Universal Line Model Research Articles

Assessment of the accuracy of the modal-domain line models with real and frequency-independent transformation matrix for electromagnetic transient simulation in lines with underbuilt cables

This paper investigates the potential inaccuracies associated with using a modal-domain model featuring a real and frequency-independent transformation matrix in simulating electromagnetic transients on power overhead lines equipped with underbuilt wires. Three transmission line configurations are examined: one without underbuilt wires, one with a single underbuilt wire, and another with two underbuilt wires. Comparisons between the simulated voltages computed using the modal-domain approach and a more accurate phase-domain model as a reference reveal that the modal-domain approach leads to inaccuracies in the simulated voltages. These inaccuracies intensify with an increase in the number of underbuilt wires and the soil resistivity. Differences of up to approximately 24 % in the peak values of the simulated voltages were observed, considering the transmission line geometry analyzed in this study. Based on these findings, it is recommended that for simulating transients in overhead lines with installed underbuilt wires, phase-domain models, such as the Universal Line Model, should be employed.

Comparison of Rational Krylov and Vector Fitting in Transient Simulation of Transmission Lines and Cables

This paper presents stringent comparisons between Vector Fitting (VFIT), Relaxed Vector Fitting (R-VFIT) and Rational Krylov Fitting (RKFIT) techniques in the fitting of transmission line and cable functions to rational forms while accounting for the frequency dependence of electrical parameters. The fitting procedures encapsulate new solution strategies to improve fitting performance. Various case studies are presented to assess the performance of RKFIT in terms of model order reduction, passivity violation and computing times. In addition, the use of RKFIT in obtaining the frequency dependent cable model (FDCM) is demonstrated for the first time together with a resolved case study that showed unstable behavior with the universal line model (ULM) methodology.

Implementation of an Alternative Frequency-Dependent Three-Phase Transmission Line Model Based on the Folded Line Equivalent Model in MatLab-Simulink

This paper proposes an alternative multiconductor transmission line model that combines the folded line equivalent with the modal transformation. The folded line equivalent decomposes the nodal admittance matrix of a transmission line into its open-circuit and short-circuit contributions. These contributions are fitted to rational functions, which are associated with Norton equivalent circuits based on their state space models. The proposed model uses an orthogonal matrix to transform voltages and currents from the phase domain to the folded line equivalent domain and vice versa. Because the transformation matrix is orthogonal, we represent it using ideal transformers in simulation software. First, we use a circuit representation of Clarke’s matrix to decompose a transmission line into its modes. Then, each mode is decomposed into its open-circuit and short-circuit contributions using a circuit implementation of the proposed matrix. The proposed approach can accurately represent short lines in simulations with time steps equal to or greater than the propagation time of the transmission line. We compare the results obtained with the proposed approach to those obtained with power systems computer-aided design/electromagnetic transients including the DC universal line model.

A modified implementation of the Folded Line Equivalent transmission line model in the Alternative Transient Program

This paper proposes a modified implementation of the Folded Line Equivalent transmission line model to represent three-phase transmission lines. The Folded line equivalent decomposes a line into its open circuit and short circuit contributions. We propose an alternative orthogonal matrix to transform single-phase transmission line parameters, voltages, and currents to the Folded Line Equivalent domain. Because the proposed matrix is orthogonal, it can directly represent bidirectional transformations using circuit components in simulation software, such as the ATP-EMTP. First, the circuit implementation of Clarke’s matrix decouples a three-phase transmission line into its modes. Then, the circuit implementation of the proposed matrix decouples each mode into its open circuit and short circuit contributions. This paper proves that the proposed approach outputs similar results as those obtained by the Universal Line Model and the JMarti model in the simulation of transmission lines under open-circuit, energization, and fault conditions. However, because the proposed model does not derive from the method of characteristics, it can run with simulation times greater than the propagation delay of the transmission line. It is shown that the proposed approach outputs accurate results for time steps that are 10%, 100%, 200%, and 400% of the propagation delay of the transmission line. Thus, one major advantage of the proposed approach is that it can be used to reduce runtimes in the representation of short transmission lines in large complex networks by increasing time steps without compromising accuracy.

