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The so-called Koopman operator framework has attracted a lot of interest in dynamical systems theory and control theory. In this note we extend the framework to systems described by nonlinear semi-explicit Differential-Algebraic Equations (DAEs). A complete spectral characterization of DAE systems with asymptotically stable equilibrium points is presented. A difficulty of the Koopman operator framework is also pointed out for a planar DAE system with forward impasse point.

The so-called short-term voltage stability is typical of nonlinear problems arising in dynamic performance of electric power systems. This paper reports a simulation result on load margin for short-term voltage stability of the IEEE 9-bus system connected to a multi-terminal DC system based on voltage-source converter (VSC). The simulation shows that the introduction of AC voltage control in VSC increases the load margin under a step change in load on the AC system, thereby showing a clear benefit of the AC voltage control for enhancing the short-term voltage stability without harnessing the AC system.

We study the scale dependence of effective diffusion of fluid tracers, specifically, its dependence on the P\'{e}clet number, a dimensionless parameter of the ratio between advection and molecular diffusion. Here, we address the case that length and time scales on which the effective diffusion can be described are not separated from those of advection and molecular diffusion. For this, we propose a new method for characterizing the effective diffusivity without relying on the scale separation. For a given spatial domain inside which the effective diffusion can emerge, a time constant related to the diffusion is identified by considering the spatio-temporal evolution of a test advection-diffusion equation, where its initial condition is set at a pulse function. Then, the value of effective diffusivity is identified by minimizing the $L_\infty$ distance between solutions of the above test equation and the diffusion one with mean drift. With this method, for time-independent gyre and time-periodic shear flows, we numerically show the scale dependence of the effective diffusivity and its discrepancy from the classical limits that were derived on the assumption of the scale separation. The kinematic origins of the discrepancy are revealed as the development of the molecular diffusion across flow cells of the gyre and as the suppression of the drift motion due to a temporal oscillation in the shear., Comment: 11 pages, 5 figures

This paper proposes a mode-in-state contribution factor for a class of nonlinear dynamical systems by utilizing spectral properties of the Koopman operator and sensitivity analysis. Using eigenfunctions of the Koopman operator for a target nonlinear system, we show that the relative contribution between modes and state variables can be quantified beyond a linear regime, where the nonlinearity of the system is taken into consideration. The proposed contribution factor is applied to the numerical analysis of large-signal simulations for an interconnected AC/multi-terminal DC power system., Comment: 11 pages, 2 figures

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., Comment: 20 pages, 8 figures

The motivation of our research is to pursue a possibility of utilizing electric vehicles for the future operation of power transmission and distribution grids. In this paper, we report a Power-Hardware-In-the-Loop (Power-HIL) testing on the provision of Ancillary Service (AS) by in-vehicle batteries. The AS of our interest is multi-objective in the sense that it simultaneously provides both primary frequency control reserve for a transmission grid and voltage support for a high-voltage distribution grid. The Power-HIL testing is crucial to validating the Multi-Objective AS because the dynamics of the grids and of power conditioning systems emerge in a common time scale, latter of which involves modeling difficulties and thus needs their inclusion as real physical devices, that is, Power-HIL. We show that the Multi-Objective AS is simulated consistently in a Power-HIL testbed and works effectively in a dynamic situation of transmission and distribution grids.

This paper pursues to construct a theoretical framework which can efficiently capture the dynamics of large-scale heterogeneous power grids. We formulate a networked nonlinear descriptor system con...

We develop a system-level design for the provision of Ancillary Service (AS) for control of electric power grids by in-vehicle batteries, suitably applied to Electric Vehicles (EVs) operated in a sharing service. An architecture for cooperation between transportation and energy management systems is introduced that enables us to design an autonomous Vehicle-to-Grid (V2G) for the provision of multi-objective AS: primary frequency control in a transmission grid and voltage amplitude regulation in a distribution grid connected to EVs. The design is based on the ordinary differential equation model of distribution voltage, which has been recently introduced as a new physics-based model, and is utilized in this paper for assessing and regulating the impact of spatiotemporal charging/charging of a large population of EVs to a distribution grid. Effectiveness of the autonomous V2G design is evaluated with numerical simulations of realistic models for transmission and distribution grids with synthetic operation data on EVs in a sharing service. In addition, we present a hardware-in-the-loop test for evaluating its feasibility in a situation where inevitable latency is involved due to power, control, and communication equipments., 28 pages, 23 figures

The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems -- the Koopman semigroup and its associated Koopman generator -- plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on non-stationary Koopman modes., Comment: 24 pages, 2 figures

This introductory chapter provides an overview of the Koopman operator framework. We present basic notions and definitions, including those related to the spectral properties of the operator. We also review numerical methods that are used to represent the operator on a finite-dimensional basis and to compute its spectral properties from data (DMD algorithm and Arnoldi-based methods). Finally, several extensions of the Koopman operator to input–output systems are reviewed.

This chapter develops a methodology for modeling of heat transfer dynamics in a building atrium via Koopman mode decomposition (KMD). We address the phenomenon of heat transfer due to movement of air inside an atrium, where the air slowly moves over the distance between individual rooms. The heat transfer is modeled as a heat advection equation with a vector-valued velocity coefficient and a nonlinear input term from heating, ventilation, and air-conditioning (HVAC) operations, which corresponds to a control variable. KMD is applied to the equation so that the heat transfer and HVAC operation are characterized in terms of frequencies and wavenumber vectors of Koopman modes (KMs). The velocity coefficient is then identified with a KM governing the heat transfer. The effectiveness of the modeling is demonstrated with measurement data on temperature field and HVAC operation of a practically used building. A possibility of how to use this modeling for control of heat transfer dynamics is discussed at the end of this chapter.

Data-driven analysis of dynamic performance has attracted a lot of interest in the modern power grid with highly accurate measurement technologies such as Synchrophasor. In this chapter, we report the research effort to do this for voltage dynamics in a rudimentary model of power grids. We use the combination of the so-called delay embedding with the extended dynamic mode decomposition, which is a technique for approximating the Koopman operator directly from time-series data. This combination enables us to approximate the spectral properties of the Koopman operator through voltage–amplitude measurement of a single bus. The spectral properties clarify not only the local information on the voltage dynamics such as frequency/damping in transient responses but also global one such as the flow structure of the underlying mathematical model.

Data-based assessment of dynamic performances has attracted a lot of interest in the modern power system with high-accurate measurement technologies such as Synchrophasor. In this paper, we report the research effort to do this for voltage dynamics in a rudimentary model of power systems. We propose to combine the so-called delay embedding with the extended dynamic mode decomposition, which is a technique for approximating the Koopman operator directly from time-series data. This combination enables us to approximate spectral properties of the Koopman operator through voltage measurement of a single bus. The spectral properties clarify local information on the voltage dynamics such as frequency/damping in transient responses and global one such as the flow structure of the underlying mathematical model.

The concept of microgrid has been proposed as an effective method for utilizing distributed multiple generation units including renewable energy. A microgrid connected to the commercial grid has a potential to contribute to not only short-term output levelling by maintaining the output power [kW] at reference but also long-term balancing of electricity supply and demand by regulating the actual amount of output energy [kWh]. In this paper, we propose a new approach to optimal supply planning of a microgrid considering the imbalance (i.e. the difference between planned and actual amounts of electric energy). By evaluating the approach numerically, we show that the minimization of imbalance contributes to not only the supply-demand balancing but also its economically-efficient operation in terms of the sum of the running and imbalance costs.