Frequency Dependent Transmission Line Model Research Articles

An Enhanced Method to Achieve Exact DC Values for Frequency-dependent Transmission lines

This paper proposes an improved method to enhance the dc response of a frequency-dependent transmission line model used in EMT studies. A modification to the rational function approximation of propagation and characteristic admittance matrices of a transmission line is introduced to enforce exact dc values at 0 Hz. Furthermore, weighting factors are applied to improve accuracy at low frequencies. Finally, the order of the propagation function is reduced to decrease the computational effort.The validity of the proposed approach is demonstrated using examples involving underground cables and overhead lines. First, the effect of dc correction is demonstrated by comparing transmission line frequency domain characteristics. In addition, time domain simulations via open and short circuit conditions show a more accurate simulation of HVDC transmission lines with the proposed method.

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 New Approach to Represent the Corona Effect and Frequency-Dependent Transmission Line Models in EMT-Type Programs

This paper presents an accurate and efficient model to represent the corona effect and frequency dependence of line parameters in electromagnetic transient simulations. The new method, named the voltage and frequency dependent line model (VFDLM), can be seen as a more general case of the well-known frequency-dependent (FD) or wideband (WB) line models, wherein the characteristic admittance and propagation function are considered voltage- and frequency-dependent parameters. The time domain traveling wave equations are solved using recursive convolutions and rational approximation through vector fitting (VF). Since the model can be represented by Norton equivalents, it is totally compatible with EMT-type programs. The model is validated through comparisons with three field measurement data available in the literature, and good agreement could be observed between them, showing that the VFDLM is a stable, efficient, and accurate approach to represent the corona effect in EMT-type programs.

GPU and CPU-Based Parallel FDTD Methods for Frequency-Dependent Transmission Line Models

Finite-difference time-domain (FDTD) is a popular method utilized for solving frequency-dependent transmission line structures. It is also conveniently applicable to nonuniform wires. The FDTD algorithm discretizes the transmission line problem into a finite number of space-segments and solve for the voltage and current of each segment at every time-step. Therefore, they inherently involve more computations per timestep than conventional terminal based models. In this letter, parallel implementations of a modified FDTD algorithm for a frequency-dependent transmission line problem using multicore CPU and GPU architectures are proposed in order to increase its computational efficiency. Accuracy and performance of the parallel FDTD methods are discussed with examples. The results indicate that a speedup of a few folds compared to serial execution is achieved by the parallel implementation using multicore CPU architecture whereas a massive speedup is achieved by using GPU. The proposed model is also suitable for modelling transmission lines in massively parallel electromagnetic transient (EMT) simulation methods.

Development of phase domain frequency-dependent transmission line model on FPGA for real-time digital simulator

The transmission line model is one of the most important components for the real-time digital simulator based on the electromagnetic transient (EMT) algorithm. In order to predict the behavior of power systems under both steady and transient states, the coupling and frequency dependence of the transmission line need to be precisely represented. This paper presents a frequency-dependent phase domain (FDPD) transmission line model on a field programmable gate array (FPGA) for an EMT type real-time digital simulator. The developed model has been fully pipelined and parallelized in hardware to achieve the lowest time-step. All the hardware developments for this model depicted in this paper are carried out in VHDL, and a customized 48-bits floating-point data representation is used for hardware implementation to improve the accuracy. This developed FPGA-based line model can be interfaced to the rest of the network to perform real-time simulation.

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.

Comprehensive analysis of the fault inception angle influence in fault-induced traveling waves

This work presents a comprehensive analysis of the fault inception angle influence in the first wavefront of the fault-induced traveling waves (TWs) for all types of faults in a transmission line. The EMTP-ATP software simulated fault-induced TWs for all transmission line fault types considering the JMarti frequency-dependent transmission line model, whereas an emulated transmission line in an experimental setup generated fault-induced TWs of single line-to-ground faults for validating the effects of the fault inception angle. This work demonstrated that the energy of the fault-induced TWs changes with the fault inception angle according to a squared sinusoidal function, and pointed out the situations where there is no TW in each type of fault. Besides the support for a better understanding of fault-induced TWs, the analysis accomplished here will be essential for further development of TW-based detection, location, and protection methods.

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

Development of an Experimental Scaled-Down Frequency Dependent Transmission Line Model

The penetration of power electronic based equipment in the power grid has increased the co-occurrence of electromechanical and electromagnetic transients. Significant research efforts have been focused in the last decade to study these interactions. Frequency dependent transmission line models are indispensable to understand such interactions. This paper describes a systematic procedure for the development of an experimental scaled-down 220 V frequency dependent transmission line model of a 230 kV transmission line. A reduced order lumped parameter frequency dependent transmission line using modal transformation is derived for the 230 kV transmission line and scaled-down to 220 V. Clarke and inverse Clarke transformations are implemented using specially designed 1- $\phi $ transformers. The inductances of the scaled-down model are realized using amorphous cores. Line energization, balanced, and unbalanced fault studies are carried out using the experimental line. The experimental results are compared with the simulation results of a universal line and a constant parameter $\pi $ -model using EMTP-RV. The proposed experimental line properly captured important features of the frequency dependent line such as smoother wave shape during energization, traveling time, currents, and voltage magnitudes during the fault and after the fault removal.

