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This chapter addresses the problem of power system modeling, with emphasis on the development of aggregate microgrid (MG) models for electromechanical stability studies. It considers both deterministic and stochastic models for distribution and transmission applications and introduces the concept of power system dynamic equivalencing. The chapter presents a systematic methodology for modeling distributed MGs for integration studies. This framework can be used to investigate the integration of high penetrations of distributed energy resources/MGs energy into the system. The chapter focuses of on the derivation of low‐frequency models to be used in stability studies. Developing MG equivalent models is of high importance when dealing with MGs‐integrated power system stability analysis. The equivalent model should consider uncertainties related to output power fluctuations and different operating modes of the sources.

This chapter describes the experience in the analysis of wind and solar integration in largescale power grids with complex dynamics and operating characteristics. It presents the methods for interpreting and understanding the impact of distributed renewable energy resources on global oscillatory behavior and discusses the processes involved in development and testing of tools for studying wind and hydro generation. The chapter also discusses the issues of network reactive control performance and tuning, inertia reduction, generation commitment, grid stability, and power system frequency response in detail. Steady‐state eigenvalue analyses and detailed step‐by‐step simulations are used for assessing issues regarding wind and solar integration. The advent of modern, inverter‐based wind and solar farms offers a direct means for controlling system voltage. Specifically, the wind farms in the south systems are expected to provide local voltage support.

Modern power grids, characterized by high penetration of inverter‐interfaced generation sources, suffer from a lack of rotational inertia and governor control, which may cause faster frequency dynamics and larger frequency deviations and even instability problem. This chapter discusses advanced inertia‐emulation‐based control strategies to enable high penetration levels of microgrid (MG). These approaches combine an adaptive energy storage system (ESS) dispatch strategy with an MG‐controlled islanding scheme to provide frequency support and enhance small‐signal stability and voltage regulation. Dispatching the ESSs in the buses associated with the generators in the desired region and disconnecting the multi‐MGs (MMGs) in other buses improve the system performance. Disconnecting the MMGs from the network in a time horizon greater than the response time of ESS increases the frequency evolution to satisfy the rolling window criterion.

The development of synchronized phasor measurement units technology has given utilities the ability to implement wide‐area voltage stability monitoring systems, which provide time‐stamped data in near real time. In this sense, selected voltage control areas can be used to implement wide‐area voltage–volt–ampere reactive control schemes, develop voltage monitoring and prediction schemes, monitor reactive power reserve requirements, and develop (localized) reactive power markets, just to mention a few potential applications. This chapter presents a novel use of measurement‐based analysis techniques to identify weakly interacting voltage control areas in power systems with high penetration of distributed energy resources. Attention is focused on three main aspects, namely the identification of critical system zones showing a coherent behavior and the associated reactive power sources, the use of spectral analysis to identify the critical buses, and the computation of reduced‐order models. Connections with modal identification methods are also investigated and numerical issues are discussed.

This chapter discusses special attributes of the specific power electronic‐based technologies to enhance system dynamics. It also discusses optimal placement of the technologies in detail. After a discussion on the application of virtual synchronous generators (VSGs) in power system stability and performance improvement, optimal placement of dispatchable inertia besides the associated realization mechanisms would be emphasized. Without the loss of generality, while distributed generations (such as wind farms), high‐voltage direct current systems, and energy storage systems (ESSs) could emulate virtual inertia, the ESSs are considered as the source of virtual inertia in this chapter. Manipulation of the inertia constant by the VSGs could enhance system dynamics and stability. This could be realized using inertia‐based control schemes. Indeed, inertia provision plays an important role in designing advanced control schemes in modern power grids. Optimal placement of virtual inertia can be studied in two perspectives: frequency stability and small‐signal stability.

Knowledge of the fundamental characteristics of global oscillation modes provides valuable information about the stability of oscillatory phenomena and may help to identify machines and wind and solar photovoltaic (PV) farms and their control systems involved in the exchange of oscillating energy and the design or modification of controllers. This chapter identifies basic issues important in understanding the oscillatory performance of wind and solar PV penetrated power systems. It presents a fundamental study of the characterization of the dynamic behavior of power systems with increased renewable generation. The chapter discusses factors that affect the performance of the method. It investigates connections with other modal identification methods. The chapter also investigates data‐driven approaches that supplement information on conventional small‐signal performance. It discusses the challenges that need to be confronted for modal characterization techniques to be more widely applied by the power system community in a large‐scale power system.

