Frequency Dependent Network Equivalent Research Articles

Three-Phase Frequency-Dependent Network Equivalents in the ATP for Lumped Parameter Systems Using Descriptor Formulation, Rational Models, and Symmetrical Component Data

Three-Phase Frequency-Dependent Network Equivalents in the ATP for Lumped Parameter Systems Using Descriptor Formulation, Rational Models, and Symmetrical Component Data

A guaranteed passive model for multi-port frequency dependent network equivalents using network synthesis approach

Frequency Dependent Network Equivalent (FDNE) models are used in both real-time and offline Electromagnetic Transient (EMT) simulation tools to capture the behavior of a large external network, thus eliminating the need to simulate the entire network in full detail. An important requirement for the FDNE is that it be passive to guarantee stability. In this work, a new approach based on Brune's and Tellegen's network realization methods is proposed for constructing FDNEs. This approach inherently guarantees the passivity of the realized network. Tellegen's method is adapted and automated so it can be applied directly to the tabulated data of the impedance as a function of frequency. The feasibility of the proposed technique is validated through example case studies.

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.

Frequency Regulation and Control of Grid-Connected Wind Farms Based on Online Reduced-Order Modeling and Adaptive Control

This article presents a frequency regulation and stabilization control architecture for a wind turbine generator (WTG) integrated power grid, which can be added to existing WTGs. The architecture uses online reduced-order modeling of the power grid with frequency-dependent network equivalent that can seamlessly evaluate the grid fluctuations with respect to the point of common coupling of the wind farm. Then, an adaptive control architecture with the help of an online identification routine is used to balance the torque control of the wind generator (here doubly fed induction generator is considered) such that the frequency of the system is balanced. The main advantage of the proposed architecture is that the approach ensures frequency stability during grid oscillations at any location due to grid-related or WTG-related problems. The proposed approach not only maintains the frequency within acceptable limits during oscillations, but also makes sure that voltages are at normal values. The approach can work without wind speed measurements. The architecture is evaluated using wind-farm-integrated Kundur two-area and IEEE 39-bus test systems using a real-time digital simulator.

Representation of multiport rational models in an electromagnetic transients program: Networks with lumped and distributed parameters

The use of equivalents which reproduce the electrical network’s behavior over a wide frequency range (Frequency Dependent Network Equivalent) is of fundamental importance to carry out electromagnetic transient studies in large-scale power systems, regarding high precision and acceptable computation time. There are several approaches for building these equivalents, being the most utilized the one based on rational models of transfer functions. Currently, due to its efficiency and robustness, the Vector Fitting has been the most used method to obtain these rational models. Thereby, the problem turns out to be how to include these rational functions of the complex frequency s in an electromagnetic transient program, particularly, in the ATP. Therefore, this paper proposes a method to represent multiport rational models (rational models of MIMO - multi-input multi-output transfer functions) in the ATP. The proposed method takes into account the passive and active behavior of the external system. Its accuracy is verified for a network with lumped and distributed parameters.

A Two-Layer Network Equivalent With Local Passivity Compensation With Applications to Hybrid Simulations of MMC-Based AC–DC Grids

A frequency-dependent network equivalent (FDNE) is essential to capture wide-band frequency dynamics in the hybrid simulation of large-scale modular multi-level converter based ac–dc grids. The FDNE model must be enforced to be passive, ensuring the numerical stability in time-domain simulations. However, existing passive enforcement techniques based on global optimization perturbations cannot guarantee convergence, accuracy, and efficiency simultaneously. To address the issues, a two-layer FDNE (T-FDNE) model is developed for the ac grids. The two layers, namely, detailed layer and equivalent layer, have their admittances derived by perturbation test and analytical approach, respectively. The passivity of the T-FDNE model is guaranteed by the proposed local passivity compensation technique using auxiliary rational functions. Since no optimization is required and the passivity is enhanced locally, the convergence, accuracy, and efficiency can be improved considerably. By incorporating the T-FDNE model into the interface model of transient (TS) and electromagnetic transient hybrid simulations, wide-band frequency interactions, especially those of very high frequency, can be reflected effectively. The performance (efficiency and accuracy) of the T-FDNE model as well as of the hybrid simulation method has been validated on a modified and practical ac–dc system in China.

