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Synergistic use of intrusive and non-intrusive model order reduction techniques for dynamical power grids
Abstract
This manuscript combines the recently developed nonlinear moment-matching (NLMM) technique with dynamic mode decomposition (DMD) to obtain a simulation-free reduction framework for power systems. Unlike the conventional model reduction methods for power systems, where the external area is linearized, we consider the nonlinear effective network (EN) and the synchronous motor (SM) power grid models. First, the reduced system is constructed from the solution of the underlying approximate Sylvester equation by exciting the system with user-defined inputs from a representative exogenous system. Then, a non-intrusive reduction is performed using DMD to approximate the nonlinear function via Koopman modes in an equation-free manner. The advantage is that a “simulation-free” nonlinear model order reduction framework is obtained to approximate the response of the large-scale power grid models. Finally, we substantiate our observations using numerical simulations of reduced EN and SM models of the IEEE 118 and IEEE 300 bus systems for realistic fault scenarios. Results show that the overall CPU times of the reduced-order models are lowered to half as compared to the original models while maintaining the fidelity. The results are also compared with POD-DEIM for reference.
... Since most of these methods rely on linear MOR techniques, they are not well suited for complex nonlinear models. Towards this direction, nonlinear MOR methods for power systems have also been reported [25,26,27,28]. Methods based on artificial neural networks (ANNs) have been used to search for equivalent models for analyzing transient states and parameter estimation in power grid networks [29,30,31,32]. ...
... On the other hand, nonlinear MOR methods rely on constructing a reduced subspace followed by a Galerkin projection to evolve the latent or reduced dynamics onto that subspace. While linear projection frameworks works very well for reducing certain type of nonlinearities [47,28], it is less so for reducing power system models. This is because these models experience a slow-decaying nature of singular values and hence using a linear projection framework can lead to less computational speedups and inaccurate results, compared to linear systems. ...
The model order reduction (MOR) enterprise has shown unprecedented applications in the area of power systems by
allowing tractable and realistic simulations of complex power grids. Nearly all model reduction techniques rely on
constructing a linear reduced trial subspace followed by a Galerkin projection to restrict the evolution of the latent
dynamics onto the reduced subspace. However, power system models often exhibit a slow decay of singular values
and, as such, are often approximated poorly due to the truncation of higher-order modes. Furthermore, employing a
linear projection for such models places a fundamental limit on the accuracy of the reduced order models (ROMs).
This paper uses techniques from machine learning to develop a dimensionality reduction framework for large-scale
power grid models to overcome these limitations effectively. In particular, we use an autoencoder (AE) network to learn
low-dimensional nonlinear trial manifold to avoid the limitation of a linear trial subspace. This is followed by the
use of long short-time memory (LSTM) networks to obtain the evolution of the reduced dynamics in a non-intrusive
manner. The resulting framework yields ROMs that are compact and several orders of magnitude more accurate
than the linear projection-based methods. We demonstrate the proposed technique on the standard IEEE 118-Bus
system, which has 54 generators, and the European high-voltage system with 260 generators.
... In order to test standard MOR methods like POD [112], TPWL [98] or moment-matching methods [95], the first-order approximation of both the EN and SM models can be obtained as follows: ...
... The desired output is voltage measured at the first node. The model can be used to test a variety of reduction routines and can also serve a parametric problem for the choice of variations in parameters I d and α [95]. ...
We provide a publicly available collection of sixteen large-scale benchmark nonlinear state-space models in this contribution. The models are written in the MATLAB language and are scalable in spatiotemporal degrees of freedom. The aim is to provide the active research community with a suite of high-dimensional nonlinear models to test the state-of-the-art nonlinear model reduction strategies. We also review some of the most-widely employed reduction methods for these models. Furthermore, we also present some parametric nonlinear models to design and validate parametric model reduction schemes.
This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.
Identifying generator coherency with respect to slow oscillatory modes has numerous power system use cases including dynamic model reduction, dynamic security analysis, or system integrity protection schemes (e.g., power system islanding). Despite their popularity in both research and industry, classic eigenvector-based slow coherency techniques may not always return accurate results. The multiple past endeavors to improve their accuracy often lack a solid mathematical foundation. Motivated by these deficiencies, we propose an alternative consistent approach to generator slow coherency. Firstly, a new approach is introduced to accurately detect slow coherent generators by effectively minimizing generic normalized cuts. As a by-product, the new approach can also guide the choice of the number of slow coherent groups. Secondly, it is shown that the combination of the the proposed slow coherency approach and an enhanced version of the inertial generator aggregation method allows to produce accurate dynamic equivalents even if the selected number of generator groups is relatively low.
