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The general frequency response program provides the frequency response of any linear feedback control system including the open loop control system. The system characteristic matrix, obtained from the Laplace transformations of the dynamic and control equations, is input to the program. A variety of outputs are available.
Frequency Response of the full length of the Prysmian Cable with 100 ft the cable placed in the oven. One set of S-parameter data was collected at room temperature (22 °C) and the second at 300 °C after 5 days of baking.
Modern power systems are increasingly vulnerable to frequency instability as inverter-based resources (IBRs) replace synchronous machines and reduce system rotational inertia. Existing small-signal frequency stability assessment methods are either computationally intensive, relying on simulation-driven approaches, or lack analytical stability regions that explicitly account for secondary frequency response (SFR). This paper introduces new analytical frameworks that enable evaluation small-signal frequency stability while explicitly incorporating tunable IBR and SFR parameters. Using Kharitonov’s theorem with an overbounding approach, explicit small-signal stability criteria are derived. In addition, based on Białas’ criterion and Hurwitz matrix, analytical stability regions are established to reveal feasible design spaces for SFR and IBR parameters tuning. Extensive Matlab/Simulink-based simulations validate the accuracy and computational efficiency of the proposed methods, demonstrating that coordinated tuning of SFR and IBR parameters can substantially enhance system resilience. By bridging analytical rigor with practical tunability, this work provides an analytical framework for assessing small-signal frequency stability in low-inertia grids, supporting the real-time, scalable, and resilient operation of sustainable power systems.
The ongoing changes in the generation resource mix, driven by the rapid adoption of inverter-based resources (IBR) as well as the early retirement of synchronous generators, are bringing new challenges to the planning and operation of bulk electric power systems. Consequently, there is an increasing need to understand, design, and quantify the reliability service provision from IBRs by performing integrated scheduling and dynamic simulations. However, test systems that have consistent scheduling and dynamic models rarely exist largely because of the decoupled nature of the two simulations on a synchronous generator-dominated system. This paper develops the dynamic model of a reduced Western Electricity Coordinating Council (WECC) system. In conjunction with the existing scheduling model, it is suitable for integrating scheduling and dynamic simulations. The reduced 240-bus WECC model reflects the generation resource mix of the Western Interconnection as of 2018. Moreover, the developed dynamic model is validated against field frequency events measured by FNET/GridEye and preserves the dominant inter-area oscillation mode in WECC.
Increasing penetration levels of solar photovoltaic (PV) and the reduced inertia of inverter-based generation technologies have caught the attention of the electric power industry and academia in the United States. This paper conducts a comprehensive frequency response study of the Western Interconnection under high penetration levels of PV, which are up to 80% at the interconnection level and 100% at the regional level. Based on the 2022 Light Spring case, PV has been projected by areas for four PV penetration levels based on collected realistic PV distribution information. The overall system frequency response has been analyzed under different scenarios. In particular, the 100% regional PV penetration case has been developed to reveal local frequency issues in the Western Interconnection. The potential of using existing resources to improve frequency response has also been studied. New insights in this paper address the implications of the rapid deployment of solar generation in the Western Interconnection.
Frequency response estimation results are presented using piloted inputs and a real-time estimation method recently developed for multisine inputs. A nonlinear simulation of the F-16 and a Piper Saratoga research aircraft were subjected to different piloted test inputs while the short period stabilator/elevator to pitch rate frequency response was estimated. Results show that the method can produce accurate results using wide-band piloted inputs instead of multisines. A new metric is introduced for evaluating which data points to include in the analysis and recommendations are provided for applying this method with piloted inputs.
Flexibility in inverter-based loads could be used to support the converter-dominated power grid by offering a rapid, autonomous, and adjustable power reserve during system transients to help maintain system stability. Based on technical potential, ancillary service (AS) value, and implementation costs, this study illustrates the cost-benefit analysis of grid-supportive loads (GSLs) for the supply of fast frequency response (FFR). The net benefit for each GSL is demonstrated using a case study and relevant data sources. The findings suggest that implementation costs for enabling GSL features are low compared to the value that grid operators get from the acquisition of responsive reserve services. The authors believe that, given the rising popularity of renewable energy sources, GSLs can be a useful tool for grid stability in low-inertia systems.