Accuracy Evaluation of Electromagnetic Transients Simulation Algorithms

This paper introduces a novel frequency domain technique to globally evaluate the accuracy of electro-magnetic transient simulations. It is shown that simulation accuracy at low frequencies can sometimes be poorer than at high frequency. A modified approach which quantifies accuracy from a driving point as a function of frequency is also introduced that uses the Bilinear Transformation and Norton equivalents, to produce a “simulation accuracy spectrum”. This approach can be applied to large systems without explicitly forming state space equations. It also permits the accuracy analysis of networks with distributed components such as frequency dependent transmission lines. Two examples are used to verify the proposed technique, a small network with a frequency dependent transmission line modeled by the universal line model; and the IEEE 39 bus system connected with an LCC-HVdc system.

Implementation of the universal line model in the alternative transients program✰

The universal line model (ULM) is a transmission line model developed directly in the phase domain that is recognized for its accuracy and generality. It is currently the reference model for transient studies, but is exclusively found on commercial electromagnetic transients programs. This paper describes an implementation of ULM in the Alternative Transients Program (ATP), which is free for licensed users, using the foreign models tool and the type-94 component available in this platform. The implemented model is validated through comparisons with EMTP-RV. Then, it is used to investigate transients on overhead transmission lines considering a rigorous representation of the ground return impedance and ground admittance assuming frequency-dependent ground parameters.

An improved passivity enforcement algorithm for transmission line models using passive filters

This paper proposes a simple but effective method based on shunt passive filters to enforce passivity on a frequency dependent transmission line model for multi-conductor cables and overhead lines. The passivity enforcement algorithm is applied to a widely-used frequency dependent line model in EMT-type software, the Universal Line Model. The passivity violating regions of the transmission line model are identified using the frequency sweep method. The passive shunt series RLC filters are added to the nodes of the transmission lines to eliminate passivity violations. Examples of multi-conductor underground cable systems are presented to demonstrate the validity of the proposed approach.

An algorithm for fitting passive equivalent circuits for lumped parameter frequency dependent transmission line models

Accurately fitting rational functions to the frequency response of modal impedances is crucial for including frequency dependency in lumped parameter models of transmission lines. Vector fitting is widely used for fitting rational functions to the frequency response of modal impedances and then an R-L equivalent circuit is obtained from the rational functions. A single-step method based on the properties of Foster equivalent circuit is proposed to directly fit an R-L equivalent circuit to the frequency response of modal impedances. The positive peaks, negative peaks and the positive zero crossings of the slope change plot of the frequency response are found to provide a good approximation of zeros, poles and the flat region locations of the frequency response. An enhanced fitting algorithm is proposed based on these observations. A close enough fitting is achieved using the proposed method with less number of passive elements. Using the proposed model in EMTP-RV, 400, 765, 1200 kV transmission lines and a 11-bus 500 kV network are simulated for switching transients. The results are compared with the constant parameter cascaded (Formula presented.) -model and the universal line model in EMTP-RV. The switching transient results of the proposed model are found to be comparable to the Marti's model in EMTP-RV. © 2021 The Authors. IET Generation, Transmission & Distribution published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology

Electromagnetic Transient Analysis of Transmission Line Based on Rational Krylov Approximation

This paper shows a comparison between Vector fitting and rational Krylov fitting techniques for the determination of rational models concerning the fitting accuracy, the computational performances and the model order. Primarily, the mathematics behind the second technique are presented. It should be noted that rational Krylov fitting have never been used in transmission line modeling. A new procedure is proposed to use rational Krylov fitting instead of vector fitting in the universal line model (ULM). Furthermore, it is demonstrated that this procedure has several advantages over the traditional one. Two illustrative examples involving a transmission system are presented for validation of the new procedure.

Alternative method using recursive convolution for electromagnetic transient analysis in balanced overhead transmission lines

This study proposes an alternative method to calculate the transient responses of multi-conductor overhead transmission lines based on ABCD representation, recursive convolutions and numerical method for rational approximation in the frequency-dependent curves. In the classical universal line model, the characteristic admittance matrix Y c ( ω ) and propagation matrix H ( ω ) are represented by rational functions. In H ( ω ) , previous and correct identification of the modal travelling times is required. However, these modal travelling times are estimated numerically, which do not offer good accuracy in some cases. This fact has a considerable effect on the rational representation of H ( ω ) itself and consequently on the transient responses. In the alternative method, a previous estimation of the modal travel times is unnecessary because when using the ABCD representation for each propagation mode, a direct relation exists between the currents and tensions of the receiving and sending ends. These relationships are fitted by smooth rational functions, and then, using recursive convolution methods, the authors obtain the explicit equations of the currents and voltages to calculate the transient responses. The results demonstrate that the alternative method is accurate when compared with the frequency-domain solution of the transmission line transformed to the time domain using the numeric Laplace transform and with the Bergeron method.