A Node-to-Node Admittance Functions Implementation of an Improved Frequency Dependent Multiconductor Transmission Line Model

The electromagnetic interference (EMI) performances of the interconnects and cables can be predicted via a standard multi-conductor transmission line (MTL) model, while the latter is not valid for the evaluation of power rail collapse and ground bounce responses. To circumvent the limitations, a more general and feasible improved MTL representation is presented in this paper. It physically incorporates the partial resistance and partial inductance parameters of all signal and reference conductors. To consider the frequency dependent behavior of the per-unit-length (PUL) electrical parameters in time domain simulations, a terminal description for this improved MTL model with any desired length is demonstrated. Subsequently, an equivalent node-to-node admittance functions (NAFs) implementation for this terminal representation is carried out. The correctness and effectiveness of the NAFs circuit model in time domain is then numerically validated by analyzing two dedicated examples.

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.

Real-Time Electromagnetic Transient Simulation of Multi-Terminal HVDC–AC Grids Based on GPU

High-fidelity electromagnetic transient (EMT) simulation plays a critical role in understanding the dynamic behavior and fast transients involved in operation, control, and protection of multiterminal dc (MTdc) grids. This article proposes a cost-effective high-performance real-time EMT simulation platform for large-scale cross-continental MTdc grids based on graphics processing unit (GPU). Fast dynamic transients from both ac and dc networks are captured in real time with 5μs time step, using advanced hybrid-discretized modular multilevel converter model, frequency-dependent transmission line model, and EMT-type model of synchronous generators. The proposed simulation platform i) assembles detailed EMT models of all components within an MTdc-ac grid into a single platform. This setup provides a complete simulation solution to capture fast transient signals required for high-bandwidth controller design and protection studies without any compromise; ii) implements the first GPU-based simulation architecture and corresponding algorithms for MTdc-ac grids with real-time performance at scales of 1s; iii) is highly efficient and balances the high utilization of GPU resources and low latency required for the simulation; and iv) outperforms the existing central processing unit- or digital signal processor (DSP)/field-programmable gate array-based simulators in terms of its higher scalability on large-scale MTdc-ac grids and superior price-performance ratio on the hardware. Accuracy and performance of the proposed platform are evaluated with respect to the reference results from power system computer aided design (PSCAD)/EMTdc environment.

Wideband Modeling of Power SiC mosfet Module and Conducted EMI Prediction of MVDC Railway Electrification System

The SiC mosfet in the medium-voltage direct-current (MVdc) transportation electrification system features faster switching performance, while simultaneously binging more significant electromagnetic interference (EMI) issues within the rolling stocks, substations, and radiated disturbance into space along the catenaries and tracks. Due to the necessity to involve both the transient characteristics of power semiconductor devices and the stray parameters of all the equipment in the analysis of EMI, it is considerably challenging to perform wideband device-level simulation on traditional commercial software for such a complex system with numerous trains and stations. A computationally efficient method for wideband modeling and simulation of the MVdc high-speed railway system for the assessment of conducted EMI during the project design stage is proposed in this article. Physical characteristics of the semiconductor devices, parasitic parameters of the mosfet package, and converter topology are all taken into consideration to provide not only accurate system-level performance of the system but also an insight into high-frequency characteristics under different operation conditions. The calculation burden is alleviated by a hierarchical circuit partitioning architecture based on the frequency-dependent time-domain transmission line model and the Norton equivalent parameter extraction of each mosfet module to split the whole system into several smaller subcircuits in terms of matrix size, and a fully parallel implementation of the MVdc system is carried out on the graphics processor. The developed program is used to study the case of Jing–Zhang high-speed railway system topology, which is compatible to be modified to the MVdc project. Simulation results show that it is essential to estimate the EMI level comprehensively considering the alternative of speed and dc voltage.

Performance Assessment of the RTDS/RSCAD VSC Model

This document presents a performance assessment of the three-phase two-level voltage source converter (VSC) model of the real-time digital simulator RTDS/RSCAD. This evaluation is conducted through two case studies. One case is a simple two-bus power electronic-based system, and the second case is a three-bus system with a frequency-dependent transmission line model and a closed-loop controlled VSC with communication delay. The results obtained reveal that the RTDS/RSCAD VSC model presents some issues on stability, harmonic spectrum, ripple, and power quality in general, as compared with the PSCAD VSC solutions. These results also suggest the need for better power electronic components models.

