Newton algorithm for fitting transfer functions to frequency response measurements

The purpose of this paper is to explore the role of Battery Energy Storage System (BESS) in improving the frequency response of a low inertia power system. For the recent years, prolific amount of renewable energy resources that include wind and photovoltaic (PV) sources have been introduced in South Australian power system. These wind generators are mostly variable speed machines, hence they neither have inertia response nor can they perform governor like response following a contingency. The demand in South Australia can be significantly catered utilizing renewable sources and considerable power import from the adjacent state Victoria. Therefore, the number of synchronous generators in the South Australian network can be small. In such situations, the frequency response following a contingency can be concerning and the network may undergo substantial under frequency load shedding. To overcome this challenge, this paper proposes a frequency responsive BESS model to improve the frequency response of a low inertia power system that resembles the South Australian network. The network is investigated under various operating conditions to explore the performance of BESS regarding frequency response enhancement.

A parallel damping arrangement is typically included on the secondary side of a capacitor voltage transformer to improve the measurement accuracy and transient response. The damping arrangement, however, can affect the frequency and transient response of the capacitor voltage transformer. This paper investigates the use of a pseudo-random impulse sequence as a perturbation signal to estimate the model parameters of a passive capacitor voltage transformer damping arrangement. Transfer functions describing the damping arrangement are derived and are used to determine the observability of the circuit. The pseudo-random impulse sequence perturbation signal is applied to the terminals of the damping system, and system responses are recorded and subsequently used to estimate the associated frequency responses. Estimated parameter values, obtained from both the time-domain and frequency-domain responses using the genetic algorithm, are presented. It is shown that the frequency responses obtained using the derived mathematical functions correlate closely with the estimated responses. The damping arrangement parameter values are also shown to be estimated with a high degree of accuracy when more that one transfer function is used in the parameter estimation procedure.

Tapered transmission line resonators can be used to achieve specific frequency responses and enhance the performance of microwave devices and impedance transformers. To develop a mathematical model (transfer function) of such a wideband resonator will facilitate the construction of an equivalent circuit and the synthesis of resonators of this type. This study discusses the system modeling of tapered transmission line (TTL) resonators using the singularity expansion method (SEM). The transfer function is analytically established for two types of wideband TTL resonators: exponential tapered transmission lines (ETTL) and linear tapered transmission lines (LTTL). The analytical transfer function is derived for the general tapered ratio (k/β). The physical poles of the structures are identified using the pole-energy matrix pencil (MP) approach. The transfer function expression is then utilized to develop an accurate equivalent lumped-element circuit for the TTL resonator. This study’s simplified approach for estimating the transfer function and associated poles and zeros from the S-parameters is the main contribution, whereas it is known that the transfer function becomes complicated for wideband frequency response. A direct synthesis approach is presented to develop two-stage and four-stage ultra-wideband (UWB) bandpass filters. The agreement between the frequency response data and the derived transfer function of the system model of the synthesized TTL filters demonstrates the reliability of the presented method. The proposed system modeling and synthesis methodology can be applied to more sophisticated filter TTL architectures.

This paper describes a practical off-line approach to system identification of very high-order, lightly-damped, multivariable systems. In particular, we address the practical and computational aspects of this problem. We also discuss the various choices that the modeler must make interactively. Our eventual objective is to enable automated modeling for such systems by minimizing the burden on the modeler. We begin with sampled measurements of the system's frequency response to obtain an initial model. This model is further refined based on time-domain data as it becomes available. The method is designed to identify very high-order systems by solving several low-order identification problems using the iterative least squares algorithm of Santhanam-Koerner, coupled with the multi-band strategy of Bayard. The individual solutions are combined to yield a transfer-function matrix model of the overall system. A succinct state-space realization of this estimate may be obtained using standard model reduction methods.

Chapter 4 - Frequency Analysis: The Fourier Series

A novel complementary filter for three-dimensional (3D) attitude estimation is proposed. In order to deals with the nonlinearity of 3D attitude, the conventional complementary filter integrated the measured angular velocity corrected by the vector measurement, but it ignored the sensor dynamics which causes the lag. Authors' previous work compensated the sensor dynamics based on the idea that it can be divided into the linear transfer function and the nonlinear coordinate transformation. The properness and stability of the total transfer function are guaranteed by the designed filter inserted between the linear and nonlinear part. However, the disturbance strongly affects the accuracy of the azimuth angle due to the coordinate transformation. This paper aims to improve the estimation accuracy by combining the idea of correcting the measured angular velocity by the vector measurement and the sensor dynamics compensation only with the proper and stable transfer function. The characteristics of the proposed method are the following two points: 1) the relaxation of the complementary condition in order to make the non-proper and unstable function proper and stable, and 2) the filtering of the estimated vector by the sensor dynamics instead of the filtering of the measured vector by the inverse dynamics in order to avoid the magnification of the high frequency noise. Through the experiment for the fast and irregular motion under 5[Hz], the validity of the proposed method is verified.