Compact Matrix Formulation for Calculating Lightning-Induced Voltages on Electromagnetic Transient Simulators

In this paper, a compact matrix formulation that simplifies the calculation of the lumped sources necessary to estimate lightning-induced effects using transmission line models available in electromagnetic transient programs is proposed. As opposed to an existing solution strategy that requires the sequential calculation of the convolution integrals involving the horizontal component of the incident electric field along multiple segments along the line, the proposed matrix solution allows such convolutions to be performed at once. This not only increases the model efficiency, but also simplifies the assembly of the solution in matrix-oriented simulation tools. The equations are described in both phase and modal domains, with formulations that are compatible with the universal line model and the frequency-dependent transmission line model proposed by Marti. It is shown that both formulations can be accurately used for calculating lightning-induced voltages on overhead lines.

Partitioned fitting and DC correction in transmission line/cable models for wideband EMT studies

This paper extends the applications of and provides further insights about partitioned fitting procedure. At the first stage of this procedure, the fitting is performed at a high frequency band by excluding frequency samples close to DC. The second stage finds a correction term for those excluded samples. This procedure achieves improved accuracy for the demonstrated case of transmission lines and cables used for HVDC transmission. The new approach complements the prevailing Universal Line Model (ULM) by reducing numerical instabilities due to large residue/pole ratios while delivering accurate DC response. This paper provides also time-domain implementation details of the state space realization obtained with partitioned fitting. The numerical stability performance of the improved fitting approach is demonstrated, through a set of test cases by applying different integration and interpolation schemes.

High performance computing engines for the FPGA-based simulation of the ULM

This paper presents a design methodology for the FPGA-based simulation of the Universal Line Model (ULM). The proposed approach yields a higher computational performance compared to alternative implementations reported in the literature. Such performance allows the use of the FPGA model in real-time simulation applications or for the acceleration of offline EMT programs. A state-space approach is used to perform the time-domain simulation of the pole-residue form of the rational fitting of the characteristic admittance and propagation functions. The paper also discusses the appropriate scheduling of the ULM computations and proper management of the history terms that lead to an optimized hardware utilization, low latency response times, and higher computational performances using floating-point arithmetic.

Discrete-Time Fourier Series to simulate transient over voltages in power systems

This paper presents a methodology based on the Discrete-Time Fourier Series (DTFS) to calculate transient over-voltages in power systems. The proposed algorithm determines the time window with sufficient width to allow the time constants of the transients to decay to a small value. This method is simpler than the traditional numerical Laplace transform (NLT) and does not require parameters adjustment to obtain accurate results. The test case is based on a study to calculate Transient Over-Voltages (TOV) for 138 kV transmission lines in British Columbia, Canada. To assess the accuracy of the results, the DTFS solution is compared with the traditional NLT solution with optimized parameters and with the frequency-dependent time-domain models: fdLINE (JMARTI Model) and Universal Line Model (ULM).

Time-Domain Procedures for Lightning-Induced Voltage Calculation in Electromagnetic Transient Simulators

In this paper, two time-domain procedures are proposed for the calculation of lightning-induced voltages using lossy transmission line models available in electromagnetic transient simulators. In both procedures, the influence of the incident electromagnetic field is represented in the form of current sources that are connected externally at the ends of the illuminated line. One of the procedures is formulated directly in the phase-domain and is compatible with the universal line model (ULM). The second procedure is formulated in the modal domain and is compatible with the transmission line model proposed by Marti. Both procedures are validated taking as reference a first order FDTD solution of telegrapher's equations considering the presence of external electromagnetic fields. It is shown that the proposed procedures can be efficiently and accurately used for the calculation of lightning-induced voltages on overhead lines.