Passivity Enforcement of FDNEs via Perturbation of Singularity Test Matrix

This paper presents a fast and efficient alternative method to enforce passivity of frequency-dependent network equivalents (FDNE). Based on the eigenvalue perturbation theory, the proposed technique relocates the eigenvalues of the half-size singularity matrix by iteratively perturbing the residues of the FDNE model until the rational function becomes passive. It is demonstrated that the proposed technique achieves significant computational savings compared to traditional techniques. Three illustrative case studies involving different conditions of passivity violations are presented to validate the proposed method. Its effectiveness is demonstrated via comparison with two commonly used techniques, i.e., Fast Residue Perturbation (FRP) and Hamiltonian Matrix Perturbation (HMP).

Development and Applicability of Online Passivity Enforced Wide-Band Multi-Port Equivalents For Hybrid Transient Simulation

This paper presents a method for developing single and multi-port frequency dependent network equivalent (FDNE) based on a passivity enforced online recursive least squares identification algorithm, which identifies the input admittance matrix in z-domain. Furthermore, with the proposed architecture, a real-time hybrid model of the reduced power system is developed that integrate transient stability analysis and FDNE. Main advantages of the proposed architecture are, it identifies the FDNE even with unknown network parameters in the frequency range of interest, and yet can be implemented directly due to discrete formulation while maintaining desired accuracy, stability, and passivity conditions. The accuracy and characteristics of the proposed method are verified by implementing on two-area, IEEE 39 and 68 bus power system models.

Research on application of FDNE based on RTDS and comparison analysis of fault transient of Shanghai Sijing power grid

As a most widely applied real-time digital simulation system, real time digital simulation (RTDS) is however quite limited on scale of simulation by the number of hardware purchased. Through frequency-dependent network equivalent (FDNE), scale of simulation of RTDS could be largely increased by frequency equivalent on limited hardware. Based on basic principles of FDNE, function limitations and problem are analysed, taking the Shanghai Sijing district power grid as an example; detailed simulation model and FDNE equivalent module are built, transient characteristics under faults before and after FDNE equivalent are analysed. Furthermore, the fault characteristics of high-voltage direct current (HVDC)-Sijing power grid is also studied, which provides a realistic means for real-time digital simulation of AC–HVDC hybrid power grid with limited hardware of RTDS.

Pole-Selective Residue Perturbation Technique for Passivity Enforcement of FDNEs

This paper presents a new passivity enforcement technique for frequency-dependent network equivalents expressed as rational models. The main feature of the proposed technique is that only a few residues of the rational model are perturbed, so that the passivity enforcement procedure becomes computationally more efficient compared to existing methods. The proposed technique is verified on practical network cases. It is demonstrated that the accuracy of the perturbed model is maintained in frequency-domain fitting and time-domain simulations of transients.

A Multiple Time-Step Formulation of Frequency-Dependent Network Equivalents

This paper describes a novel approach for electromagnetic transient simulations of frequency-dependent network equivalents featuring variable time-step length along the simulation run. This demands an inclusion of an intermediate step allowing reinitialization of state variables without a need to modify the structure of companion networks adopted by EMTP-like programs for time-domain simulations. Besides keeping the accuracy of the classic EMTP modeling, the proposed formulation provides a sensible gain in the overall computation time and also a smooth transition regardless of the change in time-step length. This approach provides a more efficient realization of admittance-based models is achieved in a straightforward way, and the computational burden is considerably reduced without a significant loss of accuracy.

Single‐port and multi‐port frequency‐dependent network equivalents with numerically stable branches

This study proposes new methods for finding frequency-dependent network equivalents of power systems. Since stability of the time-domain solution is a major concern, additional constraints are added to ensure the stability in both single-port and multi-port network equivalents. For single-port problem, an equivalent circuit consisting of only passive elements is estimated to ensure passivity and, consequently, stability of the time-domain solution. For multi-port case, additional stable branches are added to fit the frequency response data with high accuracy while maintaining stability of the equivalent. Several case studies are considered to evaluate the performance of the proposed approaches. The simulation results prove the efficacy of the proposed methods for finding stable equivalent circuits with high accuracy.

A full frequency dependent line model based on folded line equivalencing and latency exploitation

This work proposes to improve the numerical performance in the rational modeling of full frequency dependent transmission lines. In this paper, latency is used to improve the numerical efficiency of time-domain implementation of the pole-residue representation of the nodal admittance matrix in the case of frequency dependent network equivalents. The idea of latency is to use different time-steps for distinct poles, i.e., fast poles are solved using a small time-step while slower poles may be solved using larger time-steps. For the separation of the slow and fast dynamics, a technique named Multiple Companion Networks (MCN) has been developed. The technique has been applied to transient tests of a transmission line modeled with a rational model. The results obtained indicated that a gain in numerical efficiency is possible without compromising accuracy.

