Data-driven Static Equivalence with Physics-informed Koopman Operators

With deployment of measurement units,fitting static equivalent models of distribution networks(DNs)by linear regression has been recognized as an effective method in power flow analysis of a transmission network.Increasing volatility of measurements caused by variable distributed renewable energy sources makes it more difficult to accurately fit such equivalent models.To tackle this challenge,this letter proposes a novel data-driven method to improve equivalency accuracy of DNs with distributed energy resources.This letter provides a new perspective that an equivalent model can be regarded as a mapping from internal conditions and border voltages to border power injections.Such mapping can be established through 1)Koopman operator theory,and 2)physical features of power flow equations at the root node of a DN.Performance of the proposed method is demonstrated on the IEEE 33-bus and IEEE 136-bus test systems connected to a 661-bus utility system.

Data-driven Estimation for a Region of Attraction for Transient Stability Using the Koopman Operator

This paper presents a novel Koopman Operator based framework to estimate the region of attraction for power system transient stability analysis.The Koopman eigenfunctions are used to numerically construct a Lyapunov function.Then the level set of the function is utilized to estimate the boundary of the region of attraction.The method provides a systematic method to construct the Lyapunov function with data sampled from the state space,which suits any power system models and is easy to use compared to traditional Lyapunov direct methods.In addition,the constructed Lyapunov function can capture the geometric properties of the region of attraction,thus providing useful information about the instability modes.The method has been verified by a simple illustrative example and three power system models,including a voltage source converter interfaced system to analyze the large signal synchronizing instability induced by the phase lock loop dynamics.The proposed method provides an alternative approach to understanding the geometric properties and estimating the boundary of the region of attraction of power systems in a data driven manner.Index Terms-Koopman operator,lyapunov function,power system transient stability,region of attraction.

Data-driven prediction of temperature variations in an open cathode proton exchange membrane fuel cell stack using Koopman operator

In this study,a novel application of the Koopman operator for control-oriented modeling of proton exchange membrane fuel cell(PEMFC)stacks is proposed.The primary contributions of this paper are:(1)the design of Koopman-based models for a fuel cell stack,incorporating K-fold cross-validation,varying lifted dimensions,radial basis functions(RBFs),and prediction horizons;and(2)comparison of the performance of Koopman-based approach with a more traditional physics-based model.The results demonstrate the high accuracy of the Koopman-based model in predicting fuel cell stack behavior,with an error of less than 3%.The proposed approach offers several advantages,including enhanced computational efficiency,reduced computational burden,and improved interpretability.This study demonstrates the suitability of the Koopman operator for the modeling and control of PEMFCs and provides valuable insights into a novel control-oriented modeling approach that enables accurate and efficient predictions for fuel cell stacks.

Determination of Parameters of Time-delayed Embedding Algorithm Using Koopman Operator-based Model Predictive Frequency Control

Power systems around the world have been registering a degenerating inertial response in view of the growth of inverter-based resources along with the withdrawal of conventional coal units.Therefore,there is a need for swift frequency support and its control,preferably by means of power electronic-interfaced storage devices,owing to their beneficial capabilities.Despite being particularly efficient,pragmatically,the traditional model-based non-linear control techniques are not highly popular in power system control design,primarily due to the complications faced in obtaining accurately suitable models for certain power system components.Lately,the modelfree Koopman operator-based model predictive control(KMPC)has proven to be highly conducive for data-driven non-linear control design.The principle behind KMPC is to change the coordinates in a manner to get an approximately linear model,which can then be controlled using a linear model predictive control.In this study,we employed time-delayed embedding of measurements to reconstruct a new set of preferable coordinates,thereby suggesting an approach for finding the optimal number of time lags and the embedding dimensions which are the key parameters of this algorithm.The efficacy of this KMPC framework is established by adopting a decentralized frequency control problem through a decoupled synchronous machine system,which we proposed for both the Kundur two-area system as well as the IEEE 39-bus test system.

Information flow between stock markets:A Koopman decomposition approach

Stock markets in the world are linked by complicated and dynamical relationships into a temporal network.Extensive works have provided us with rich findings from the topological properties and their evolutionary trajectories,but the underlying dynamical mechanism is still not in order.In the present work,we proposed a technical scheme to reveal the dynamical law from the temporal network.The index records for the global stock markets form a multivariate time series.One separates the series into segments and calculates the information flows between the markets,resulting in a temporal market network representing the state and its evolution.Then the technique of the Koopman decomposition operator is adopted to find the law stored in the information flows.The results show that the stock market system has a high flexibility,i.e.,it jumps easily between different states.The information flows mainly from high to low volatility stock markets.And the dynamical process of information flow is composed of many dynamic modes distribute homogenously in a wide range of periods from one month to several ten years,but there exist only nine modes dominating the macroscopic patterns.

GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators partial derivative(t) and its dual, creation operators partial derivative(t)*.

Koopman analysis of nonlinear systems with a neural network representation

The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields.The intrinsic complexity of their dynamics defies many existing tools based on individual orbits,while the Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits,which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator.However,it is difficult to identify and represent the most relevant eigenfunctions in practice.Here,combined with the Koopman analysis,a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems.By invoking the error minimization,a fundamental set of Koopman eigenfunctions are derived,which may reproduce the input dynamics through a nonlinear transformation provided by the neural network.The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.