Model order reduction (MOR) also known as dimension reduction is a computational tool to obtain cost-effective lower order approximations of large scale dynamical systems. This paper presents a detailed yet simplified MOR approach using nonlinear moment matching (NLMM) in conjuncture with the Discrete Empirical Interpolation Method (DEIM). NLMM avoids the expensive simulation of the underlying nonlinear Sylvester partial differential equation by reducing it to a system of nonlinear algebraic equations using proper step-by-step simplifications. This reduces the offline computational cost of generating the orthonormal projection basis substantially. This is followed by the DEIM algorithm, resulting in comprehensive savings in computational resources. The proposed algorithms are tested on two benchmark problems and the results so obtained are compared with proper orthogonal decomposition for different test inputs.
The paper proposes a new adaptive approach to power system model reduction for fast and accurate time-domain simulation. This new approach is a compromise between linear model reduction for faster simulation and nonlinear model reduction for better accuracy. During the simulation period, the approach adaptively switches among detailed and linearly or nonlinearly reduced models based on variations of the system state: it employs unreduced models for the fault-on period, uses weighted column norms of the admittance matrix to decide which functions to be linearized in power system differential-algebraic equations for large changes of the state, and adopts a linearly reduced model for small changes of the state. Two versions of the adaptive model reduction approach are introduced. The first version uses traditional power system partitioning where the model reduction is applied to a defined large external area in a power system and the other area defined as the study area keeps full detailed models. The second version applies the adaptive model reduction to the whole system. The paper also conducts comprehensive case studies comparing simulation results using the proposed adaptively reduced models with the linearly reduced model and coherency-based reduced model on the Northeast Power Coordinating Council 140-bus 48-machine system.
Theory and methods to obtain reduced order models by moment matching from input/output data are presented. Algorithms for the estimation of the moments of linear and nonlinear systems are proposed. The estimates are exploited to construct families of reduced order models. These models asymptotically match the moments of the unknown system to be reduced. Conditions to enforce additional properties, e.g. matching with prescribed eigenvalues, upon the reduced order model are provided and discussed. The computational complexity of the algorithms is analyzed and their use is illustrated by two examples: we compute converging reduced order models for a linear system describing the model of a building and we provide, exploiting an approximation of the moment, a nonlinear planar reduced order model for a nonlinear DC-to-DC converter.
This paper concerns the application of reduced order modeling techniques to power grid simulation. Swing dynamics is a complex non-linear phenomenon due to which model order reduction of these problems is intricate. A multi point linearization based model reduction technique trajectory piece-wise linearization (TPWL) method is adopted to address the problem of approximating the nonlinear term in swing models. The method combines proper orthogonal decomposition with TPWL in order to build a suitable reduced order model that can accurately predict the swing dynamics. The method consists of two stages, an offline stage where model reduction and selection of linearization points is performed and an online stage where the reduced order multi-point linear simulation is performed. An improvement of the strategy for point selection is also proposed. The TPWL method for a swing dynamics model shows that the method provides accurate reduced order models for non-linear transient problems.
A data-driven algorithm recently proposed to solve the problem of model reduction by moment matching is extended to multi-input, multi-output systems. The algorithm is exploited for the model reduction of large-scale interconnected power systems and it offers, simultaneously, a low computational complexity approximation of the moments and the possibility to easily enforce constraints on the reduced order model. This advantage is used to preserve selected slow and poorly damped modes. The preservation of these modes has been shown to be important from a physical point of view and in obtaining an overall good approximation. The problem of the choice of the so-called tangential directions is also analyzed. The algorithm and the resulting reduced order model are validated with the study of the dynamic response of the NETS-NYPS benchmark system (68-Bus, 16-Machine, 5-Area) to multiple fault scenarios.