A novel, general-purpose recursive linearization (RL) methodology for the analysis of small-signal stability that combines the use of chain rule-based techniques with perturbation theory is proposed. The method provides a straightforward and fast alternative to the linearization of complex systems described by ordinary differential equations (ODE) models that can be used to reduce the computational effort as well as to extend the applicability of perturbed Koopman mode analysis (PKMA) and the method of normal forms (MNF). First, the concept of recursive linearization is introduced to aid in the computation of higher-order approximations. Techniques for exploiting the structure of the expansions are then provided and relationships for computing second- and higher-order terms of the power series expansion of a real- or complex-valued nonlinear analytical model are developed. A synthetic nonlinear system is first used to illustrate the application of the RL method and provide computational complexity comparisons against perturbation-based methods. The application to multi-machine power systems is then considered to demonstrate that the RL has the accuracy of the analytic linearization (AL) method with less computational burden than the perturbation-based methods widely used in power systems analysis tools.

In this paper, the problem of optimal placement of virtual inertia is considered as a techno-economic problem from a frequency stability point of view. First, a data driven-based equivalent model of battery energy storage systems, as seen from the electrical system, is proposed. This experimentally validated model takes advantage of the energy storage system special attributes to contribute to inertial response enhancement, via the virtual inertia concept. Then, a new framework is proposed, which considers the battery storage system features, including annual costs, lifetime and state of charge, into the optimal placement formulation to enhance frequency response with a minimum storage capacity. Two well-known dynamical frequency criteria, the frequency nadir and the rate of change of frequency, are utilized in the optimization formulation to determine minimum energy storage systems. Moreover, a power angle-based stability index is also used to assess the effect of virtual inertia on transient stability. Sensitivity and uncertainty analyses are further conducted to assess the applicability of the method. The efficiency of the proposed framework is demonstrated on a linearized model of a three-area power system as well as two nonlinear systems. Simulation results suggest that the proposed method gives improved results in terms of stability measures and less ESS capacity, when compared with other methods proposed in the literature.

In this study, a data-driven framework to determine the optimal sensor locations for power system oscillation monitoring and state reconstruction is proposed. In this approach, candidate sensor locations are selected sequentially using proper orthogonal decomposition (POD) analysis of a reduced-order binary measurement matrix. Using criteria inferred from the spatial structure of the POD modes, sensor locations having large magnitudes and small cross couplings between modes at the candidate locations are selected from the measurement matrix. The method can be used to reconstruct global system dynamics using a few sensor measurements as well as for the prognosis of power system oscillatory behaviour. The developed procedures are applied to the IEEE 39-bus test system and to a realistic 5449-bus model of a large power system.

A novel control scheme based on a new stochastic equivalent model of the power system, which provides flexible inertia constant to enable high penetration levels of microgrid (MG) generation, is proposed. The approach combines an adaptive energy storage system (ESS) dispatch strategy with an MG-controlled islanding scheme to provide frequency support for the host power grid. First, a center-of-gravity (COG) formulation of the inertia-control scheme is proposed, in which the unbalance torques between the control areas and the COG are used to reduce frequency deviations. An application example-based equivalent model of ESSs as seen from the grid side is then proposed that takes advantage of the ESS special attributes to contribute to inertial response. Simulation results on two test systems demonstrate the efficiency of the proposed control scheme to increase the host grid ability to efficiently integrate multiple MGs. Numerical tests suggest that the proposed control scheme increases system penetration level at least by 12%, which justifies the use of inertia emulation techniques to accommodate higher levels of distributed generation.

