[Paper Review] On the region of attraction of phase-locked states for swing equations on connected graphs with inhomogeneous dampings

Young-Pil Choi, Zhuchun Li|arXiv (Cornell University)|Oct 26, 2016

Nonlinear Dynamics and Pattern Formation12 references3 citations

TL;DR

This paper develops a novel energy-based method to estimate the region of attraction for phase-locked states in lossless power grids modeled as second-order Kuramoto oscillators on connected graphs with inhomogeneous dampings. By constructing a refined energy functional, the authors derive a sufficient condition for synchronization that relaxes prior assumptions of homogeneous damping and bounded graph diameter, enabling analysis of general network topologies and realistic power grid dynamics.

ABSTRACT

We consider the synchronization problem of swing equations, a second-order Kuramoto-type model, on connected networks with inhomogeneous dampings. This was largely motivated by its relevance to the dynamics of power grids. We focus on the estimate of the region of attraction of synchronous states which is a central problem in the transient stability of power grids. In the recent literature, Dörfler, Chertkov, and Bullo [Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 2005-2010] found a condition for the synchronization in smart grids. They pointed out that the region of attraction is an important unsolved problem. In [SIAM J. Control Optim., 52 (2014), pp. 2482-2511], only a special case was considered where the oscillators have homogeneous dampings and the underlying graph has a diameter less than or equal to 2. There the analysis heavily relies on these assumptions; however, they are too strict compared to the real power networks. In this paper, we continue the study and derive an estimate on the region of attraction of phase-locked states for lossless power grids on connected graphs with inhomogeneous dampings. Our main strategy is based on the gradient-like formulation and energy estimate. We refine the assumptions by constructing a new energy functional which enables us to consider such general settings.

Motivation & Objective

• Address the critical challenge of transient stability in power grids by estimating the region of attraction for phase-locked states.
• Overcome limitations of prior work that required homogeneous damping and small graph diameter (≤2), which are unrealistic for real power networks.
• Develop a general framework applicable to connected graphs with arbitrary topology and heterogeneous damping coefficients.
• Provide a quantitative, energy-based sufficient condition for synchronization that is computable and analyzable under general settings.
• Enable the analysis of transient stability for complex, large-scale power systems with stochastic renewable integration.

Proposed method

• Propose a gradient-like formulation of the swing equations to model second-order Kuramoto dynamics on connected graphs.
• Construct a new energy functional that accounts for inhomogeneous damping and network connectivity, enabling energy-based analysis.
• Apply energy estimates to derive a sufficient condition for convergence to phase-locked states, based on initial phase and frequency deviations.
• Introduce parameters $ D_0 $ and $ \varepsilon $ to control the size of the estimated region of attraction, with $ D_0 $ bounding phase differences and $ \varepsilon $ controlling damping heterogeneity.
• Use numerical simulations with a fourth-order Runge-Kutta method to validate and visualize the estimated region of attraction for two-oscillator networks.
• Leverage the energy method to avoid reliance on phase-difference analysis, which fails under general connectivity and heterogeneity.

Experimental results

Research questions

• RQ1 What is a sufficient condition for the existence of a region of attraction for phase-locked states in lossless power grids with inhomogeneous dampings?
• RQ2 How can the region of attraction be estimated in connected networks without assuming homogeneous damping or diameter ≤2?
• RQ3 Can a refined energy functional be constructed to handle general network topologies and heterogeneous damping coefficients?
• RQ4 How do the parameters $ D_0 $ and $ \varepsilon $ influence the size and shape of the estimated region of attraction?
• RQ5 To what extent is the proposed estimate conservative, and can it be improved by refining the energy functional?

Key findings

• The proposed energy-based method successfully estimates the region of attraction for phase-locked states in connected graphs with inhomogeneous dampings, relaxing prior restrictive assumptions.
• The size of the region of attraction increases with larger values of $ D_0 $, which bounds the initial phase differences, indicating that larger initial phase deviations can still lead to synchronization.
• Smaller values of $ \varepsilon $, which quantify damping heterogeneity, yield larger regions of attraction, suggesting that less variation in damping improves robustness to initial conditions.
• Numerical simulations confirm that the method is conservative: initial conditions outside the estimated region still lead to synchronization, indicating room for improvement.
• The method enables analysis of transient stability in realistic power grids with stochastic renewable sources, where large disturbances and heterogeneous dynamics are common.
• The framework is general and does not rely on phase-difference analysis, making it suitable for arbitrary connected network topologies and non-uniform damping parameters.