In this paper, nonlinear model reduction for power systems is performed by the balancing of empirical controllability and observability covariances that are calculated around the operating region. Unlike existing model reduction methods, the external system does not need to be linearized but is directly dealt with as a nonlinear system. A transformation is found to balance the controllability and observability covariances in
order to determine which states have the greatest contribution to the input-output behavior. The original system model is then reduced by Galerkin projection based on this transformation. The proposed method is tested and validated on a system comprised of a 16-machine 68-bus system and an IEEE 50-machine 145-bus system. The results show that by using the proposed model reduction the calculation efficiency can be greatly improved; at the same time, the obtained state trajectories are close to those for directly simulating the whole system or partitioning the system while not performing reduction. Compared with the balanced truncation method based on a linearized model, the proposed nonlinear model reduction method can guarantee higher accuracy and similar calculation efficiency. It is shown that the proposed method is not sensitive to the choice of the matrices for calculating the empirical covariances.
The dynamics of power-grid networks is becoming an increasingly active area of research within the physics and network science communities. The results from such studies are typically insightful and illustrative, but are often based on simplifying assumptions that can be either difficult to assess or not fully justified for realistic applications. Here we perform a comprehensive comparative analysis of three leading models recently used to study synchronization dynamics in power-grid networks—a fundamental problem of practical significance given that frequency synchronization of all power generators in the same interconnection is a necessary condition for a power grid to operate. We show that each of these models can be derived from first principles within a common framework based on the classical model of a generator, thereby clarifying all assumptions involved. This framework allows us to view power grids as complex networks of coupled second-order phase oscillators with both forcing and damping terms. Using simple illustrative examples, test systems, and real power-grid datasets, we study the inherent frequencies of the oscillators as well as their coupling structure, comparing across the different models. We demonstrate, in particular, that if the network structure is not homogeneous, generators with identical parameters need to be modeled as non-identical oscillators in general. We also discuss an approach to estimate the required (dynamical) system parameters that are unavailable in typical power-grid datasets, their use for computing the constants of each of the three models, and an open-source MATLAB toolbox that we provide for these computations.
This paper proposes a new model reduction algorithm for power systems based on an extension of balanced truncation. The algorithm is applicable to power systems which are divided into a study area which requires a high-fidelity model and an external area, making up most of the power system, which is to be reduced. The division of the power system can be made arbitrarily and does not rely on the identification of coherent generators. The proposed algorithm yields a reduced order system with a full nonlinear description of the study area and a reduced linear model of the external area.
The problem of using the Karhunen–Loève transform with partial data is addressed. Given a set of empirical eigenfunctions, we show how to recover the modal coefficients for each gappy snapshot by a least-squares procedure. This method gives an unbiased estimate of the data that lie in the gaps and permits gaps to be filled in a reasonable manner. In addition, a scheme is advanced for finding empirical eigenfunctions from gappy data. It is shown numerically that this procedure obtains spectra and eigenfunctions that are close to those obtained from unmarred data.
Vulnerability assessment is one of the main tasks in a Self-Healing Grid structure, since it has the function of detecting the necessity of performing global control actions in real time. Due to the short-time requirements of real time applications, the eligible vulnerability assessment methods have to consider the improvement of calculation time. Although there are several methods capable of performing quick assessment, these techniques are not fast enough to analyze real large power systems in real time. Based on the fact that vulnerability begins to develop in specific regions of the system exhibiting coherent dynamics, large interconnected power systems can be reduced through dynamic equivalence in order to reduce the calculation time. A dynamic equivalent should provide simplicity and accuracy sufficient for system dynamic simulation studies. Since the parameters of the dynamic equivalent cannot be easily derived from the mathematical models of generators and their control systems, numerical identification methods are needed. Such an identification task can be tackled as an optimization problem. This paper introduces a novel heuristic optimization algorithm, namely, the Mean-Variance Mapping Optimization (MVMO), which provides excellent performance in terms of convergence behavior and accuracy of the identified parameters. The identification procedure and the level of accuracy that can be reached are demonstrated using the Ecuadorian-Colombian interconnected system in order to obtain a dynamic equivalent representing the Colombian grid.
Originally introduced in the fluid mechanics community, dynamic mode
decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics
of nonlinear systems. However, existing DMD theory deals primarily with
sequential time series for which the measurement dimension is much larger than
the number of measurements taken. We present a theoretical framework in which
we define DMD as the eigendecomposition of an approximating linear operator.
This generalizes DMD to a larger class of datasets, including nonsequential
time series. We demonstrate the utility of this approach by presenting novel
sampling strategies that increase computational efficiency and mitigate the
effects of noise, respectively. We also introduce the concept of linear
consistency, which helps explain the potential pitfalls of applying DMD to
rank-deficient datasets, illustrating with examples. Such computations are not
considered in the existing literature, but can be understood using our more
general framework. In addition, we show that our theory strengthens the
connections between DMD and Koopman operator theory. It also establishes
connections between DMD and other techniques, including the eigensystem
realization algorithm (ERA), a system identification method, and linear inverse
modeling (LIM), a method from climate science. We show that under certain
conditions, DMD is equivalent to LIM.