This chapter discusses the analysis of the impact of high levels of microgrid (MG) penetration on power system stability from various points of view such as frequency stability, small‐signal stability and voltage performance. Methods for interpreting system dynamics in terms of simplified system representations are developed, and criteria to determine maximum penetration levels are given using deterministic and statistical sensitivity analyzes. A systematic methodology for the analysis of the impact of distributed MGs on power system dynamic behavior is proposed. It is well known that MGs are realized in the low and medium voltage levels. Therefore, following the calculation of maximum penetration levels of MGs, the chapter tries to determine the maximum active power that MGs can inject into each bus of the distribution system without causing steady‐state voltage violations.

Power systems exhibit nonlinear dynamics when load stress is high. Conventional linear wide-area controllers will fail in such scenarios, and instead nonlinear controllers will become necessary. This paper develops a state-feedback nonlinear quadratic controller based on perturbed Koopman mode analysis (PKMA) considering a sparse wide-area communication net-work for saving communication cost while guaranteeing closed-loop stability. The sparsity is decided based on the modal residues of the generators on selected nonlinear oscillations modes. The theoretical developments behind this sparse nonlinear control are first established, including the definition of residues for Koopman modes. Bilinear effects in the controller structure arising from quadratic PKMA-based extended models, structural constraints on the quadratic gain matrix design, and second-order phenomena in the closed-loop dynamics are all explained in detail. The design is validated using the IEEE 16-machine, 68-bus power system. The advantages and draw-backs of the proposed approach are illustrated using simulation results.

The inclusion of higher-order terms in small-signal (modal) analysis augments the information provided by linear analysis and enables better dynamic characteristic studies on the power system. This can be done by applying Normal Form theory to simplify the higher order terms. However, it requires the preliminary expansion of the nonlinear system on the normal mode basis, which is impracticable with standard methods when considering large scale systems. In this paper, we present an efficient numerical method for accelerating those computations, by avoiding the usual Taylor expansion. Our computations are based on prescribing the linear eigenvectors as unknown field in the initial nonlinear system, which leads to solving linear-only equations to obtain the coefficients of the nonlinear modal model. In this way, actual Taylor expansion and associated higher order Hessian matrices are avoided, making the computation of the nonlinear model up to third order and nonlinear modal analysis fast and achievable in a convenient computational time. The proposed method is demonstrated on a single-machine-infinite-bus (SMIB) system and applied to IEEE 3-Machine, IEEE 16- Machine and IEEE 50-Machine systems. This work was supported by TETfund, Nigeria

A new model-based framework for studying nonlinear dynamic behavior of stressed power systems that combines Koopman mode analysis and perturbation theory is proposed. First, a systematic approach to deriving high-dimensional representations of nonlinear dynamical systems based on perturbation theory is presented. Nonlinear behavior is interpreted as a projection of the Koopman operator eigenfunctions of an extended coordinate system onto the physical variables of the system. Analytical formulations are then obtained for the construction of the extended coordinate system based on perturbation theory. The analysis methodology is demonstrated on three test systems: A single-machine, infinite-bus system, a three-machine test system, and the IEEE 50-machine test system.

In this study, a comprehensive approach for model order reduction based on a Fourier series of a discrete system representation is proposed. The developed method represents an alternative to model reduction of large-scale dynamical systems and can be used for the analysis of unstable dynamic systems and the study of interactions among power system controls. Drawing on the principle of conformal mapping, the linearised system model is first transformed into its discrete equivalent. Then, the Fourier series of the discrete-time transfer function is used to obtain a reduced-order model (ROM). Using this model, approximate Hankel-based interaction measures are proposed to efficiently analyse large, unstable linear system representations and determine pairs of input outputs to design controller. The high degree of applicability and accuracy offered by the method, and its ability to extract ROMs from stable and unstable systems is demonstrated on three test systems.

In this paper, a semi-distributed tool for monitoring and analysis of power systems disturbances using wide-area sensors is proposed. The proposed framework can be used to integrate and process large amounts of multimodal, multi-type observational data from various monitoring technologies or data concentrators to assess the power system health in near-real time. First, a framework for fusing data from multiple sensors based on multivariate statistical tools is introduced and a model of the collected data is developed. Drawing on this model, a novel a data-driven strategy based on both, local and global energy metrics for the analysis of major system disturbances is proposed. The approach is tested using time domain simulations on the IEEE 118 bus system under various scenarios and different levels of observability of the system. The approach is complemented with a visualization technique to provide further means to analyze the system after a disturbance.

