We perform modal analysis of short-term swing dynamics in multi-machine power systems. The analysis is based on the so-called Koopman operator, a linear, infinite-dimensional operator that is defined for any nonlinear dynamical system and captures full information of the system. Modes derived through spectral analysis of the Koopman operator, called Koopman modes, provide a nonlinear extension of linear oscillatory modes. Computation of the Koopman modes extracts single-frequency, spatial modes embedded in non-stationary data of short-term, nonlinear swing dynamics, and it provides a novel technique for identification of coherent swings and machines.

The loss of stability-an instability-can become a critical cause of emergent cascading outages leading to wide-spread blackouts. Penetration of renewable energy sources makes the problem of instability more urgent because of the highly fluctuating nature of such sources. Here we show a data-based approach to stability assessment of power systems without models. This approach is enabled by Koopman mode analysis for nonlinear dynamical systems, which detects an instability based on the properties of the point spectrum of the Koopman operator. We apply the technique to data on physical power flows sampled from the two major accidents, the 2011 Arizona-Southern California grid outage and the 2006 system disturbance of the European interconnected grid, and successfully detect unstable power flow patterns that govern the complex dynamics occurring during the accidents.

We interpret and explain a phenomenon in short-term swing dynamics of multi-machine power grids that we term the Coherent Swing Instability (CSI). This is an undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines in a sub-grid coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance. We develop a minimal mathematical model of CSI for synchronous machines that are strongly coupled in a loop transmission network and weakly connected to the infinite bus. This model provides a dynamical origin of CSI: it is related to the escape from a potential well, or, more precisely, to exit across a separatrix in the dynamical system for the amplitude of the weak nonlinear mode that governs the collective motion of the machines. The linear oscillations between strongly coupled machines then act as perturbations on the nonlinear mode. Thus we reveal how the three different mode oscillations—local plant, inter-machine, and inter-area modes—interact to destabilize a power grid. Furthermore, we present a phenomenon of short-term swing dynamics in the New England (NE) 39-bus test system, which is a well-known benchmark model for power grid stability studies. Using a partial linearization of the nonlinear swing equations and the proper orthonormal decomposition, we show that CSI occurs in the NE test system, because it is a dynamical system with a nonlinear mode that is weak relative to the linear oscillatory modes.

We suggest a precursor to phenomena of loss of transient stability in multi-machine power systems. This precursor is based on discovery of [Y. Susuki, I. Mezić , and T. Hikihara, J. Nonlinear Sci., vol. 21, no. 3, pp. 403-439, June 2011], an emergent transmission path of energy from many oscillatory modes to one oscillatory mode that represents an instability phenomenon of interest. The pathway from high frequency modes to the lowest frequency mode is called the coherent swing instability (CSI). The modes are extracted from sensor data or data provided by simulation outputs of power system oscillations by using the Koopman mode analysis that is based on a fully nonlinear spectral theory and represents an extension of linear oscillatory mode analysis. The CSI transmission path is identified by computation of the so-called action transfer operator that is derived by refining a mathematical model of transient stability using the Koopman mode analysis. This provides a new technique for monitoring the loss of transient stability by a combination of practical data, mathematical modeling, and computation.

This paper applies a new technique for modal decomposition based solely on measurements to test systems and demonstrates the technique's capability for partitioning a power network, which determines the points of separation in an islanding strategy. The mathematical technique is called the Koopman mode analysis (KMA) and stems from a spectral analysis of the so-called Koopman operator. Here, KMA is numerically approximated by applying an Arnoldi-like algorithm recently first applied to power system dynamics. In this paper we propose a practical data-driven algorithm incorporating KMA for network partitioning. Comparisons are made with two techniques previously applied for the network partitioning: spectral graph theory which is based on the eigenstructure of the graph Laplacian, and slow-coherency which identifies coherent groups of generators for a specified number of low-frequency modes. The partitioning results share common features with results obtained with graph theory and slow-coherency-based techniques. The suggested partitioning method is evaluated with two test systems, and similarities between Koopman modes and Laplacian eigenvectors are showed numerically and elaborated theoretically.