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Novel Data-Driven Indices for Early Detection and Quantification of Short-Term Voltage Instability from Voltage Trajectories

Mohammad Almomani, Muhammad Sarwar, Venkataramana Ajjarapu

TL;DR

The paper tackles the challenge of early detection and quantification of short‑term voltage instability caused by delayed voltage recovery and undamped oscillations. It introduces STVSI, a data‑driven index that decomposes voltage trajectories via Empirical Mode Decomposition and evaluates stability through Lyapunov Exponents and KL divergence against a Gompertz reference. The method yields two interpretable indices, $D_{ ext{KL}}^{r}$ for recovery and $D_{ ext{KL}}^{ ext{imf}}$ for oscillations, with thresholds $D_{ ext{critical}}^{r}$ and $D_{ ext{critical}}^{ ext{imf}}$, and includes a procedure to tune $\gamma_1$ for discriminating recovery profiles. Validation on the Nordic system demonstrates early warnings within seconds and a gradated stability measure, offering practical benefits for real‑time monitoring and proactive protection against OEL LVRT triggered instability.

Abstract

This paper presents a novel Short-Term Voltage Stability Index (STVSI), which leverages Lyapunov Exponent-based detection to assess and quantify short-term stability triggered by Over Excitation Limiters (OELs) or undamped oscillations in voltage. The proposed method is measurement-based and decomposes the voltage trajectory into two key components using Empirical Mode Decomposition (EMD): a residual part, which indicates delayed voltage recovery, and an oscillatory part, which captures oscillations. The residual component is critical, as it can detect activation of OELs in synchronous generators or Low Voltage Ride-Through (LVRT) relays in inverter-based resources, potentially leading to instability within the quasisteady-state time frame. Meanwhile, the oscillatory component may indicate either a stable or unstable state in the short term. To accurately assess stability, STVSI employs an entropy-based metric to measure the proximity of the system to instability, with specific indices for short-term voltage stability based on oscillations and recovery. Simulations on the Nordic power system demonstrate that STVSI effectively identifies and categorizes voltage stability issues. Moreover, STVSI not only detects voltage stability conditions but also qualitatively assesses the extent of stability, providing a nuanced measure of stability.

Novel Data-Driven Indices for Early Detection and Quantification of Short-Term Voltage Instability from Voltage Trajectories

TL;DR

The paper tackles the challenge of early detection and quantification of short‑term voltage instability caused by delayed voltage recovery and undamped oscillations. It introduces STVSI, a data‑driven index that decomposes voltage trajectories via Empirical Mode Decomposition and evaluates stability through Lyapunov Exponents and KL divergence against a Gompertz reference. The method yields two interpretable indices, for recovery and for oscillations, with thresholds and , and includes a procedure to tune for discriminating recovery profiles. Validation on the Nordic system demonstrates early warnings within seconds and a gradated stability measure, offering practical benefits for real‑time monitoring and proactive protection against OEL LVRT triggered instability.

Abstract

This paper presents a novel Short-Term Voltage Stability Index (STVSI), which leverages Lyapunov Exponent-based detection to assess and quantify short-term stability triggered by Over Excitation Limiters (OELs) or undamped oscillations in voltage. The proposed method is measurement-based and decomposes the voltage trajectory into two key components using Empirical Mode Decomposition (EMD): a residual part, which indicates delayed voltage recovery, and an oscillatory part, which captures oscillations. The residual component is critical, as it can detect activation of OELs in synchronous generators or Low Voltage Ride-Through (LVRT) relays in inverter-based resources, potentially leading to instability within the quasisteady-state time frame. Meanwhile, the oscillatory component may indicate either a stable or unstable state in the short term. To accurately assess stability, STVSI employs an entropy-based metric to measure the proximity of the system to instability, with specific indices for short-term voltage stability based on oscillations and recovery. Simulations on the Nordic power system demonstrate that STVSI effectively identifies and categorizes voltage stability issues. Moreover, STVSI not only detects voltage stability conditions but also qualitatively assesses the extent of stability, providing a nuanced measure of stability.

Paper Structure

This paper contains 17 sections, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature Review and Research Gap
  3. Contribution of This Paper
  4. Background
  5. Lyapunov exponent (LE)
  6. Empirical Mode Decomposition
  7. KL Divergence Measure
  8. Proposed Short-Term Voltage Stability Index (STVSI)
  9. Oscillation stability index threshold ($D_{\text{critical}}^{\text{imf}}$)
  10. Recovery-caused stability index threshold ($D_{\text{critical}}^r$)
  11. Step-by-Step Implementation
  12. Step 1: Stability Assessment Using KL Divergence
  13. Recovery Index
  14. Oscillation Index
  15. Step 2: Tuning of $\gamma_1$
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Relationship between residual signals $s_1$ and $s_2$ and the tripping characteristic $T$ of OEL or LVRT.
  • Figure 2: Step-by-step calculation of the proposed stability index for two scenarios: the upper row illustrates a recovery case for Motor D with dynamic load levels of 80% (unstable due to OEL triggering) and 75% (stable), while the lower row presents an oscillatory stable case for Motor A. Each row includes the original voltage signal (V), EMD decomposition into residual (R) and IMFs, corresponding Lyapunov Exponents (LEs), probability distributions of LEs, and KL divergence calculations, demonstrating the index's ability to differentiate stability conditions based on both oscillatory and recovery components.