Koopman mode analysis has shown considerable promise for the analysis and characterization of global behavior of power system transient processes recorded using wide-area sensors. In this paper, a framework for feature extraction and mode decomposition of spatiotemporal data based on the Koopman operator is presented. A physical interpretation of the Koopman modes as columns of a matrix of observability measures is derived, and criteria for selecting a reduced set of Koopman modes are proposed. This approach allows to selectively isolate and quantify the dominant physical mechanisms underlying the recorded power system data, and it can be used for wide-area monitoring and assessment. Simulation results on both simulated and measured data show that the method can accurately identify the dominant spatial patterns or shapes and temporal patterns associated with specific system behavior.

Frequency control of power grids has become a relevant research topic due to the increasing penetration of renewable energy sources, changing system structure, and the integration of new storage systems, controllable loads and power electronics technologies. The advances in control, communication, and computation technologies also contribute to the development of new techniques and solutions. This paper provides an updated review of most important frequency stability concerns, applied modern control strategies, and existing challenges for the integration of renewable energy sources. The paper also discusses the current trends, recent achievements, and new research directions.

In this paper, a general approach for constructing reduced-order models of power system models based on a truncated balanced realization (TBR) linear reduction procedure is described. The method enables efficient computation of Gramian covariance matrices in terms of the system frequency response, and it can be used for control design and sensitivity analysis. The method guarantees convergence of the controllability and observability Gramians and avoids the need for frequency sweep of the transfer function matrix. A key innovation for the TBR-based reduced-order models introduced in this paper is the use of modal information to approximate the controllability and observability Gramians. With this approach, the connection between the linear system modes and the Gramians is clarified, and the contribution of critical system motion modes to the collective dynamics is readily determined. Conditions for convergence of the model and approximation methods are then investigated, and the issues of accuracy...

Reduction of overall system inertia in response to increasing penetration level of Micro-Grids (MGs) makes frequency control more challenging. In this paper, a new analytical approach to mathematically investigate the impact of Micro-Grids penetration level on power system frequency stability is proposed. The method interprets the frequency dynamic behavior of the penetrated system based on the conventional system in terms of minimum instantaneous frequency, steady state deviation and a 15-second rolling window. Then, new indices are proposed to determine the maximum allowable MGs penetration level. These indices are derived by adapting the developed analytical approach to the required standards regarding the aforementioned criteria. Each index determines an upper bound on reduction of system inertia, which, in turn specifies the associated maximum penetration level. The maximum value of the inertia constant related to the developed indices (whichever is the larger) determines the system maximum allowable penetration level. The effectiveness of the proposed analytical approach and indices is demonstrated on a 16-machine 68-bus system and the IEEE 50-machine systems. Simulation results for large-scale power systems with high stiffness reveal that based on the operating condition, one of the minimum instantaneous frequency or 15-second rolling window impose an upper bound on penetration level.

A mathematically rigorous framework combining diffusion map analysis techniques with simulation data from electromagnetic transient programs is proposed to characterize the collective dynamics of transient processes. The framework can be used to compress and analyze large and complex data sets obtained from transient simulations. Through this formulation, the structures that dominate the system response, and the global measurements that contribute most to this response can be directly calculated. The resulting coordinates can be correlated with the time scales of specific motion or be effectively used to extract spatiotemporal features of high-dimensional data. The application of the method is illustrated on both, electromagnetic transient simulations and transient stability data. Simulation studies reveal that diffusion-map data-analysis techniques generate accurate estimates of modal parameters even in the presence of measurement noise.

A global multiscale method based on a dynamic mode decomposition (DMD) algorithm to characterize the global behavior of transient processes recorded using wide-area sensors is proposed. The method interprets global dynamic behavior in terms of both, spatial patterns or shapes and temporal patterns associated with dynamic modes containing essentially single-frequency components, from which the mode shapes, frequencies and growth and decay rates of the modes can be extracted simultaneously. These modes are then used to detect the coherent and dominant structures within the data. The technique is well suited for fast wide-area monitoring and assessment of global instability in the context of modern data fusion-based estimation techniques. Results of the application of the proposed method to large, high-dimensional data sets are encouraging.

