Transfer Function Residues Research Articles

A Novel Approach for Evaluating Nonstationary Response of Dynamic Systems to Stochastic Excitation

The transient state of a dynamic system, such as offshore structures, to random excitation is always nonstationary. Many studies have contributed to evaluating response covariances at the transient state of a linear multi-degree-of-freedom (MDOF) system to random excitations, but a closed-form solution was not available unless the excitation was assumed to be a physically unrealizable white noise process. This study derives explicit, closed-form solutions for the response covariances at the transient state by using a pole-residue (PR) approach operated in the Laplace domain when the excitations are assumed to be stationary random processes described by physically realizable spectral density functions. By using the PR method, we can analytically solve the triple integral in evaluating the nonstationary response covariance. As this approach uses the poles and residues of system transfer functions, rather than the conventional mode superposition technique, the method is applicable to MDOF systems with non-classical damping models. Particular application of the proposed method is demonstrated for multi-story shear buildings to stochastic ground acceleration characterized by the Kanai-Tajimi spectral density function model, and a numerical example is provided to illustrate the detailed steps. No numerical integrations are required for computing the response covariances as the exact closed-form solution has been derived. The correctness of the proposed method is numerically verified by Monte Carlo simulations.

Computation of the controllable swing mode spectrum of FACTS and HVDC in large power systems

This paper presents computation of swing modes of a large power system that could be significantly affected by power swing damping controllers in FACTS or HVDC devices at a given location. Modal controllability is a suitable measure to isolate these modes for analysis. Computation of the controllable swing mode spectrum is useful, especially in situations where the controller structure and feedback signals are not frozen (e.g., at the planning stage). This paper proposes two important steps that allow us to map the problem of finding highly controllable swing modes to the problem of finding the swing modes that have high transfer function residues (for which efficient algorithms are available). The steps are: (a) normalization of the eigenvectors corresponding to different modes and (b) identification of specific feedback signals for each type of FACTS/HVDC device such that the modal observability and modal controllability are tightly coupled. Once the mapping is done, a computationally efficient method like the Subspace Accelerated Dominant Pole Algorithm [16] (SADPA) can be adapted to find the highly controllable swing modes. The effectiveness of this approach is demonstrated by case studies of FACTS and HVDC devices in a 16-machine system and the Indian power grid.

A Real-Time Attack Localization Algorithm for Large Power System Networks Using Graph-Theoretic Techniques

We develop a graph-theoretic algorithm for localizing the physical manifestation of attacks or disturbances in large power system networks using real-time synchrophasor measurements. We assume the attack enters through the electro-mechanical swing dynamics of the synchronous generators in the grid as an unknown additive disturbance. Considering the grid to be divided into coherent areas, we pose the problem as to localize which area the attack may have entered using relevant information extracted from the phasor measurement data. Our approach to solve this problem consists of three main steps. We first run a phasor-based model reduction algorithm by which a dynamic equivalent of the clustered network can be identified in real-time. Second, in parallel, we run a system identification in each area to identify a transfer matrix model for the full-order power system. Thereafter, we exploit the underlying graph-theoretic properties of the identified reduced-order topology, create a set of localization keys, and compare these keys with a selected set of transfer function residues. We validate our results using a detailed case study of the two-area Kundur model and the IEEE 39-bus power system.

Hybrid modal‐balanced truncation method based on power system transfer function energy concepts

The combined use of modal and balanced truncations methods is proposed for model order reductions. To efficiently combine these methods, a stopping criterion based on spectral energy concepts is also proposed. This criterion was implemented into the code of the widely known subspace accelerated dominant pole algorithm (SADPA), designed to compute a set of dominant poles and associated residues of transfer functions from large-scale, sparse, linear descriptor systems. The resulting enhanced SADPA code automatically stops once the computed set of dominant poles and associated residues is sufficient to build a modal reduced order model (ROM) whose energy content approaches that of the complete model within a specified tolerance and considering a frequency window of interest. The number of dominant poles in this set is much smaller than the number of poles of the full system model. Hence, their state-space realisation usually has a small enough dimension for the efficient application of the square root balanced truncation method. This new method, named hybrid modal-balanced truncation, produces ROMs whose order are much smaller than that of the modal ROMs and, most importantly, can also be applied to unstable models.