A frequency-domain approach is described for estimating parameters, such as stability and control derivatives, in aircraft flight dynamic models from measured input and output data. The approach uses orthogonal phase-optimized multisines for moving the aircraft control effectors, Fourier analysis for computing multiple-input multiple-output frequency responses, and a maximum likelihood estimator called frequency response error (FRE) for determining values and uncertainties for the model parameters. The approach is demonstrated using flight test data for two subscale airplanes: the T-2 generic transport model and the X-56A aeroelastic demonstrator. Results and comparisons with the output-error method indicated that the approach produced accurate estimates of stability and control derivatives and their uncertainties from flight test data.
Premixed flames are sensitive to flow disturbances, which can arise from acoustic or vortical fluctuations. For transverse instabilities, it is known that a dominant mechanism for flame response is “injector coupling,” whereby pressure oscillations associated with transverse waves excite axial flow disturbances. These axial flow disturbances then excite heat release oscillations. The objective of this paper is to consider another mechanism—the direct sensitivity of the unsteady heat release to transverse acoustic waves—and to compare its significance relative to the induced axial disturbances, in a linear framework. The rate at which the flame adds energy to the disturbance field is quantified using the Rayleigh criterion and evaluated over a range of control parameters, such as flame length and swirl number. The results show that radial modes induce heat release fluctuations that always add energy to the acoustic field, whereas heat release fluctuations induced by mixed radial-azimuthal modes can add or remove energy. These amplification rates are then compared to the flame response from induced axial fluctuations. For combustor-centered flames, these results show that the direct excitation mechanism has negligible amplification rates relative to the induced axial mechanism for radial modes. For transverse modes, the fact that the nozzle is located at a pressure node indicates that negligible induced axial velocity disturbances are excited; as such, the direct mechanism dominates. As a result, for flames that are not centered on pressure nodes, the direct mechanism for mixed modes dominates for certain nozzle locations and flame angles.
Data-reduction methods using general-purpose analog computer equipment and compatible testing techniques for determining the frequency response of linear physical systems are examined. The techniques considered may be classed as steady state or transient depending on the method of excitation. The relative merits of periodic, slow sweep, and transient (rapid sweep) forcing functions are discussed and applications are given that relate to dynamic-response tests of aeroelastic systems. Two frequency-sweep-input methods are considered in detail. one case the sweep rate is sufficiently slow that the response is approximately the same as that for steady-state conditions. With this input the frequency response can be evaluated and displayed in real time while the test is in progress. Errors due to treating sweep data as steady state can be eliminated, when desired, by reanalyzing tape-recorded time histories of the input and output as transient rather than as periodic data. In the second method the frequency-response function is deter- mined from the system's transient response to a very rapid sweep input. The purpose of frequency sweep in this case is to provide sufficient harmonic content in the input to overcome noise while keeping the test time as short as possible. tests and limited flight-test data presented herein, it appears that a transient-type rapid-sweep forcing function offers a considerable saving in test time while preserving the accuracy possible with steady-state sinusoidal inputs.
Individual microgrids can improve the reliability of power grids during extreme events, and networked microgrids can further improve efficiency through resource sharing and increase the resilience of critical end-use loads. However, networked microgrid operations are subject to large switching transients, which can cause dynamic instability and lead to system collapse. These transients are especially prevalent in microgrids with high penetrations of inverter-connected renewable energy resources, which do not provide the system inertia needed to mitigate the transients. Existing generator controls can be modified to invoke a drop in terminal voltage in response to a frequency deviation, thereby reducing load and improving frequency response. This paper investigates the use of a reinforcement-learning--based controller trained over several switching transient scenarios to modify generator controls during large frequency deviations. Compared to previously used proportional integral controllers, the proposed controller can improve primary frequency response while adapting to changes in system topologies and events.
The Multi-Area Frequency Response Integration Tool (MAFRIT) is the only software tool of its kind that integrates primary frequency response (turbine governor control) with secondary frequency response (automatic generation control). It simulates the power system dynamic response in full time spectrum with variable time steps from millisecond to minutes to hours and days. Capable of simulating both normal and event conditions, this tool can represent real power system operations and thus evaluate the primary and secondary reserves adequacy. This unique interaction of a turbine governor model and a novel automatic generation control model places special emphasis on electric power systems with high penetrations of renewable generation.