Adaptive Time-Stepping Universal Line and Machine Models for Real Time and Faster-Than-Real-Time Hardware Emulation

Transmission lines and rotating machines that widely exist in power systems should be accurately modeled in real-time electromagnetic transient (EMT) simulation for obtaining precise results for hardware-in-the-loop applications. In the conventional EMT simulator, the time-step is fixed, which may lead to inefficiencies when the time constants of the system change. The adaptive time-stepping (ATS) method can efficaciously solve this problem; however, the ATS schemes for the universal transmission line model (ULM) and universal machine (UM) model remain to be investigated. This article derives the ATS models for ULM and UM, and the proposed ULM model is more stable than the traditional model. Both ATS models are emulated on the parallel and pipelined architecture of the field-programmable gate array (FPGA). The proposed subsystem-based ATS scheme and the local truncation error (LTE) based time-step control enable the large-scale systems to be simulated in real time and “faster-than-real-time” modes. The IEEE 39-bus system with ATS models is emulated on two interconnected FPGA boards, and the emulation results compared with PSCAD/EMTDC and fixed time-stepping (FTS) hardware emulator verify the effectiveness of the proposed models and show that the LTE of ULM and UM can be reduced by 76.5% and 62.0%, respectively, compared with the FTS simulation.

Accurate and Reduced Order Identification of Propagation Function for Electromagnetic Transient Analysis of Cables

This paper proposes a set of fitting features that enables identifying propagation function of cables accurately in the form of multi-delay rational transfer functions while maintaining reduced order of approximation. Frequency domain partitioning and adaptive weighting techniques are applied directly in phase domain for the evaluation of poles and residues simultaneously to ensure the precision of fitting of all entries including the low-magnitude off-diagonal elements. The objective is not only to obtain a precise fitting in phase domain but also to account for intrinsic modal decomposition so that integration errors in time domain are not magnified. The order of approximation is considerably reduced by post-processing the fitting using the balanced realization technique. When the proposed fitting approach is combined with a precise integration technique in time domain, it leads to accurate evaluation of transients as demonstrated in this paper, eliminating the spurious oscillations or numerical instabilities that may be encountered in the universal line model regardless of the integration or interpolation technique.

Partitioned Fitting and DC Correction for the Simulation of Electromagnetic Transients in Transmission Lines/Cables

This letter proposes a two-stage fitting procedure for transmission line/cable functions in which low-frequency samples are exclusively considered. At the first stage, fitting is performed for a reduced band by excluding frequencies close to DC. Reducing the fitting range improves the numerical conditioning of the overall system of equations and relieves fitting. The second stage consists of finding a correction term for the out-of-band samples close to DC. The procedure, when used with the recently introduced frequency-dependent cable model approach, allows modeling transmission lines and cables with improved fitting precision at low frequencies. Overall, the new approach is called frequency-dependent model (FDM) with DC correction, i.e., FDM/DC. It can be used to complement the prevailing universal line model in two ways: for reducing the integration errors due to unbalanced fitting and for improving the precision of DC response. Two examples are provided to demonstrate the utility of the new approach.

Frequency-Dependent Pi Model of a Three-Core Submarine Cable for Time and Frequency Domain Analysis

In this paper, a Frequency-Dependent Pi Model (FDPi) of a three-core submarine cable is presented. The model is intended to be used for the representation of submarine cables in an Offshore Wind Power Plant (OWPP) scenario for both time and frequency domain analysis. The frequency-dependent variation of each conductive layer is modeled by a Foster equivalent network whose parameters are tuned by means of Vector Fitting (VF) algorithm. The complete formulation for the parameterization of the model is presented in detail, which allows an easy reproduction of the presented model. The validation of the model is performed via a comparison with a well-established reference model, the Universal Line Model (ULM) from PSCAD/EMTDC software. Two cable system case studies are presented. The first case study shows the response of the FDPi Model for a three-core submarine cable. On the other hand, the second case study depicts the response of three single-core underground cables laying in trefoil formation. This last case shows the applicability of the FDPi Model to other types of cable systems and indirectly validates the response of the aforementioned model with experimental results. Additionally, potential applications of the FDPi model are presented.

Time-Window-Based Discrete-Time Fourier Series for Electromagnetic Transients in Power Systems

This paper approaches the solution of electromagnetic transients in the frequency domain from the point of view of choosing a time window with sufficient width to allow the time constants of the transients to decay to a small value. This criterion, and the required maximum frequency content of the simulation, fully determine the simulation parameters. Satisfying this criterion, a very simple methodology, like the discrete time Fourier series (DTFS), can provide very accurate results without the inherent difficulties of time-warping solutions based on the numerical Laplace transform (NLT). Even though the NLT can give very accurate results, it requires expert knowledge to correctly specify the damping factors, sampling schemes, and windowing functions. The DTFS solution is compared with the NLT and with analytical solution for a circuit with lumped elements, and with the most-accepted frequency-dependent transmission line models in the Electromagnetic Transients Program: the JMARTI (fdLINE) model and the Universal Line Model for a double-circuit transmission line.