Complex eigenvector scaling from mass perturbations

This paper presents an approach to normalize experimentally extracted complex eigenvectors so that their outer product gives transfer function residues. The approach, an implementation of the mass perturbation strategy, is exact for arbitrary perturbation magnitudes and number of sensors when the modal space is complete and is robust against modal truncation. It is shown that improvements over a sensitivity solution are significant when the relation between the eigenvalue and the perturbation magnitude is strongly nonlinear.

Design of WAMS-Based Multiple HVDC Damping Control System

High voltage direct current (HVDC) links are interties, characterized by good controllability, large adjustable capacity and fast responses, and have inherent advantage in damping interarea oscillations over power system stabilizers (PSS) and flexible alternating current transmission systems (FACTS). In this paper, a systematic and practical means to design the multiple HVDC damping control system, which aims to enhance the damping of certain interarea modes and minimize the interactions between different control loops, is provided. In the proposed procedure, the wide-area feedback signals that have the good observability of target modes are preselected by an index based on residue analysis firstly. Then the interactions between the candidate control loops, in which the setting power reference of the HVDC as the input and the preselected wide-area signals as the output, are investigated by relative gain array (RGA) analysis. According to the RGA results, a suitable input-output pairing for multiple HVDC damping controller is determined. The transfer function residues used in the feedback signal preselection and the RGA calculation are identified by Prony analysis in the transient stability program and electromagnetic transient program respectively. Finally, the damping controller parameters are optimized by time-domain simulations. The application in China Southern Power Grid (CSG) is illustrated as the case study.

Identification of Electromechanical Modes and Placement of PSSs Using Relative Gain Array

The paper investigates the possibility to use relative gain array (RGA) for power system small-disturbance stability analysis. The RGA has been successfully used in the past as steady state measure of interactions for multivariable, decentralized control in chemical and process engineering. The effectiveness of the RGA in identifying electromechanical modes and selecting the optimal location for placement of PSSs is compared with the conventional approach that uses transfer function residues, eigenvalues and participation factors. Two different multimachine power systems are used for the case studies. Limitations and advantages of the use of RGA in this purpose are clearly indicated in the paper.

Modal Expansion of Structural Systems with Time Delays

Conventional modal analyses are not applicable to structural systems with time delays because time delays destroy any possible orthogonality among system eigenvectors. A new series solution method is proposed for transient response analysis of a class of delayed linear structural systems. In this method, inverse Laplace transform is combined with a root locus sensitivity analysis, leading to a relation between the transfer function residues and eigensolutions of the delayed system. With this relation, closed-form time-domain response can be accurately and efficiently estimated in series of eigenvectors, without involving transfer function singularities and s-domain integration, which are usually encountered in inverse Laplace transform. The proposed method is illustrated on a gyroscopic dynamic system under delayed feedback control.

An investigation on the use of power system stabilizers for damping inter-area oscillations in longitudinal power systems

This paper presents the progress of an investigation on the use of power system stabilizers (PSSs) to enhance the damping of low-frequency inter-area modes in longitudinal power systems. A dynamic model of the Mexican Interconnected System (MIS) is adopted to evaluate the potential for small signal stability improvement by means of PSSs. Power oscillation flow studies, transfer function residues and controllability and observability characteristics are used to understand the nature of oscillation patterns, mode identification and PSSs siting. It is shown that PSSs can significantly enhance the damping of modal oscillations, especially under high stress conditions.

Prony based methods for simultaneous identification of transfer functions and initial conditions

This work extends previous results on transfer function identification using methods based on Prony signal analysis. With the Prony based method of this paper, input signals are piecewise continuous and are characterized by sets of input eigenvalues between points of discontinuity. Previously developed Prony based methods for transfer function identification are limited in terms of the class of input signals considered. The advantage of the more general input form of this paper is that it can be tailored to excite system modes in frequency ranges of interest. In addition to generating system eigenvalues and transfer function residues, the approach also can be used to obtain a feedthrough gain and initial condition residues. In the first phase of the solution, system eigenvalues are estimated; and in the second phase, transfer function residues and initial condition residues are estimated. Methods of model order selection also are described, and a detailed example with noise included is given to illustrate the approach.

