[Paper Review] Lyapunov modal analysis and participation factors with applications to small-signal stability of power systems

Alexey B. Iskakov, Igor B. Yadykin|arXiv (Cornell University)|Sep 5, 2019

Vibration and Dynamic Analysis5 citations

TL;DR

This paper introduces Lyapunov Modal Analysis (LMA), a novel framework that combines selective modal analysis with spectral decomposition of specially chosen Lyapunov functions to estimate time-integrated energy from modal interactions in power systems. It proposes Lyapunov participation factors (LPFs) that identify resonant interactions, mode merging, and instability onset by associating these phenomena with specific state variables, enabling rapid real-time stability assessment in large-scale systems.

ABSTRACT

When random disturbances are regularly introduced into a dynamical system over time, its small-signal stability is determined by the energy of perturbations accumulated in the system. To analyze this perturbation energy, this paper proposes a novel physically motivated Lyapunov modal analysis (LMA) framework, which combines selective modal analysis with the spectral decompositions of specially chosen Lyapunov functions. This approach allows the modal interactions in dynamical systems to be characterized and estimated in connection with specific state variables. Conventional participation factors characterize the relative contribution of the system modes and state variables to the evolution of states and modes, respectively. In contrast, the proposed Lyapunov participation factors characterize similar contributions to corresponding Lyapunov functions, which determine the integral energy associated with the states and modes on an infinite or finite time interval. This allows the estimation of modal interactions in terms of total energy produced by their mutual actions over time. Using a two-area four-machines power system, we demonstrate that LMA reliably identifies resonant modal interactions, merging of modes, and loss of stability, even for a linear model, and associates them with certain state variables. The Lyapunov participation factors corresponding to the selected part of the system spectrum can be calculated independently and serve as a basis for rapid real-time calculations of critical mode behaviors in large-scale dynamical systems.

Motivation & Objective

• To address the limitations of conventional modal analysis in capturing time-integrated energy accumulation from repeated random disturbances in power systems.
• To develop a physically motivated framework that links modal interactions to total energy over time, rather than instantaneous dynamics.
• To enable rapid, real-time identification of critical stability phenomena—such as resonance, mode merging, and instability—by associating them with specific state variables.
• To provide a computationally efficient method for analyzing critical modes in large-scale systems by focusing on a selected part of the spectrum independently.

Proposed method

• Proposes a Lyapunov modal analysis (LMA) framework that integrates selective modal analysis with spectral decomposition of custom Lyapunov functions to quantify time-integrated energy from system modes.
• Introduces Lyapunov participation factors (LPFs) that measure the contribution of modes and state variables to the energy of Lyapunov functions over finite or infinite time intervals.
• Uses spectral decomposition of Lyapunov functions to estimate the total energy produced by mutual actions of modes over time, enabling analysis of modal interactions in terms of energy accumulation.
• Defines pair-wise interaction indicators such as Lyapunov Modal Interaction Functions (LMIFs) and Modal Interaction Lyapunov Participation Factors (MISLPFs) to detect resonant interactions and mode merging.
• Applies the method to a two-area four-machine power system to validate detection of instability, resonance, and mode merging through LPFs and MISLPFs.
• Demonstrates that Lyapunov indicators can be computed independently for critical spectral components, enabling fast real-time stability monitoring without full-spectrum computation.

Experimental results

Research questions

• RQ1 How can modal interactions in power systems be characterized in terms of accumulated energy over time rather than instantaneous dynamics?
• RQ2 What is the role of specific state variables in the energy buildup associated with resonant or unstable modes?
• RQ3 Can Lyapunov participation factors reliably detect the onset of instability, mode merging, or resonant interactions in small-signal stability analysis?
• RQ4 To what extent can Lyapunov-based indicators be computed independently for critical modes, enabling real-time stability monitoring in large-scale systems?

Key findings

• The proposed Lyapunov modal analysis (LMA) successfully identifies resonant modal interactions, mode merging, and loss of stability in a two-area four-machine power system using only a linear model.
• Lyapunov participation factors (LPFs) associated with unstable modes such as S1 show a participation approaching unity as instability is approached, confirming loss of stability.
• When aperiodic modes S4 and S5 merge into a single oscillation, their participations in other modes increase before merging and sharply rise afterward, indicating a transition phase.
• Resonant interaction between oscillatory modes S6 and S7 is detected by a characteristic increase in mutual participation with opposite signs as their frequencies approach each other.
• State variables such as Δ(ω₂−ω₃) and Δ(ω₄−ω₃) are identified as highly sensitive to the resonant interaction between S6 and S7 near αc ≈ 1.375, with MISLPFs and state participations in LMIEs confirming their dominant role.
• The magnitude of state participations in Lyapunov energy (LMIEs) is robust to the presence of unstable modes, as the Lyapunov energy of S6 and S7 is independent of S1, ensuring reliable detection of interactions.