Implicitly Coupled Electromechanical and Electromagnetic Transient Analysis Using a Frequency-Dependent Network Equivalent

This paper presents an improved solution of the implicitly coupled electromechanical and electromagnetic transient analysis problem using a frequency-dependent network equivalent (FDNE). In previous work, the fundamental frequency equivalent was used as the network equivalent of the external system. While simple, this fundamental frequency equivalent is only accurate for a single frequency, the fundamental frequency. This may lead to an inaccurate equivalent of the transient stability (TS) network. This paper extends previous work by using an FDNE derived from the vector-fitting technique. The FDNE is able to resolve the transient behavior over a wider frequency spectrum leading to a more accurate representation of the TS network. Results demonstrating the accuracy of the proposed scheme are presented on a 3-bus test system and the IEEE 118-bus system.

An optimal vector fitting method for estimating frequency-dependent network equivalents in power systems

Abstract Vector fitting (VF) algorithms have become popular and powerful tools for estimating models formed by rational basis function (RBF) expansions. In this paper, we first translate the well-known continuous time-domain VF method (cTD-VF) to a discrete time-domain framework. We denote this new domain VF method by dTD-VF. Differently from the cTD-VF, the dTD-VF formulation does not rely on a numerical approximation of convolution integrals and, as a result, it can be easily implemented with a variety of RBF sets. The second part of this paper shows that the proposed dTD-VF can also be transformed into a novel instrumental variable (IV)-dTD-VF technique, which is shown to have a guaranteed optimal solution at convergence. Moreover, this important optimality property does not depend on the nature of the noise that corrupts the data (for instance, if it is white or colored). Two case studies highlight the advantages of using the proposed methods. One of these examples consists of modeling the admittance characteristics of a power system implemented as a frequency-dependent network equivalent (FDNE).

A Guaranteed and Efficient Method to Enforce Passivity of Frequency-Dependent Network Equivalents

Rational models of frequency-dependent network equivalents (FDNEs) must be passive to ensure numerical stability in time domain simulations. Therefore, passivity enforcement is an essential step in FDNE rational modeling procedures; however, current approaches are either not guaranteed or not efficient. Here, a positive real lemma based semidefinite programming model is first implemented to guarantee the passivity of the result obtained. Second, this paper incorporates an approach to significantly improve efficiency while guaranteeing passivity by exploiting an important feature of rational models: that constituent complex-pole terms corresponding to dominant resonance peaks can be adjusted to be passive through minor changes, and a partition-based scheme is proposed. Numerical investigations demonstrate that the proposed method is efficient while introducing little additional modeling error.

Application of Balanced Realizations for Model-Order Reduction of Dynamic Power System Equivalents

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.

A Semidefinite Programming Model for Passivity Enforcement of Frequency-Dependent Network Equivalents

This letter presents a semidefinite programming model for passivity enforcement of frequency-dependent network equivalents. This model has some advantages over the existing quadratic programming model. Numerical tests show the validity of the proposed model.

A Fast Method to Identify the Order of Frequency-Dependent Network Equivalents

Rational approximation of frequency-dependent network equivalents with vector fitting (VF) is widely used in power system analysis; however, a detailed examination of methods to determine the order of the model has not been reported. The traditional order identification method of the frequency response resonance peak is conservative, as demonstrated in this paper. We discuss the essential relationships between VF and Prony analysis (PA). A PA-based method is then proposed to identify the order of model, considering the resonance peaks. The method is computationally inexpensive, and the precision can be adjusted easily. Numerical investigations demonstrate the validity and efficiency of the method.

Compacting and partitioning‐based simulation solution for frequency‐dependent network equivalents in real‐time digital simulator

Rational models of frequency-dependent network equivalents (FDNEs) have been used in real-time digital simulator (RTDS) for power-system simulation. However, this can lead to a computational burden issue; the application of FDNEs may result in a loss of real-time simulation features because the computational cost of the FDNE component exceeds the limits of RTDS. The authors describe a solution that combines compacting and partitioning of the FDNEs, whereby the former reduces the redundancy in the mathematical model and the latter allows us to exploit parallel computer architectures. Then they describe the results of numerical simulations that demonstrate the effectiveness of the approach. Moreover, the proposed simulation solution is not limited to the applications of FDNEs in RTDS, it solves a set of subsistent computational issues when apply rational models in real-time electromagnetic transients programs tools.