A novel technique to determine the best pairing of inputs and outputs for the design of wide-area damping controllers (WADCs) is proposed. The method extends the conventional use of the relative gain array (RGA) for interacting control systems, to the block relative gain (BRG) to analyze interactions among multi-input multi-output (MIMO) controllers. First, geometric measures of controllability and observability are used to screen the most effective stabilizing signals for partially centralized control of large interconnected power systems. Then, the BRG analysis is used to quantify the potential for adverse interactions between a given set of controllers or subsystems. This gives candidate sets for supplementary controllers which are solved using linear matrix inequality (LMI) techniques and are evaluated in closed-loop mode to detect and quantify process interactions. In general, this approach also allows better coordination of control capabilities and the use of interaction measures to help in the choice of control location and structure. The effectiveness of the proposed control scheme on system behavior is tested on a realistic 5-area model of the Mexican interconnected power system.

Wide-area monitoring substantially improves modal identification and characterization under noisy conditions. This paper discusses the use of blind source separation (BSS) techniques to extract and identify modal responses and mode shapes from simulated data. The method has advantages over other global analysis methods in that it allows for the analysis of both, transient and ambient data and is thus well suited for global system monitoring of power system oscillatory behavior using wide-area measurement systems (WAMS) data. Methods for analysis of complex datasets using BSS techniques are developed, and a physical explanation is offered. The developed procedures are tested on simulated data, and the impact of various parameters such as noise and time lags on the quality of signal separation are examined.

In this paper, a frequency control strategy that overcomes some of the problems associated with the integration of large wind farms into the Mexican Interconnected System (MIS) is presented. In recent years, several wind farms have been installed in the MIS, especially in the southeastern network of the system. Installed wind energy capacity is around 2.2 GW, representing about 3.5 % of the total installed electric power in the grid. Increase of this kind of renewable generation is expected to have a strong effect on the dynamic performance at the MIS, especially when the penetration level of wind farms will be an important contribution of total generation. Transient stability studies conducted on a 3500-bus system of the Mexican grid shows that the developed procedures can greatly improve system frequency performance of the entire Mexican interconnected system.

This paper discusses the application of a spatiotemporal analysis method based on the dynamic mode decomposition (DMD) framework to analyze harmonic distortion in a microgrid test system with several distributed generation units. The proposed technique extracts spatiotemporal coherent patterns from which the frequency components, modal energy, mode-shape and participation factors can be determined simultaneously. Due to its multiscale nature, the method is well suited for the analysis and monitoring of harmonic distortion in complex microgrid systems. The explicit representation of switching processes in the power electronic converters and non-linear loads is fully considered to assess the robustness of the proposed framework.

In this paper, a wide-area, measurement-based islanding detection technique based on nonlinear, nonparametric data reduction methods and statistical quality control charts is presented. The method exploits the spectral properties of a probability transition matrix of similarities measures on updated observation window, and accounts for nonlinearities in the measured signals. When combined with a spectral change detection algorithm, changes in the measured data leading to island formation and deviations from data-driven (upper and lower) control limits are computed in an automated manner. Detailed time-domain simulations are conducted to test the ability of the method to detect islanding conditions.

Large-scale integration of renewable energy generation into the Mexican electric grid poses significant challenges and opportunities. Wind power development, particularly in the southeastern and northern portions of the Mexican interconnected system (MIS), may affect generation scheduling and impose additional uncertainty on available transfer capability and grid operations. Currently installed wind energy capacity in the MIS is around 2.5 GW, representing about 4% of total installed generation in the system. Large increases in wind generation capacity in the system are expected to have a significant impact on the dynamic stability and reliability of the system, especially as the amount of wind generation represents a significant portion of the local hydrogeneration and wind farms (WFs) grow in size and complexity. This chapter examines the impact of increased penetration of wind resources in the southeastern region of the Mexican grid on the dynamic performance of the system. Major technical challenges for power system operation are examined and rigorous analytical methods for analyzing and assessing the effect of large-scale wind and other resources on system dynamic behavior are proposed. Using recent planning and operation wind data, detailed studies on the effects of new wind generation on power system dynamic behavior are carried out to determine the potential operational impacts and mitigation measures. Integration issues related to large-scale wind generation development include power system frequency and voltage control, small-signal rotor angle stability and subsynchronous torsional interactions with static VAR compensators (SVCs) and series-compensated transmission lines.

