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Eigenvalue Tracking of Large-Scale Systems Impacted by Time Delays

Andreas Bouterakos, Georgios Tzounas

TL;DR

This work develops a continuation-based framework to track eigenvalue trajectories in time-delayed power-system models described by DDAEs, enabling efficient assessment of small-signal stability on large networks. It formulates a parameter-dependent pencil $P(s,p) = s E(p) - A_0(p) - \sum_j A_j(p) e^{-s \tau_j}$ and uses a predictor-corrector scheme to follow eigenpairs as system parameters and delay magnitudes vary, while preserving sparsity. The method extends to multiple delays and to delay-variations and noisy communications, employing spectral discretization and Newton-corrector refinement, and is validated on a modified IEEE 39-bus system and a large all-island Irish transmission model, demonstrating accuracy and scalability. The approach provides insight into how delays and DER participation affect stability margins, supporting design of delay-robust control strategies for inverter-dominated grids.

Abstract

The paper focuses on tracking eigenvalue trajectories in power system models with time delays. We formulate a continuation-based approach that employs numerical integration to follow eigenvalues as system parameters vary, in the presence of one or multiple delayed variables. The formulation preserves the sparsity of the delay differential-algebraic equation (DDAE) system model and allows treating the delay magnitude itself as a varying parameter with implementation aspects discussed in detail. Accuracy is demonstrated on a modified IEEE 39-bus system with distributed energy resources. Scalability is discussed using a realistic dynamic model of the Irish transmission network.

Eigenvalue Tracking of Large-Scale Systems Impacted by Time Delays

TL;DR

This work develops a continuation-based framework to track eigenvalue trajectories in time-delayed power-system models described by DDAEs, enabling efficient assessment of small-signal stability on large networks. It formulates a parameter-dependent pencil and uses a predictor-corrector scheme to follow eigenpairs as system parameters and delay magnitudes vary, while preserving sparsity. The method extends to multiple delays and to delay-variations and noisy communications, employing spectral discretization and Newton-corrector refinement, and is validated on a modified IEEE 39-bus system and a large all-island Irish transmission model, demonstrating accuracy and scalability. The approach provides insight into how delays and DER participation affect stability margins, supporting design of delay-robust control strategies for inverter-dominated grids.

Abstract

The paper focuses on tracking eigenvalue trajectories in power system models with time delays. We formulate a continuation-based approach that employs numerical integration to follow eigenvalues as system parameters vary, in the presence of one or multiple delayed variables. The formulation preserves the sparsity of the delay differential-algebraic equation (DDAE) system model and allows treating the delay magnitude itself as a varying parameter with implementation aspects discussed in detail. Accuracy is demonstrated on a modified IEEE 39-bus system with distributed energy resources. Scalability is discussed using a realistic dynamic model of the Irish transmission network.

Paper Structure

This paper contains 21 sections, 43 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature Review
  4. Contribution
  5. Paper Organization
  6. Small-Signal Stability Analysis
  7. DDAE Power System Model
  8. Eigenvalue Analysis
  9. Eigenvalue tracking
  10. Tracking in the Presence of a Single Delay
  11. Generalization to Multiple Delays
  12. Delay Magnitude as Varying Parameter
  13. Communication Delays with Noise and Data Dropouts
  14. Numerical Integration
  15. Case Study
  16. ...and 6 more sections

Figures (5)

  • Figure 1: 39-bus system: tracking in the presence of a single delay.
  • Figure 2: 39-bus system: tracking in the presence of multiple delays.
  • Figure 3: 39-bus system, SM at bus 31: PSS with input signal delay.
  • Figure 4: AIITS: tracking poorly damped electromechanical modes in the presence of multiple delays.
  • Figure 5: AIITS, SM at bus 717: increasing PSS input signal delay.