[Paper Review] A Hierarchical Approach to Stability Assessment of Large Scale Interconnected Networks

Thanh Long Vu, Konstantin Turitsyn|arXiv (Cornell University)|Mar 17, 2016

Control and Stability of Dynamical Systems10 references3 citations

TL;DR

This paper proposes a scalable hierarchical framework for assessing transient stability in large-scale nonlinear interconnected networks, such as power grids, by independently analyzing subsystem stability and enforcing diagonal dominance on a structure matrix of subsystem dynamics and interconnection gains. The method guarantees stability via a linear combination of subsystem Lyapunov functions when the structure matrix is diagonally dominant, enabling efficient, low-complexity analysis of systems with multiple equilibria and local stability only.

ABSTRACT

Interconnected networks describe the dynamics of important systems in a wide range such as biological systems and electrical power grids. Some important features of these systems were successfully studied and understood through simplified model of linear interconnection of linear subsystems, where provably global properties, e.g. global convergence to a specific state, usually hold true. However, in severely disturbed conditions many of those systems exhibit strongly nonlinear behaviour. Particularly, multiple equilibrium points may coexist and make the dynamical behavior of the system difficult to predict. Aiming at understanding the fragility of interconnected systems, we will provide a hierarchical framework to assess the metastability and resilience of such systems. This framework is based on independently characterizing stability of individual subsystems when they are uncoupled from the network, and then enforcing the diagonal dominance property on a structure matrix capturing the subsystems stability and the input-to-output gains of interconnection network. Since the subsystems are usually of low order and the structure matrix has size equal to the number of subsystems, this framework is easy to implement and thus scalable to large scale interconnected systems. Possible application of this framework in assessing stability of microgrids will be discussed at the end of this paper.

Motivation & Objective

• To address the challenge of stability assessment in large-scale interconnected systems that exhibit strongly nonlinear behavior and multiple equilibria, especially under severe disturbances.
• To overcome the limitations of global stability analysis in nonlinear systems where only local stability can be guaranteed.
• To develop a computationally scalable method for stability certification in systems with thousands to millions of components, such as power grids.
• To provide a practical tool for assessing metastability and resilience in complex networks, particularly in microgrids and power systems.
• To reduce conservativeness in stability certificates by leveraging a family of Lyapunov functions and adaptive optimization.

Proposed method

• The method decouples the analysis into two steps: (1) characterizing the stability of each individual nonlinear subsystem in isolation using Lyapunov functions, and (2) estimating the input-to-output gains of interconnections between subsystems.
• A structure matrix M is constructed to encode subsystem stability properties and interconnection gains, with the matrix derived from subsystem dynamics and coupling strengths.
• Diagonal dominance of the structure matrix M is enforced as a sufficient condition for the existence of a common Lyapunov function for the entire network.
• The stability condition is formulated as a linear matrix inequality (LMI) with matrix size equal to the number of subsystems, enabling efficient computation.
• The framework uses a family of Lyapunov functions to construct inner approximations of the region of attraction, improving accuracy and reducing conservativeness.
• An adaptation algorithm based on convex optimization (e.g., using CVX in MATLAB) is applied to compute the optimal positive diagonal matrix C that certifies stability for a given initial state.

Experimental results

Research questions

• RQ1 Can a scalable framework be developed to assess transient stability in large-scale interconnected networks with multiple equilibria and local stability only?
• RQ2 How can the stability of nonlinear interconnected systems be guaranteed without relying on global stability assumptions?
• RQ3 What conditions on the interconnection structure ensure the existence of a composite Lyapunov function for the entire network?
• RQ4 How can the computational complexity be reduced while maintaining accuracy in stability certification for large systems?
• RQ5 Can the conservativeness of stability certificates be reduced through adaptive selection of Lyapunov functions?

Key findings

• The proposed framework guarantees stability of the interconnected system if the structure matrix M is diagonally dominant, ensuring the existence of a composite Lyapunov function via a positive diagonal matrix C.
• For a 20-node network with time-varying interconnections, the method successfully certified convergence to the origin from an initial state with V(x(0)) = 14.4952 and Vmin = 17.9277, confirming asymptotic stability.
• The method is computationally scalable, as the LMI size depends only on the number of subsystems, not the full system dimension.
• The framework enables explicit inner approximation of the region of attraction using a family of Lyapunov functions, improving the precision of stability certificates.
• The approach was validated numerically on a looped network with 20 subsystems, demonstrating convergence of the Lyapunov function V(x) to zero over time.
• The method is applicable to uncertain interconnected networks and can be extended to descriptor systems used in power system modeling.