—Time series models provide a powerful tool to extract nonstationary features from measured data. In this article, a statistical framework based upon a dynamic harmonic regression model for examining modal behavior is provided. In this model, temporal patterns in measured data are modeled within a stochastic state space setting. Estimates of the states or time-varying parameters are then obtained using an optimal estimation method based on the Kalman filter. Techniques to estimate future values of the unobserved signal are also analyzed. The widely applicable technique is illustrated on both simulated and measured data. Factors that affect the performance of the method are discussed, including the effects of non-linear trends, data quality, and sampling design. Connections with other modal identification methods are also investigated.

The study of low-frequency electromechanical modes in power systems has experienced much progress in the past few years. In this research, a nonstationary recursive least-squares (RLS) algorithm with variable forgetting factor is combined with a Kalman filter to simultaneously estimate low-frequency electromechanical modes from measured ambient power system data. Extensions and generalizations to current adaptive filtering algorithms to account for nonstationarity are implemented and tested and the correspondence between the Kalman and RLS variables is examined. Applications of the proposed nonstationary RLS algorithm to track the evolving dynamics of critical power system electromechanical modes in both, simulated and measured data, are presented. Comparison with other RLS and least-mean squares algorithms demonstrate the accuracy of the proposed framework in tracking changes in modal parameters over time. The issues of computational efficiency and memory requirements are discussed in detail.

A multi-sensor multi-temporal data-fusion technique for power swing monitoring is proposed and evaluated. The method combines the ability of statistical techniques to identify the dominant structures in an ensemble of observations with that of time–frequency techniques to extract temporal features. Due to its multi-scale nature, the proposed framework is well suited for the analysis and monitoring of wide-area signals obtained using synchronized phasor measurement units. Features relevant for wide-area inter-area oscillation monitoring are systematically obtained from decentralized data concentrators, and techniques to analyze the distribution of the extracted features are proposed. Methods to quantify the spatial and temporal structure of critical modes are also described, and key research directions in the area of data fusion are discussed. The application of the proposed framework to real data is studied, focusing on the issues of data compression and feature extraction and classification.

A number of recent studies have examined the problem of modal reduction of large-scale power system linear models. In this paper, an optimal model reduction strategy for the analysis and control of inter-area oscillations in power systems that uses concepts from proper orthogonal decomposition and modal analysis and preserves exactly the input–output properties of the original system is presented. The technique uses a new projection matrix that minimizes the error between the states of the original system and the states of its reduced-order model, and is suitable for analyzing large power system models described by differential–algebraic equation (DAE) systems. Based on this theoretical framework, an efficient algorithm for model reduction is proposed which combines the ease of use of linear modal analysis, with the flexibility of extracting relevant behavior that statistical multivariate methods can offer. Methods for analysis of reduced system representations using linear approximations are critically reviewed and compared. The proposed technique is tested on a 50-machine, 145-bus, and 453-line test system. The effects of system dimension, accuracy of the approximations to capture inter-system oscillations, and the generality of the techniques are discussed in detail in the parametric study. Accuracy of the solution and weaknesses of the model reduction are also investigated. Direct numerical simulation of the power system dynamic model is performed to investigate the accuracy and robustness of the method.

A Fourier, series-based method for model reduction of large linear power system models is introduced. The method approximates the discrete-time system transfer function by a Fourier expansion and preserves the stability properties of the original system. An iterative procedure to determine both, the Fourier coefficients and the reduced-order model (ROM) is proposed. The developed model represents an alternative to model reduction of large-scale dynamical systems and is used to assess the effect of voltage control by means of static VAR compensators on system damping in a real-world power system.