The extraction of dynamically relevant structures from time-resolved flow data has commonly be restricted to numerically generated flow fields. Equivalent structures, however, could not be obtained from experimental measurements, since the commonly used mathematical techniques required the explicit or implicit availability of an underlying model equation. A numerical scheme based on a Krylov subspace method for the extraction of dynamic modes directly from flow fields --- without the need to resort to a model equation --- will be introduced. This technique can be applied equally to numerically generated or experimental data and thus provides a means to decompose time-resolved measurements into dynamically dominant structures. The treatment of subdomains, spatially evolving flows, PIV data and simple flow visualizations will be demonstrated; a connection to the proper orthogonal decomposition (POD) technique, which is a byproduct of the dynamic mode decomposition, will be pointed out.
Phase models describing self-synchronization phenomena in populations of globally coupled oscillators are generalized including ``inertial'' effects. This entails that the oscillator frequencies also vary in time along with their phases. The model can be described by a large set of Langevin equations when noise effects are also included. Also, a description of such systems can be given in the thermodynamic limit of infinitely many oscillators via a suitable Fokker-Planck-type equation. Numerical simulations confirm that simultaneous synchronization of phases and frequencies is possible when the coupling strength goes to infinity.
An imperative condition for the functioning of a power-grid network is that its power generators remain synchronized. Disturbances can prompt desynchronization, which is a process that has been involved in large power outages. Here we derive a condition under which the desired synchronous state of a power grid is stable, and use this condition to identify tunable parameters of the generators that are determinants of spontaneous synchronization. Our analysis gives rise to an approach to specify parameter assignments that can enhance synchronization of any given network, which we demonstrate for a selection of both test systems and real power grids. Because our results concern spontaneous synchronization, they are relevant both for reducing dependence on conventional control devices, thus offering an additional layer of protection given that most power outages involve equipment or operational errors, and for contributing to the development of "smart grids" that can recover from failures in real time.
We study synchronization dynamics in networks of coupled oscillators with
bimodal distribution of natural frequencies. This setup can be interpreted as a
simple model of frequency synchronization dynamics among generators and loads
working in a power network. We derive the minimum coupling strength required to
ensure global frequency synchronization. This threshold value can be
efficiently found by solving a binary optimization problem, even for large
networks. In order to validate our procedure, we compare its results with
numerical simulations on a realistic network describing the European
interconnected high-voltage electricity system, finding a very good agreement.
Our synchronization threshold can be used to test the stability of frequency
synchronization to link removals. As the threshold value changes only in very
few cases when aplied to the European realistic network, we conclude that
network is resilient in this regard. Since the threshold calculation depends on
the local connectivity, it can also be used to identify critical network
partitions acting as synchronization bottlenecks. In our stability experiments
we observe that when a link removal triggers a change in the critical
partition, its limits tend to converge to national borders. This phenomenon,
which can have important consequences to synchronization dynamics in case of
cascading failure, signals the influence of the uncomplete topological
integration of national power grids at the European scale.
The problem of model reduction by moment matching for nonlinear systems is addressed and solved using the recently introduced notion of moment for nonlinear systems. It is shown that reduced order models can be parameterized by a free mapping which, in turn, can be used so that the model possesses specific properties, e.g. it has an asymptotically stable equilibrium or given relative degree, it is minimum phase, it is passive. In addition, a nonlinear enhancement of the notion of Markov parameters is provided. The theory is illustrated by means of simple examples.
MATPOWER is an open-source Matlab-based power system simulation package that provides a high-level set of power flow, optimal power flow (OPF), and other tools targeted toward researchers, educators, and students. The OPF architecture is designed to be extensible, making it easy to add user-defined variables, costs, and constraints to the standard OPF problem. This paper presents the details of the network modeling and problem formulations used by MATPOWER, including its extensible OPF architecture. This structure is used internally to implement several extensions to the standard OPF problem, including piece-wise linear cost functions, dispatchable loads, generator capability curves, and branch angle difference limits. Simulation results are presented for a number of test cases comparing the performance of several available OPF solvers and demonstrating MATPOWER's ability to solve large-scale AC and DC OPF problems.
