Yoshihiko Susuki

Yoshihiko Susuki, Igor Mezic, Fredrik Raak et al.

Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.

Yoshihiko Susuki, Igor Mezić

This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set. We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction. We provide a characterization of invariant sets in the quasiperiodically forced systems. A theoretical result on uniform boundedness of the invariant sets is presented. The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages. Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability. We show that our theoretical results can be used to understand stability regions in such complex systems.

Naoto Mizuta, Yoshihiko Susuki, Yutaka Ota et al.

We develop an algorithm for synthesizing a spatial pattern of charging/discharging operations of in-vehicle batteries for provision of Ancillary Service (AS) in power distribution grids. The algorithm is based on the ODE (Ordinary Differential Equation) model of distribution voltage that has been recently introduced. In this paper, firstly, we derive analytical solutions of the ODE model for a single straight-line feeder through a partial linearization, thereby providing a physical insight to the impact of spatial EV charging/discharging to the distribution voltage. Second, based on the analytical solutions, we propose an algorithm for determining the values of charging/discharging power (active and reactive) by in-vehicle batteries in the single feeder grid, so that the power demanded as AS (e.g. a regulation signal to distribution system operator for primary frequency control reserve) is provided by EVs, and the deviation of distribution voltage from a nominal value is reduced in the grid. Effectiveness of the algorithm is established with numerical simulations on the single feeder grid and on a realistic feeder grid with multiple bifurcations.

Yoshihiko Susuki, Ryo Hamasaki, Atsushi Ishigame

We report a new approach to estimating power system inertia directly from time-series data on power system dynamics. The approach is based on the so-called Koopman Mode Decomposition (KMD) of such dynamic data, which is a nonlinear generalization of linear modal decomposition through spectral analysis of the Koopman operator for nonlinear dynamical systems. The KMD-based approach is thus applicable to dynamic data that evolve in nonlinear regime of power system characteristics. Its effectiveness is numerically evaluated with transient stability simulations of the IEEE New England test system.

Qiang Wu, T. John Koo, Yoshihiko Susuki

Dynamic security analysis is an important problem of power systems on ensuring safe operation and stable power supply even when certain faults occur. No matter such faults are caused by vulnerabilities of system components, physical attacks, or cyber-attacks that are more related to cyber-security, they eventually affect the physical stability of a power system. Examples of the loss of physical stability include the Northeast blackout of 2003 in North America and the 2015 system-wide blackout in Ukraine. The nonlinear hybrid nature, that is, nonlinear continuous dynamics integrated with discrete switching, and the high degree of freedom property of power system dynamics make it challenging to conduct the dynamic security analysis. In this paper, we use the hybrid automaton model to describe the dynamics of a power system and mainly deal with the index-1 differential-algebraic equation models regarding the continuous dynamics in different discrete states. The analysis problem is formulated as a reachability problem of the associated hybrid model. A sampling-based algorithm is then proposed by integrating modeling and randomized simulation of the hybrid dynamics to search for a feasible execution connecting an initial state of the post-fault system and a target set in the desired operation mode. The proposed method enables the use of existing power system simulators for the synthesis of discrete switching and control strategies through randomized simulation. The effectiveness and performance of the proposed approach are demonstrated with an application to the dynamic security analysis of the New England 39-bus benchmark power system exhibiting hybrid dynamics. In addition to evaluating the dynamic security, the proposed method searches for a feasible strategy to ensure the dynamic security of the system in face of disruptions.

Yoshihiko Susuki, Natsuki Katayama, Alexandre Mauroy et al.

This paper proposes a novel formulation of frequency response for nonlinear systems in the Koopman operator framework. This framework is a promising direction for the analysis and synthesis of systems with nonlinear dynamics based on (linear) Koopman operators. We show that the frequency response of a nonlinear plant is derived through the Laplace transform of the output of the plant, which is a generalization of the classical approach to LTI plants and is guided by the resolvent theory of Koopman operators. The response is a complex-valued function of the driving angular frequency, allowing one to draw the so-called Bode plots, which display the gain and phase characteristics. Sufficient conditions for the existence of the frequency response are presented for three classes of dynamics.

Zhicheng Zhang, Yoshihiko Susuki, Atsushi Okazaki

Convective features, represented here as warm bubble-like patterns, reveal essential high-level information about how short-term weather dynamics evolve within a high-dimensional state space. In this paper, we introduce a data-driven framework that uncovers transient dynamics captured by Koopman modes responsible for these structures and traces their emergence, growth, and decay. Our approach applies the sparsity-promoting dynamic mode decomposition to weather simulations, yielding a few number of selected modes whose sparse amplitudes highlight dominant transient structures. By tuning the sparsity weight, we balance reconstruction accuracy and model complexity. We illustrate the methodology on weather simulations, using the magnitude of velocity and vorticity fields as distinct observable datasets. The resulting sparse dominant Koopman modes capture the transient evolution of bubble-like pattern and can reduce the dimensionality of weather system model, offering an efficient surrogate for diagnostic and forecasting tasks.

Natsuki Katayama, Alexandre Mauroy, Yoshihiko Susuki

The Koopman operator framework enables global analysis of nonlinear systems through its inherent linearity. This study aims to clarify spectral properties of the Koopman operators for nonlinear systems with control inputs. To this end, we treat the inputs as parameters throughout this paper. We then introduce the Koopman operator for a parameterized dynamical system with a globally exponentially stable equilibrium point and analyze how eigenfunctions of the operator depend on the parameter. As a main result, we obtain a global linearization, which enables one to transform the nonlinear system into a finite-dimensional linear system, and we show that it depends continuously on the parameter. Subsequently, for a control-affine system, we investigate a condition under which the transformation providing a global bilinearization does not depend on the parameter. This provides the condition under which the global bilinearization for the control-affine system is independent of the parameter.

Akitoshi Masuda, Yoshihiko Susuki, Manel Martínez-Ramón et al.

Koopman Mode Decomposition (KMD) is a technique of nonlinear time-series analysis that originates from point spectrum of the Koopman operator defined for an underlying nonlinear dynamical system. We present a numerical algorithm of KMD based on Gaussian process regression that is capable of handling noisy finite-time data. The algorithm is applied to short-term swing dynamics of a multi-machine power grid in order to estimate oscillatory modes embedded in the dynamics, and thereby the effectiveness of the algorithm is evaluated.

Marcos Netto, Yoshihiko Susuki, Lamine Mili

This paper develops a novel data-driven technique to compute the participation factors for nonlinear systems based on the Koopman mode decomposition. Provided that certain conditions are satisfied, it is shown that the proposed technique generalizes the original definition of the linear mode-in-state participation factors. Two numerical examples are provided to demonstrate the performance of our approach: one relying on a canonical nonlinear dynamical system, and the other based on the two-area four-machine power system. The Koopman mode decomposition is capable of coping with a large class of nonlinearity, thereby making our technique able to deal with oscillations arising in practice due to nonlinearities while being fast to compute and compatible with real-time applications.