Transient numerical calculation and differential protection algorithm for HVDC transmission lines based on a frequency-dependent parameter model

Being free from the influences of the transmission characteristics, the current and voltage near a fault point can better reflect the fault features. Thus, these electrical quantities along a transmission line can be used to formulate the protection criteria to improve the sensitivity and selectivity of the protection. A numerical calculation method for transient electrical quantities along a transmission line based on a frequency-dependent parameter transmission line model is studied. Differential protection uses currents from both terminals and can protect transmission lines with excellent selectivity. However, the differential protection currently used for high-voltage direct-current (HVDC) transmission lines requires an increase in the time delay and the threshold to avoid mis-operation induced by the distributed capacitance current along the transmission lines, which results in low operation speeds and poor sensitivity. This paper presents a new differential protection algorithm using the calculated current at compensation points to overcome this problem. Because the parameters involved in the differential protection can be precalculated offline, this algorithm has low computational complexity and can be easily implemented. Unaffected by the distributed, frequency-dependent, and coupling characteristics of the transmission line parameters, the proposed differential protection approach can operate correctly with high sensitivity throughout the entire process under fault. Simulations demonstrate the feasibility and validity of the differential protection algorithm.

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.

Frequency-Dependent Multi-Conductor Transmission Line Model for Shielded Power Cables Considering Geometrical Dissymmetry

In this paper, a broadband equivalent node-to-node admittance functions (NAFs) model for shielded power cables are presented aiming at the direct time-domain simulation. The finite-element method is employed to obtain the frequency-dependent per-unit-length resistances, self- and mutual-inductances, and the self- and mutual-capacitances of the power cables. The matrix rational approximations are then applied to obtain the rational model (in pole-residue form) of the terminal admittance matrix of the multi-conductor transmission line (MTL) model. Next, the numerical stability of the rational model is guaranteed by the terminal (port) passivity enforcement technique. The circuit representation of the equivalent NAFs of the full cable with the length of interest can be synthesized based on the pole-residue description. It is validated by comparing the frequency response with the one computed by applying the original MTL model.

Frequency-Dependent Multiconductor Transmission Line Model With Collocated Voltage and Current Propagation

This paper reviews the classical multiconductor transmission line (MTL) equations and proposes additional constraints on these equations. A fundamental physical constraint is that the voltage and current waves must be collocated and travel together with the same propagation function. Based on this condition, the Revised Multiconductor Transmission Line (RMTL) equations are proposed. As opposed to the classical MTL equations that require complex frequency-dependent transformation matrices for their diagonalization, the RMTL equations can be diagonalized very accurately using a single real constant transformation matrix. A new Frequency-Dependent Line Model (FDLM) is proposed based on the RMTL equations. FDLM is compared with the two most accepted frequency-dependent line models in the Electromagnetic Transients Program (EMTP): The JMARTI model (fdLine) that uses a constant transformation matrix as an approximation, and the phase-coordinates Universal Line Model (ULM) that fits the frequency dependence of the transformation matrices. These time-domain models are compared with a reference frequency-domain solution for a double-circuit vertical line.

Multi-Scale and Frequency-Dependent Modeling of Electric Power Transmission Lines

A frequency-dependent transmission line model for multi-scale simulation of diverse transients over a wide range of frequencies is developed, implemented, and validated. It makes use of the concept of frequency-adaptive simulation of transients in which the Fourier spectra are adaptively shifted in the frequency domain to reduce the discretization time-steps in the time domain. The transients are modeled through dynamic phasors comprising the real and imaginary parts of analytic signals to facilitate the frequency shifting. In the proposed line model, all mathematical operations such as numerical recursive convolutions are, therefore, expressed in terms of analytic signals. A modal decomposition is performed to attain decoupled modes for the multi-phase case. The transition from the representation of electromagnetic traveling waves with time-steps below the wave propagation time to the tracking of slower electromechanical transients at time-steps above the wave propagation time is achieved by the automatic insertion of a $\pi$ -segment to represent the galvanic coupling within one time-step. Accurate and efficient simulation of both electromagnetic and electromechanical transients within a simulation run is so supported. The validation is verified through comparison with a staged field test covering the diverse transients of line energization, transient recovery voltage, and steady state.

Key techniques in real time digital simulation for closed-loop testing of HVDC systems

This paper introduces several key techniques in realtime digital simulation for closed-loop testing of high voltage direct current (HVDC) systems, including line commutated converter (LCC) HVDC valve group models, modular multilevel converter (MMC) HVDC models, phase domain frequency dependent transmission line models, and switchable filter models. Various issues such as background, algorithms, implementations, applications, and validation of these models are introduced. These techniques represent the most advanced technology in modelling HVDC systems on a real time simulator. The models have been implemented on the RTDS Simulator, a comprehensive real time power system digital simulation tool.