have already changed it, causing the whole orchestra to get out of sync, wobble, or even crash (a blackout).

The Problem: Tracking the "Wobbles"

Engineers need to know exactly how much delay the system can handle before it starts to wobble dangerously. They do this by looking at the system's "eigenvalues."

Think of an eigenvalue as a heartbeat or a vibration frequency of the orchestra.

If the heartbeat is steady and calm, the system is stable.
If the heartbeat starts racing or skipping, the system is unstable.

Usually, engineers calculate these heartbeats by taking a snapshot of the system at one specific moment. But the real world is messy. The amount of delay changes (like network traffic getting worse), and the settings of the machines change. If you only take a snapshot, you miss the whole movie. You don't see how the heartbeat changes as the delay gets worse.

The Solution: The "GPS for Heartbeats"

This paper introduces a new method called Eigenvalue Tracking. Instead of taking static snapshots, the authors created a GPS system for these heartbeats.

Here is how their method works, using simple analogies:

1. The "Hiker" Analogy (Continuation Methods)
Imagine you are hiking up a mountain (the mountain represents the changing stability of the grid).

Old Way: You take a photo of the view every 100 meters. You might miss a sudden cliff or a hidden path between the photos.
New Way (This Paper): You are a hiker with a GPS that constantly updates your position. You take one step, look at the map, take another step, and update the map again. This allows you to trace the entire path of the heartbeat as the delay changes, even if the path twists and turns unexpectedly.

2. The "Rubber Band" Analogy (Time Delays)
Think of the time delay as a rubber band connecting the conductor to the musicians.

The paper asks: "What happens if we slowly stretch this rubber band?"
Their method tracks the heartbeat as the rubber band stretches from 0.01 seconds to 0.6 seconds.
They discovered something surprising: Sometimes, stretching the rubber band makes the system unstable, but stretching it even further actually makes it stable again! It's like a rubber band that gets wobbly in the middle but snaps back into shape if pulled hard enough. Without their "GPS tracking," engineers would have missed this second stable zone.

3. The "Traffic Jam" Analogy (Real-World Noise)
In the real world, data doesn't just get delayed; it gets lost (packet dropouts) or noisy (static).

The authors updated their GPS to handle "traffic jams" and "static." They showed that even with messy data, their method can still predict exactly when the orchestra will start to wobble.

Why Does This Matter?

The authors tested this on two things:

A Model System: A standard 39-bus power grid (like a training simulator).
A Real System: The entire transmission network of Ireland (a massive, real-world orchestra with over 1,500 buses).

The Results:

Their method is fast. Traditional methods are like trying to solve a puzzle by moving every single piece one by one; this new method moves the whole picture at once.
Their method is accurate. It correctly predicted when the Irish power grid would become unstable due to delays.
It allows engineers to design better controllers. Instead of guessing how much delay is safe, they can now see the exact "safe zone" and the "danger zone" on a map.

The Bottom Line

This paper gives power grid engineers a smart, moving map to navigate the tricky world of time delays. Instead of guessing if a system will crash when the internet gets slow, they can now trace the exact path of the system's stability, ensuring that our lights stay on even when the digital signals get a little jittery.

1. Problem Statement

The paper addresses the challenge of performing Small-Signal Stability Analysis (SSSA) in modern power systems characterized by high penetration of inverter-based resources and extensive communication networks.

The Core Issue: Time delays, arising from data processing and communication latencies (e.g., Wide Area Measurement Systems - WAMS), significantly impact system dynamics. In closed-loop control settings, these delays can weaken damping and reduce stability margins.
Mathematical Complexity: When time delays are present, power system models transform from standard Differential-Algebraic Equations (DAEs) into Delay Differential-Algebraic Equations (DDAEs).
- The characteristic equation of a DDAE is transcendental, resulting in an infinite number of eigenvalues.
- Standard eigenvalue tracking methods (repeatedly solving the eigenvalue problem as parameters change) are computationally expensive for large-scale systems, especially when using sparse techniques or when delays are treated as varying parameters.
- Existing continuation methods often fail to handle multiple delays efficiently, neglect the delay magnitude as a variable, or require the elimination of algebraic variables (destroying sparsity).

2. Methodology

The authors propose a continuation-based approach that employs numerical integration to track eigenvalue trajectories as system parameters vary. The method is designed to work directly with the sparse DDAE formulation.

