Balanced Realization Theory Research Articles

Analysis of Frequency-Dependent Network Equivalents in Dynamic Harmonic Domain

Rational function-based models have proved to be very efficient for accurate frequency-dependent modeling of power system components. These models are able to characterize the components terminal behaviours (analysing the admittance matrix) for nodal analysis. This provides a fast convergence and inherent stability to the solution routine of the model. This work presents a general framework for interfacing the dynamic phasor method to the rational models. That would be promising for the electromagnetic transient analysis (under harmonic distortion), in the frequency domain. Therefore, Y-element rational pole-residue models (employing the vector fitting method) are developed. Moreover, the pole-residue model is converted into the state-space representation. Next, the dynamic harmonic approach (in the frequency domain) is employed for harmonic analysis. It is shown that the order of state-space system can become a major concern for frequency-dependent networks analysis. Therefore, to generate a reduced-order model, the balanced realization theory is applied. Moreover, (for the sake of simplicity and efficiency) the trapezoidal integration rule is employed to discretise the state-space equations. For validation of the modelling, it is applied on three test case studies and results of these studies are compared with their time-domain analysis results.

Reduced-Order Small-Signal Models of Modular Multilevel Converter and MMC-Based HVdc Grid

Generally, the small-signal model (SSM) of a modular multilevel converter (MMC) needs to consider the dynamics of equivalent arm capacitors, which inevitably increases the model order. In view of this, this paper proposes a general order-reducing approach based on balanced realization theory to reduce the SSM order of the MMC. In particular, two reduced-order models of an MMC-based HVdc grid are proposed with respect to different research objectives. To determine the potential minimum order of single MMC or MMC-HVdc grid, the criteria including singular values (SVs) of the frequency response, dynamic response in the time domain, and the largest absolute error of SVs are fully investigated. The simulation results of a four-terminal MMC-HVdc grid show that reduced-order models of the MMC and the HVdc grid can precisely predict both the stability and instability while the dynamic characteristics are retained.

A new approach to physics-based reduction of power system dynamic models

This paper proposes a new approach to physics-based reduction of large-scale power system dynamic models, described by a set of differential and algebraic equations (DAEs). The proposed technique, based on the balanced realization theory (BRT), replaces the original model by one of reduced order, while preserving the physical meaning of retained state variables. Input and output variables of the model are selected in accordance with the goal of a particular study. State variables removed from the model are those that do not dominantly affect the nature of the phenomenon to be analyzed, as evidenced by the magnitude of the frequency response. The proposed technique provides insight into steps of the mathematical description of reduction for the considered phenomenon. In a given study, the number of state variables that can be eliminated depends on the power system model itself and on the required accuracy. Described approach is applied to two (benchmark and real-world) power systems.

An Energy-based Modal Approximation Method for Reduced-order Modeling Analysis of Power System Models

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 existing approaches.

A new framework for convergence analysis and algorithm development of adaptive IIR filters

A parameterization of an adaptive infinite impulse response (IIR) filter's poles is developed, based on balanced realization theory. From this, we develop a local approximation of the actual adapted pole parameters, in which convergence speed is related to a certain eigenvalue spread. This, in turn, is shown to relate to the Hankel singular values of the system to be identified, as well as certain coefficient sensitivity functions of the adapted filter. The local approximation is not restricted to stationary points. At these points, however, it is equivalent to a Hessian approximation, with the benefit of decomposing the Hessian matrix into terms related to the aforementioned singular values and sensitivity functions. The description of the adaptation process by means of the developed approximation leads to a greater understanding of the effects on convergence speed of factors, such as the Hankel singular values of the system, its order, the distribution of its poles, and the choice of adapted parameters. In particular, the use of direct form and lattice parameters are compared in detail. Based on these properties, a new adaptive IIR algorithm with faster convergence and relatively low computational complexity has recently been proposed, which is briefly mentioned. Results also indicate a potential for variable gain algorithms.

Balanced Model Reduction via the Proper Orthogonal Decomposition

A new method for performing a balanced reduction of a high-order linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshotsisused to obtainlow-rank,reduced-rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain.Theapproximationsarethenusedtoobtainabalancedreducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized high-order system that models unsteady motion of a two-dimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.

On Similarity Invariance of Balancing for Nonlinear Systems

A previously obtained balancing method for nonlinear systems is investigated on similarity invariance by generalization of the observations on the similarity invariance of the linear balanced realization theory. For linear systems it is well known that the Hankel singular values are similarity invariants. It is shown that under some additional conditions a similar statement on the similarity invariance holds for nonlinear systems. To be able to do so the concepts of local zero-state observability and local strong accessibility are considered. The local strong accessibility is an additional condition, which is needed to consider a nonlinear generalization of the Kalman decomposition. In the local coordinates that correspond to this decomposition the controllability and observability functions are investigated.

Selection of component modes for flexible multibody simulation

The assumed-modes method in multibody dynamics allows the elastic deformation of each component in the system to be approximated by a sum of products of spatial and temporal functions commonly known as modes and modal coordinates, respectively. This paper focuses on the choice of component modes used to model articulating and nonarticulating flexible multibody systems. Attention is directed toward three classical component mode synthesis methods whereby component normal modes are generated by treating the component interface as either fixed, free, or loaded with mass and stiffness contributions from the remaining components. The fixed and free interface normal modes are augmented by static shape functions termed constraint and residual modes, respectively. In this paper, a mode selection procedure is outlined whereby component modes are selected from the Craig-Bampton (fixed interface plus constraint), MacNeal-Rubin (free interface plus residual), or Benfield-Hruda (loaded interface) mode sets in accordance with a modal ordering scheme derived from balanced realization theory. The success of the approach is judged by comparing the actuator-to-sensor frequency response of the reduced-order system with that of the full-order system over the frequency range of interest. A finite element model of the Galileo spacecraft serves as an example in demonstrating the effectiveness of the proposed mode selection method.

Estimating Order Reduction for Dynamic Equivalents

The estimation of the accuracy of reduced order models or dynamic equivalents, for use in transient stability studies is investigated. Balanced realization theory is combined with modal coherency analysis to provide accurate estimates for dynamic equivalents based on coherency aggregation techniques. Test results for 23 and 86 generator systems are provided.