The slow coherency concept is introduced and an algorithm is developed for grouping machines having identical slow motions into areas. The singular perturbation method is used to separate the slow variables which are the area center of inertia variables and the fast variables which describe the intermachine oscillations within the areas. The areas obtained by this method are independent of fault locations. Three types of simulation approximations illustrated on a nonlinear 48 machine system indicate the validity of this algorithm.
This paper describes the first algorithm to efficiently compute
the dominant poles of any specified high order transfer function. As the
method is closely related to Rayleigh iteration (generalized Rayleigh
quotient), it retains the numerical properties of global and ultimately
cubic convergence. The results presented are limited to the study of low
frequency oscillations in electrical power systems, but the algorithm is
completely general
In this manuscript, we propose a novel reduction framework for obtaining Reduced Order Models (ROMs) of large-scale, nonlinear dynamical systems. We advocate the use of Nonlinear Moment Matching (NLMM) with the Dynamic Mode Decomposition (DMD) to get a much efficient dimensionality reduction scheme. While NLMM does not require the expensive computation of the time-displaced snapshot ensemble of the Full Order Model (FOM) in the offline stage, DMD avoids the evaluation of the nonlinear term in the online stage. Thus, significant savings in terms of the overall CPU-time is achieved. To substantiate our observations, we test the proposed scheme on a variety of benchmark models and compare the results with POD–DEIM for reference.
Reduced order models, or model reduction, have been used in many technologically advanced areas to ensure the associated complicated mathematical models remain computable. For instance, reduced order models are used to simulate weather forecast models and in the design of very large scale integrated circuits and networked dynamical systems. For linear systems, the model reduction problem has been addressed from several perspectives and a comprehensive theory exists. Although many results and efforts have been made, at present there is no complete theory of model reduction for nonlinear systems or, at least, not as complete as the theory developed for linear systems. This monograph presents, in a uniform and complete fashion, moment matching techniques for nonlinear systems. This includes extensive sections on nonlinear time-delay systems; moment matching from input/output data and the limitations of the characterization of moment based on a signal generator described by differential equations. Each section is enriched with examples and is concluded with extensive bibliographical notes. This monograph provides a comprehensive and accessible introduction into model reduction for researchers and students working on non-linear systems.
This paper transfers the concept of moment matching to nonlinear structural systems and further provides a simulation-free reduction scheme for such nonlinear second-order models. After first presenting the steady-state interpretation of linear moment matching, we then extend this reduction concept to the nonlinear second-order case based on Astolfi [2010]. Then, similar simplifications as in Cruz Varona et al. [2019a] are proposed to achieve a simulation-free nonlinear moment matching algorithm. A discussion on the simplifications and their limitations is presented, as well as a numerical example which illustrates the efficiency of the algorithm.
This paper proposes a method for the Model Order Reduction (MOR) of large scale power system models that produces Reduced Order Models (ROM) which preserve the access to selected parameters exactly like the original Full Order Model (FOM). The preserved parameters are related to decentralized power system devices, including, but not limited to, power system stabilizers that are used for the damping control of system electromechanical oscillations. The problem formulation of this Parametric MOR (PMOR) is described, as well as the implementation details regarding efficiency and flexibility. Test results are described for large practical power system models used in small-signal electromechanical stability studies. The reduced order models produced sufficiently accurate responses following parameters changes, when equivalently compared to the original FOM results. The proposed method allows preserving specified nonlinearities in chosen devices of the parameterized ROMs. It can also be applied to the reduction of Differential Algebraic Equation (DAE) models related to other areas of engineering.
This paper proposes a closed-loop identification method to estimate the equivalent inertia constant of a power system at the connection bus. A micro-perturbation is first performed with a well-designed multi-sine signal probed through any power electronic devices in the internal system. Then responses of frequency and active power measured by the phase measurement unit (PMU) at the connection bus are used for the closed-loop identification. Compared to the conventional transient signal based method, the proposed method has simple implementation and minimum impacts on the system security, and thus could be carried out in real-time to identify the time-varying and nonlinear equivalent inertia constant in modern power systems with complex heterogeneous components. The effectiveness of the proposed method is validated in an 8-generator 36-bus simulation system and an actual power system.