The increase in both, the volume (dimensionality), and the complexity and variety of measured data, requires advances in methodology to systematically extract the important dynamical motions from the observed dynamic trajectories. In this paper, methods for analysis or large volumes of data based on Koopman mode analysis (KMA) are investigated. Three specific methods are compared and tested: Koopman mode decomposition (KMD), Dynamic Mode Decomposition (DMD), and multi-channel Prony analysis. The methods are implemented and compared in terms of their ability to accurately estimate power system electromechanical modes. Numerical aspects associated with application of these approaches are investigated and recommendations for improving the analysis tools are provided. Case studies arising from the analysis of transient stability swings are presented. The efficiency and accuracy of the methods is demonstrated by comparison to other multivariate techniques.

Time series models provide a very flexible approach to nonstationary time series analysis. In this paper, a statistical framework based upon dynamic harmonic regression for examining modal behavior, trend extraction and forecasting is provided. Both, synthetic and observational data are used to assess the performance of the model. The practical application of this technique is tested on wind power generation measurements under various conditions of data acquisition.

A method for spatiotemporal analysis of high-dimensional dynamic datasets is presented. The method combines the use of the Laplace-Beltrami operator with time-lagged embedding to represent spatial and temporal patterns from measured data. The proposed method has the potential to render a significantly improved spatiotemporal representation of complex datasets and greatly facilitates extracting dynamic patterns. Because the temporal structures of the time-dependent means are modeled, the new approach displays increased statistical power and precision of parameter estimation. As an illustrative example, transient stability data is used to examine the potential usefulness of the proposed approach to characterize spatiotemporal dynamics.

A technique for extracting the dominant spatiotemporal patterns of variability from high-dimensional simulation (measurement) trajectories is presented. The proposed framework integrates concepts from nonlinear dimensionality reduction methods with nonlinear Laplacian spectral analysis, to extract physically meaningful information about the temporal evolution of low-frequency electromechanical modes. Due to its multiscale and multitemporal nature, the general techniques developed here are also useful for estimating mode shapes, as well as to identify coherent behavior and monitoring power system health. The use and interpretation of the proposed framework is demonstrated using transient stability data.

The time evolution of many physical processes in power systems is described by non-linear differential-algebraic equations. In this article, a new reduced-order modeling technique for constructing explicit state-space approximations of non-linear power system models described by differential-algebraic equation models is developed. The proposed technique takes into account the interconnection structure and provides information on the effect of control actions on system behavior. A general second-order theory is presented for treating non-linear behavior in large non-linear power system models. In this approach, the algebraic equations are first reduced by projecting the original system model onto a truncated basis of the subspace generated by the right and left eigenvectors of the linear system model. A technique for constructing an explicit state-space approximation of the non-linear differential-algebraic equation model is then developed. The method is highly efficient and capable of developing ...

Large interconnected power systems are exposed to many system disturbances that may impair system security and result in wide-spread damage. This paper discusses the development of advanced health monitoring systems for large interconnected power systems. The use of adaptive, data processing techniques to provide automatic early detection of system deterioration in large power systems is investigated. Areas of improvement in real-time system monitoring are identified and algorithms to detect both, oscillatory and abrupt changes in system behavior are proposed and tested. The applicability of the proposed monitoring system is evaluated using actual measurements from significant events.

Recent works have examined the problem of modal reduction of power system linear models. In this article, a new model reduction technique for the analysis and control of inter-area oscillations in power systems which preserves exactly the inputs and outputs of interest of the original system, is presented. The technique combines the proper orthogonal decomposition and concepts from the balanced realization theory and can be used to analyze large power system models described by differential–algebraic equation systems. Methods for analysis of reduced system representations using linear approximations are critically reviewed and compared. The procedures developed are tested on a 16-machine, 68-bus dynamic equivalent of the New England test system. The effects of system dimension, accuracy of the approximations to capture inter-system oscillations, and generality of the techniques are discussed in detail in the parametric study. It is shown that the proposed techniques are more accurate than existin...