A. Mathematical Formulation

System Model: The system is modeled as a semi-implicit DDAE involving state variables ( $x$ ), algebraic variables ( $y$ ), and delayed variables ( $x_d, y_d$ ).
Linearization: The system is linearized around an equilibrium point, leading to a matrix function $P(s, p)$ where $s$ is the eigenvalue and $p$ is a continuation parameter. The eigenvalue problem is defined as $P(s, p)\phi = 0$ .
Differentiation: Instead of solving the eigenvalue problem from scratch at every step, the authors differentiate the characteristic equation with respect to the parameter $p$ . This yields a system of equations describing the rate of change of the eigenvalue ( $\dot{s}$ ) and eigenvector ( $\dot{\phi}$ ).
Normalization: To close the system of equations (which has more unknowns than equations), an eigenvector normalization constraint ( $\phi^T \phi = c$ ) is imposed.
Real/Imaginary Splitting: The complex equations are split into real and imaginary parts to form a system of ordinary differential equations (ODEs) in the form:
$M(y)\dot{y} = h(y)$
where $y$ contains the real and imaginary parts of the eigenvalue and eigenvector. This system is solved using numerical integration (e.g., Forward Euler with Newton correction).

B. Key Extensions

Multiple Delays: The formulation is generalized to handle $\mu$ simultaneous delays without eliminating algebraic variables, preserving the sparsity of the Jacobian matrices.
Delay Magnitude as a Parameter: A novel contribution allows the time delay magnitude ( $\tau$ ) itself to be the continuation parameter. This enables direct tracing of stability margins relative to delay variations.
Stochastic Delays (Noise & Dropouts): The method is extended to model realistic WAMS effects, including packet dropouts and noise, using composite time-varying delay models (Gamma distribution-based functions) rather than assuming constant delays.

3. Key Contributions

Sparse DDAE Tracking: The method retains the sparsity structure of large-scale power system models, making it applicable to systems with thousands of buses where dense-matrix methods (like QZ) are infeasible.
Delay as a Continuation Parameter: Unlike previous works that treat delay as a fixed constant, this approach treats delay magnitude as a varying parameter, allowing for precise stability margin estimation.
Handling Real-World Effects: The framework incorporates models for communication noise and packet dropouts, moving beyond idealized constant delay assumptions.
Robustness at Singularities: The method handles "folds" (where eigenvalues coalesce and split) by re-initializing the eigendecomposition when necessary, ensuring continuous tracking of specific branches.

4. Results and Case Studies

The proposed method was validated on two distinct power system models:

A. Modified IEEE 39-Bus System

Scenario 1 (Single Delay): Tracked the Frequency Regulation (FR) mode as the DER droop constant varied. The method accurately captured the trajectory, including the point where complex eigenvalues coalesced into a defective real eigenvalue (a quadratic fold) and subsequently split.
Scenario 2 (Multiple Delays): Introduced 20 delayed variables (AVRs, PSSs, DER controllers). The method successfully tracked electromechanical modes as SM droop constants varied, matching results from repeated QR/QZ solvers but with higher efficiency.
Scenario 3 (Delay as Parameter): Varied the PSS input signal delay ( $\tau_{PSS}$ $τ_{P S S}$ ) from 0.01s to 0.6s.
- Finding: The system became unstable for $\tau_{PSS} > 0.1$ s, regained stability for $\tau_{PSS} > 0.5$ s (a non-monotonic stability behavior).
- Validation: Time-domain simulations of a three-phase fault confirmed the eigenvalue tracking predictions (e.g., divergent rotor speed at $\tau_{PSS}=0.3$ s).

B. All-Island Irish Transmission System (AIITS)

Scale: A real-world model with 1,502 buses, 1,629 states, and 9,897 algebraic variables.
Challenge: The dimension of the linearized problem (approx. $11,834 \times 11,834$ ) makes standard dense eigensolvers impractical.
Performance: The continuation method efficiently tracked poorly damped electromechanical modes as PSS gains increased and as the delay of a specific PSS signal was varied from 0.01s to 0.5s.
Outcome: The method successfully traced tightly clustered eigenvalues in the complex plane, demonstrating scalability and accuracy where traditional methods would struggle computationally.

5. Significance

Computational Efficiency: By using numerical integration of the sensitivity equations rather than repeated full eigenvalue decompositions, the method offers a computationally efficient alternative for large-scale systems.
Design Guidance: The ability to treat delay as a variable allows engineers to directly determine the maximum allowable communication latency before a system becomes unstable, which is critical for the design of Wide Area Control Systems (WACS).
Future-Proofing: The inclusion of packet dropouts and noise models makes the analysis more relevant to the practical deployment of inverter-dominated grids with communication-dependent controls.
Scalability: The preservation of sparsity ensures the method is viable for future power grids that will be significantly larger and more complex than current models.

In conclusion, this paper provides a robust, scalable, and mathematically rigorous framework for analyzing the stability of power systems under the influence of time delays, bridging the gap between theoretical DDAE analysis and practical large-scale grid operation.