In this paper, nonlinear model reduction for power systems is performed by the balancing of empirical controllability and observability covariances that are calculated around the operating region. Unlike existing model reduction methods, the external system does not need to be linearized but is directly dealt with as a nonlinear system. A transformation is found to balance the controllability and observability covariances in
order to determine which states have the greatest contribution to the input-output behavior. The original system model is then reduced by Galerkin projection based on this transformation. The proposed method is tested and validated on a system comprised of a 16-machine 68-bus system and an IEEE 50-machine 145-bus system. The results show that by using the proposed model reduction the calculation efficiency can be greatly improved; at the same time, the obtained state trajectories are close to those for directly simulating the whole system or partitioning the system while not performing reduction. Compared with the balanced truncation method based on a linearized model, the proposed nonlinear model reduction method can guarantee higher accuracy and similar calculation efficiency. It is shown that the proposed method is not sensitive to the choice of the matrices for calculating the empirical covariances.
This paper reviews and applies the balanced realization (BR) theory to obtain reduced-order models from dynamic system equivalents of electric power networks. BR allows obtaining reduced-order models via a process in which the original asymptotic stable system is internally balanced. The balanced system is truncated according to its dominant dynamics. It can be proven that the truncated, that is, reduced-order, system is also stable. This paper applies the BR method to dynamic system equivalents represented as frequency-dependent network equivalents (FDNEs). Furthermore, it shows that an assumed reduced-order FDNE can be further reduced via the BR process. Four case studies, involving one transmission network and three wind power plants, are presented.
Let A be a real m×n matrix with m≧n. It is well known (cf. [4]) that
(1)
where
The matrix U consists of n orthonormalized eigenvectors associated with the n largest eigenvalues of AAT, and the matrix V consists of the orthonormalized eigenvectors of ATA. The diagonal elements of ∑ are the non-negative square roots of the eigenvalues of ATA; they are called singular values. We shall assume that
Thus if rank(A)=r, σr+1 = σr+2=⋯=σn = 0. The decomposition (1) is called the singular value decomposition (SVD).
This paper presents an empirical Gramian balanced reduction method which efficiently solves nonlinear power system model reduction problems. This method projects the nonlinear power system dynamic model to a lower dimension subspace and the reduced model can retain the original nonlinear system input and output dynamic behaviors. A four-generator nonlinear power system, combining different factors which form empirical controllable and observable Gramian matrices, is used to analyze how external excitations and outputs affect balanced empirical Gramian reduction. The case verifies the feasibility and effectiveness of balanced empirical Gramian reduction method.
The regulator servomechanism problem is to design a feedforward and feedback control law to make the output of a given system called the plant, track a signal from a given class. There are various ways to make this precise, we shall follows Francis [5] and Isidori and Byrnes [11]. The class of signals to be tracked are described as the output of a second system called the signal generator (or exosystem). The control law consists of feedforward terms involving the state of signal generator, feedback terms involving the state of the plant and mixed terms involving both.
In interconnected power systems, dynamic model reduction can be applied to generators outside the area of interest (i.e., study area) to reduce the computational cost associated with transient stability studies. This paper presents a method of deriving the reduced dynamic model of the external area based on dynamic response measurements. The method consists of three steps, namely dynamic-feature extraction, attribution, and reconstruction (DEAR). In this method, a feature extraction technique, such as singular value decomposition (SVD), is applied to the measured generator dynamics after a disturbance. Characteristic generators are then identified in the feature attribution step for matching the extracted dynamic features with the highest similarity, forming a suboptimal “basis” of system dynamics. In the reconstruction step, generator state variables such as rotor angles and voltage magnitudes are approximated with a linear combination of the characteristic generators, resulting in a quasi-nonlinear reduced model of the original system. The network model is unchanged in the DEAR method. Tests on several IEEE standard systems show that the proposed method yields better reduction ratio and response errors than the traditional coherency based reduction methods.
In this paper, a hybrid dynamic equivalent consisting of both a coherency-based conventional equivalent and an artificial neural network (ANN)-based equivalent is developed and analyzed. The ANN-based equivalent complements the coherency-based equivalent at all the boundary buses of the retained area. It is designed to compensate for the discrepancy between the full system model and the reduced equivalent developed using any commercial software package, such as the dynamic reduction program (DYNRED), by providing appropriate power injections at all the boundary buses. These injections are provided by the ANN-based equivalent which is trained using the outputs from a trajectory sensitivity simulation of the system response to a candidate set of disturbances. The proposed approach is tested on a system representing a portion of the WECC system. The case study shows that the hybrid dynamic equivalent method can enhance the accuracy of the coherency-based dynamic equivalent without significantly increasing the computational effort.