Stabilization of inter-area oscillation modes through excitation systems

This paper presents new analysis and design methods, based on transfer function residues to: (a) classify oscillation modes as either local or interarea; (b) determine feasibility of a given control structure such as conventional decentralized power system stabilizers (PSSs) or PSSs with remote input signals; (c) identify optimal locations for PSSs; and (d) design robust PSSs. A two-area four-generator power system is used to illustrate the proposed algorithms. >

Identifying linear reduced-order models for systems with arbitrary initial conditions using Prony signal analysis

A method of identifying reduced-order linear models for systems operating in the neighborhood of an equilibrium point is presented. The method is based on Prony signal analysis, which has recently received considerable attention in the study of power system electromechanical oscillations. Prior to the application of the input test signal, the system can be in a transient state. The system input test signal is piecewise continuous and allows several Prony analyses to be performed during a transient, with each analysis conducted between input discontinuities. Results of these Prony analyses can be combined in various ways to obtain system eigenvalues, transfer-function residues, and initial condition residues. Two examples are given to illustrate the use of the method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Determination of suitable locations for power system stabilizers and static VAR compensators for damping electromechanical oscillations in large scale power systems

Efficient algorithms are presented for the solution of two important problems in the area of damping control of electromechanical oscillations in large-scale systems. The proposed algorithms allow the determination of: the most suitable generators for installing power system stabilizers; the most suitable buses in the system for placing static VAr compensators in order to damp the critical modes of oscillation. These algorithms involve the calculation of transfer function residues and represent an important extension of the powerful methodology described by V. Arcidiaconos et al. (see IEEE Trans. Power Apparatus and Systems, vol.PAS-99, p.769-78, 1980), whose use was up to now restricted to power systems of limited size. A major advantage of this methodology is that there is no limitation on the degree of modeling of the power system being studied.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Eigensolutions of non‐conservative continuous mechanical systems computed by use of a matrix iteration method

AbstractTwo‐sided Ostrowski matrix iteration is applied to solve eigenproblems of non‐conservative continuous systems in structural dynamics starting from known eigensolutions of a related conservative system. Eigensolution paths are followed as the parameters of the system governing its non‐conservative behaviour are increased from zero. No discretization is performed and no truncation errors are introduced. The real eigenvalues of the related conservative problem are calculated by use of the Wittrick–Williams algorithm. The corresponding eigenvectors are obtained with an inverse iteration method using eigenfrequency‐dependent trial eigenvectors. Residues of system harmonic transfer functions are computed using the eigenvalues and eigenvectors together with eigenvector normalization constants (modal Foss dampings). These residues are needed in a transient analysis. The method is applied to the generalized Beck column stability problem. Critical loads are calculated. Harmonic flexibilities of subcritically loaded columns in transverse vibration are studied. Applications to control problems are foreseen.

Method for Computing Time Response of Systems Described by Transfer Functions

A simple numerical method for computing the time domain response of linear time invariant systems described by their transfer functions is presented. The method does not require computation of transfer function poles or residues; it is not influenced by the multiplicity of poles or zeroes, nor does it require computation of the matrix exponential. Rather, it is based on a numerical method for inverting Laplace transforms. It is equivalent to very high order, absolutely stable numerical integration. Stiff systems present no problems.

On the stability of convolution feedback systems with dynamical feedback

This paper considers distributed n-input n-output convolution feedback systems characterized by y = G 1∗e , z = G 2∗y and e = u − z, where the forward path transfer function G ̂ 1 and the feedback path transfer function G ̂ 2 both contain a real single unstable pole at different locations. Theorem 1 gives necessary and sufficient conditions for both input-error and input-output stability. In addition to usual conditions that guarantee input-error stability a new condition is found which results in the fact that input-error stability will guarantee input-output stability. These conditions require to investigate only the open-loop characteristics. A basic device is the consideration of the residues of different transfer functions at the open-loop unstable poles. An example is given.