The authors present a system performance analysis of an optical single sideband system (OSSB) when phase errors occur in the modulating signal driving the dual-arm Mach–Zehnder modulator (MZM). The performance is evaluated in terms of the optical sideband suppression, received subcarrier power, frequency response, and power-penalty measurements. Experiments were carried out and a mathematical model of the OSSB system, incorporating the phase error, was developed. Both the theoretical and experimental results show that, although optical sideband suppression reduces with phase errors, the received subcarrier power can increase as high as 3 dB for positive phase errors. However, for transmission over longer fiber, the frequency response of the OSSB system is not only dependent on the phase error, but also on the phase shift due to chromatic dispersion (CD). The power penalty was found to be dependent on the frequency response of the OSSB system. At any frequency-response null point, the power-penalty slope is reversed as the received subcarrier power is reduced. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1574–1578, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21724

An efficient method for calculating the frequency response of a proportional damping system over a given frequency interval

Frequency response has received a lot of attention in recent years at the national level, which culminated in the development and approval of North American Electricity Reliability Corporation (NERC) BAL-003-1 Frequency Response and Frequency Bias Setting Reliability Standard. This report is prepared to describe the details of the work conducted by Pacific Northwest National Laboratory (PNNL) in collaboration with the Bonneville Power Administration and Western Electricity Coordinating Council (WECC) Joint Synchronized Information Subcommittee (JSIS) to develop a frequency response analysis tool (FRAT). The document provides the details on the methodology and main features of the FRAT. The tool manages the database of under-frequency events and calculates the frequency response baseline. Frequency response calculations are consistent with frequency response measure (FRM) in NERC BAL-003-1 for an interconnection and balancing authority. The FRAT can use both phasor measurement unit (PMU) data, where available, and supervisory control and data acquisition (SCADA) data. The tool is also capable of automatically generating NERC Frequency Response Survey (FRS) forms required by BAL-003-1 Standard.

Current methods to study atomic force microscope (AFM) cantilever dynamics use model simplification or are based on the non-trivial solution of the equation of motion. As an alternative method, transfer function analysis gives a more complete description of system dynamics. In this work a transfer function study of two different AFM configurations, the point force and base driven cantilever, is presented. Exact analytical expressions of the infinite dimensional transfer function are derived for cantilever deflection and slope along the cantilever. Frequency response and transfer function infinite product expansion are obtained for the case where system outputs are set at the free end of the cantilever. The frequency response reflects the full complexity of cantilever dynamics affected by the presence of an infinite number of poles and zeros. An analytical expression for all the zeros and poles of the system is obtained. From the frequency response and pole-zero investigations it is shown how cantilever actuation and output measurement affect AFM operation and cantilever dynamics modelling. Transfer function analysis of AFM cantilevers opens the possibility of model based AFM operation to increase imaging and manipulation performance.

This study describes the use of a biofeedback method for the noninvasive study of baroreflex mechanisms. Five previously untrained healthy male participants learned to control oscillations in heart rate using biofeedback training to modify their heart rate variability at specific frequencies. They were instructed to match computer-generated sinusoidal oscillations with oscillations in heart rate at seven frequencies within the range of 0.01-0.14 Hz. All participants successfully produced high-amplitude target-frequency oscillations in both heart rate and blood pressure. Stable and predictable transfer functions between heart rate and blood pressure were obtained in all participants. The highest oscillation amplitudes were produced in the range of 0.055-0.11 Hz for heart rate and 0.02-0.055 Hz for blood pressure. Transfer functions were calculated among sinusoidal oscillations in the target stimuli, heart rate, blood pressure, and respiration for frequencies at which subjects received training. High and low target-frequency oscillation amplitudes at specific frequencies could be explained by resonance among various oscillatory processes in the cardiovascular system. The exact resonant frequencies differed among individuals. Changes in heart rate oscillations could not be completely explained by changes in breathing. The biofeedback method also allowed us to quantity characteristics of inertia, delay, and speed sensitivity in baroreflex system. We discuss the implications of these findings for using heart rate variability biofeedback as an aid in diagnosing various autonomic and cardiovascular system disorders and as a method for treating these disorders.

It is shown that a p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sup> m transfer function G(s) with p⩾m can be decomposed as G = (I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ...(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r+2</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r+2</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</sub> where Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> are stable all-pass transfer functions, 1=ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ..=ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ≫ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> ..≫ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ≫0 are the Hankel-singular-values of GW* <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> where G*G=WW* with W stable and minimum phase. Results on the McMillan degree of G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</sup> :=(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> )..(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> )G then show that G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</sup> gives a good low order approximation to G in the sense of relative error.

Conditions for a Class of Rational Approximants of Fractional Differentiators/Integrators to Enjoy the Interlacing Property

Stray free switched capacitor loop biquad that realizes different generic transfer functions

A new method-based on transformations from 1-variable transfer functions is presented for the design of stable 2-variable transfer functions. As compared with already existing methods, this method results in a good response obtained through fewer numbers of iterations. The designed filters are always stable without the need of any constraints during minimization.