The large-scale power systems are liable to performance deterioration due to the presence of sudden small load perturbations, parameter uncertainties, structural variations, etc. Due to this, modern control aspects are extremely important in load frequency control (LFC) design of power systems. In this paper, the LFC problem is illustrated as a typical disturbance rejection as well as large-scale system control problem. For this purpose, simple approach to LFC design for the power systems having parameter uncertainty and load disturbance is proposed. The approach is based on two-degree-of-freedom, internal model control (IMC) scheme, which unifies the concept of model-order reduction like Routh and Padé approximations, and modified IMC filter design, recently developed by Liu and Gao [24]. The beauty of this paper is that in place of taking the full-order system for internal-model of IMC, a lower-order, i.e., second-order reduced system model, has been considered. This scheme achieves improved closed-loop system performance to counteract load disturbances. The proposed approach is simulated in MATLAB environment for a single-area power system consisting of single generating unit with a non-reheated turbine to highlight the efficiency and efficacy in terms of robustness and optimality.
This paper presents a novel approach to the coherency identification technique in interconnected power system using independent component analysis (ICA). The ICA is applied to the generator speed and bus angle data to identify the coherent areas of the system. The results of the application of ICA using simulated data from 16-machine 68-bus system model and on data gathered through U.K. University-based Wide-Area Measurement System are presented. The approach is able to identify the cluster of generators and buses following a disturbance in the system. It is also demonstrated that the approach is robust in the presence of noise in measured signal, which is an important factor to be considered for assessing the effectiveness of any measurement-based technique.
A new intelligent system (IS) is developed for real-time dynamic security assessment (DSA) of power systems. Taking an ensemble learning scheme, the IS structures a series of extreme learning machines (ELMs) and generalizes the randomness of single ELMs during the training. Benefiting from the unique properties of ELM and the strategically designed decision-making rules, the IS learns and works very fast and can estimate the credibility of its DSA results, allowing an accurate and reliable pre-fault DSA mechanism: credible results can be directly adopted while incredible results are decided by alternative tools such as time-domain simulation. This makes the IS promising for practical application since the potential unreliable results can be eliminated for use. Case studies considering classification and prediction are, respectively, conducted on an IEEE 50-machine system and a dynamic equivalent system of a real-world large power grid. The efficiency, robustness, accuracy, and reliability of the IS are demonstrated. In particular, it is observed that the IS could provide 100% classification accuracy and very low prediction error on its decided instances.
We present an efficient reduced-basis discretization procedure for partial differential equations with nonaffine parameter dependence. The method replaces nonaffine coefficient functions with a collateral reduced-basis expansion which then permits an (effectively affine) offline–online computational decomposition. The essential components of the approach are (i) a good collateral reduced-basis approximation space, (ii) a stable and inexpensive interpolation procedure, and (iii) an effective a posteriori estimator to quantify the newly introduced errors. Theoretical and numerical results respectively anticipate and confirm the good behavior of the technique. To cite this article: M. Barrault et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).RésuméNous présentons dans cette Note une méthode rapide de base réduite pour la résolution d'équations aux dérivées partielles ayant une dépendance non affine en ses paramètres. L'approche propose de remplacer le calcul des fonctionelles non affines par un développement en base réduite annexe qui conduit à une évaluation en ligne effectivement affine. Les points essentiels de cette approche sont (i) un bon système de base réduite annexe, (ii) une méthode stable et peu coûteuse d'interpolation dans cette base, et (iii) un estimateur a posteriori pertinent pour quantifier les nouvelles erreurs introduites. Des résultats théoriques et numériques viennent anticiper puis confirmer le bon comportement de cette technique. Pour citer cet article : M. Barrault et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).
The problem of controlling the output of a system so as to achieve asymptotic tracking of prescribed trajectories and/or asymptotic rejection of disturbances is one of the most fundamental problems in control theory. In linear system theory, a powerful approach to the solution of this problem is the design of a controller incorporating an internal model of all exogenous inputs. The extension of this approach to nonlinear systems has been thoroughly investigated in the past 10 years. The purpose of this paper is to review the major achievements in this area as well as to outline the major problems on which the research is presently progressing. Copyright © 2000 John Wiley & Sons, Ltd.
This paper presents a new method of missing point estimation (MPE) to derive efficient reduced-order models for large-scale parameter-varying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projection-based model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of data-dependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25% of the spatial grid points without compromising the accuracy of the reduced model.