This paper presents an impedance-based noninvasive approach for the characterization of power system frequency response in real time in the absence of a transient event. It measures transfer function response from the injected active power to the frequency at the point of interconnection for the estimation of inertia and primary frequency response of the system. The so-called frequency response function also shows the effects of the speed of primary frequency control. The paper also derives the relationship between the frequency response function and the network impedance seen from the point of interconnection. This relationship highlights how impedance captures the frequency response behavior of a network. The proposed methodology is demonstrated on a modified IEEE 9-bus system with 25% penetration of wind and PV generation; a 5-MW battery energy storage system is used for the injection of active power perturbations for the measurement of the frequency response function.
The identification of aircraft flight dynamics is often performed using frequency responses, which are nonparametric models that quantify the steady-state magnitude and phase of a dynamic system response to sinusoidal inputs, as a function of frequency. Frequency responses are computed from measured input and output data, and then model parameters, such as stability and control derivatives, are estimated to best fit a parametric model to the empirical frequency response data. The utility of this approach is due to the familiarity of engineers with frequency responses, a number of theoretical and practical advantages under specific conditions, the availability of software packages, and many other reasons.
A computer program called FRFit (Frequency Response Fitting) for matching single-input single-output (SISO) transfer function models to empirical frequency response data is described. The program was written in MATLAB and has a graphical user interface (GUI). It is interactive in that the user manually builds the transfer function model using ``elementary factors'' (gain, delay, differentiators and integrators, and first- and second-order poles and zeros) and adjusts their values with sliders or entry fields. A nonlinear optimization can also be used to determine maximum-likelihood estimates of the transfer function parameters and their associated uncertainties. The program has some usefulness as a teaching aid, and can be applied to model structure determination, reduced-order modeling, preliminary analysis, and other system identification problems. FRFit is demonstrated using example problems, including the identification of aircraft transfer functions and rational function approximations of Theodorsen's function.
A computer program called FRFit (Frequency Response Fitting) is described for fitting single-input single-output transfer function models to empirical frequency response data. The program is interactive in that the user specifies ``elementary factors'' (gain, delay, pure differentiators and integrators, and first- and second-order zeros and poles) by entering numerical values or moving sliders in a graphical user interface. A nonlinear optimization can then be performed to obtain maximum likelihood estimates of transfer function parameters and uncertainties to provide feedback on the modeling and refine estimates. Several examples are discussed, including the identification of aircraft pitch dynamics from simulation data and data reported in the literature, approximating Theodorsen's function of unsteady aerodynamics, and obtaining a reduced-order model of a computational fluid dynamics code describing the unsteady aerodynamics around an aeroelastic wing. The program has some usefulness as a teaching aid, and can be applied to model structure determination, reduced-order modeling, preliminary analysis, and simple system identification problems. The program was written in MATLAB and is planned for public release through the NASA Software Catalog.
We report a theoretical study of the dependence of the beat frequency of counterpropagating waves on the rotation velocity in a gas ring laser with a periodic alternating frequency bias. Rectangular biases of two types are considered: with constant amplitude and with periodic amplitude modulation. We have found dynamic zones for locking-in the counterpropagating wave frequencies and determined deviations of the frequency response from the ideal one that arise when measuring frequency nonreciprocities, close in magnitude to the amplitude of an alternating frequency bias. It is shown that in the case of a rectangular bias with periodic amplitude modulation, nonlinear distortions of the frequency response can be reduced by an order of magnitude. (paper)
Several methods of obtaining the time response of Linear systems to either a unit impulse or an arbitrary input from frequency-response data are described and compared. Comparisons indicate that all the methods give good accuracy when applied to a second-order system; the main difference is the required computing time. The methods generally classified as inverse Laplace transform methods were found to be most effective in determining the response to a unit impulse from frequency-response data of higher order systems. Some discussion and examples are given of the use of such methods as flight-data-analysis techniques in predicting loads and motions of a flexible aircraft on the basis of simple calculations when the aircraft frequency response is known.