The model reduction problem by moment matching for linear systems is revisited. The proposed approach provides an alternative to the well-established projection methods. A parameterized family of reduced order models achieving model matching is characterized, together with methods to select an element of the family to enforce specific properties of the reduced order model. The theory is illustrated by means of several worked-out examples.
Wide-area analysis and control of large-scale electric power systems are highly dependent on the idea of aggregation. For example, one often hears power system operators mentioning how northern Washington oscillates against southern California in response to various disturbance events. The main question here is whether we can analytically construct dynamic electromechanical models for these conceptual, aggregated generators representing Washington and California, which in reality are some hypothetical combinations of thousands of actual generators. In this paper we address this problem, and present a concise overview of several new results on how to construct simplified interarea models of large power networks by using dynamic measurements available from phasor measurement units (PMUs) installed at specific points on the transmission line. Our examples of study are motivated by widely encountered power transfer paths in the Western Electricity Coordinating Council (WECC), namely, a two-area radial system representing the WA-MT flow, and a star-connected three-area system resembling the Pacific AC Intertie.
The model reduction problem for (single-input, single-output) linear and nonlinear systems is addressed using the notion of moment. A re-visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment. This, in turn, is used to pose and solve the model reduction problem by moment matching for nonlinear systems, to develop a notion of frequency response for nonlinear systems, and to solve model reduction problems in the presence of constraints on the reduced model. Connections between the proposed results, projection methods, the covariance extension problem and interpolation theory are presented. Finally, the theory is illustrated by means of simple worked out examples and case studies.
A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.
A new technique is described for the automatic formation of dynamic equivalents of generating units represented by detailed models. The method applies to groups of generating units that are coherent. A coherent group of generating units, for a given perturbation, is a group of generators oscillating with the same angular speed, and terminal voltages in a constant complex ratio. Thus, the generating units belonging to a coherent group can be attached to a common bus, if necessary through an ideal, complex ratio transformer. The dynamic equivalent of a coherent group of generating units is a single generating unit that exhibits the same speed, voltage and total mechanical and electrical power as the group during any perturbation where those units remain coherent. The parameters of an equivalent model of governor, turbine, synchronous machine, excitation system and power system stabilizer are identified for each group of coherent units, by means of a least-square fit of their transfer functions. The technique is demonstrated with a large system stability data base. A brief discussion of paper is appended.
A new model for the study of power system stability via Lyapunov functions is proposed. The key feature of the model is an assumption of frequency-dependent load power, rather than the usual impedance loads which are subsequently absorbed into a reduced network. The original network topology is explicitly represented. This approach has the important advantage of rigorously accounting for real power loads in the Lyapunov functions. This compares favorably with existing methods involving approximations to allow for the significant transfer conductances in reduced network models. The preservation of network topology can be exploited in stability analysis, with the concepts of critical and vulnerable cutsets playing central roles in dynamic and transient stability evaluation respectively. Of fundamental importance is the feature that the Lyapunov functions give a true representation of the spatial distribution of stored energy in the system
This paper presents an analytical approach to the construction of power system dynamic equivalents for use in stability calculations and dynamic simulations. The method presented is capable of accurately representing the dynamical effects of generator field and amortisseur windings, voltage regulators, and speed governors. Experimental evaluation of equivalents constructed by this method is reported in a companion paper. Copyright © 1971 by The Institute of Electrical and Electronics Engineers, Inc.
This paper describes a numerical linear algebraic algorithm to compute the dominant poles of multi-input-multi-output (MIMO) high-order transfer functions. The results presented are related to the study of electromechanical oscillations in large electrical power systems, but the algorithm is completely general. The computed dominant poles may then be used to build modal equivalents for MIMO transfer functions of large linear systems, among other applications.
This paper explores synchrony, a generalization of the concept of slow-coherency, and outlines how it can form the basis for efficient construction of dynamic equivalents by aggregation. The paper describes a novel approach for selecting the inter-area modes that are to be represented by the aggregate model. A clustering algorithm for recognizing approximate synchrony is presented, and improvements over the standard slow-coherency recognition algorithm are noted. Using for illustration a 23-generator power system model with 325 state variables, the paper demonstrates the effectiveness of a synchrony-based approach to decomposing the eigenanalysis of the electromechanical modes, separating the computation of inter-area and intra-area modes in the style of multi-area selective modal analysis.