The paper describes a general methodology for the fitting of measured or calculated frequency domain responses with rational function approximations. This is achieved by replacing a set of starting poles with an improved set of poles via a scaling procedure. A previous paper (Gustavsen et al., 1997) described the application of the method to smooth functions using real starting poles. This paper extends the method to functions with a high number of resonance peaks by allowing complex starting poles. Fundamental properties of the method are discussed and details of its practical implementation are described. The method is demonstrated to be very suitable for fitting network equivalents and transformer responses. The computer code is in the public domain, available from the first author.

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Rational approximation of frequency domain responses by vector fitting

This paper describes a modification of the vector fitting (VF) procedure for rational function approximation of frequency-domain responses. The modification greatly improves the ability of VF to relocate poles to better positions, thereby improving its convergence performance and reducing the importance of the initial pole set specification. This is achieved by replacing the high-frequency asymptotic requirement of the VF scaling function with a more relaxed condition. Calculated results demonstrate a major improvement of performance when fitting responses that are contaminated with noise. The procedure is also shown to be advantageous for wideband modeling of transmission lines, network equivalents, and transformers.

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Improving the pole relocating properties of vector fitting

Voltage-source-converter (VSC) technologies present a bright opportunity in a variety of fields within the power system industry. New modular multilevel converters (MMCs) are expected to supersede two- and three-level VSC-based technologies for HVDC applications due to their recognized advantages in terms of scalability, performance, and efficiency. The computational burden introduced by detailed modeling of MMC-HVDC systems in electromagnetic-transients (EMT)-type programs complicates the study of transients especially when these systems are integrated into a large network. This paper presents a novel average-value model (AVM) for efficient and accurate representation of a detailed MMC-HVDC system. It also develops a detailed 401-level MMC-HVDC model for validating the AVM and studies the performance of both models when integrated into a large 400-kV transmission system in Europe. The results show that the AVM is significantly more efficient while maintaining its accuracy for the dynamic response of the overall system.

Detailed and Averaged Models for a 401-Level MMC–HVDC System

Summary form only given. Voltage Source Converter (VSC) technologies present a bright opportunity in a variety of fields within the power system industry. New Modular Multilevel Converters (MMCs) are expected to supersede two- and three-level VSC-based technologies for HVDC applications due to their recognized advantages in terms of scalability, performance and efficiency. Computational burden introduced by detailed modeling of MMC-HVDC systems in EMT-type programs complicates the study of transients especially when such systems are integrated into a large network. This paper presents a novel average-value model (AVM) for efficient and accurate representation of a detailed MMC-HVDC system. It also develops a detailed 401-level MMC-HVDC model for validating the AVM and studies the performance of both models when integrated into a large 400 kV transmission system in Europe. The results show that the AVM is significantly more efficient while maintaining its accuracy for the dynamic response of the overall system.

Detailed and averaged models for a 401-level MMC-HVDC system

A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix Y by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part of Y to be positive definite (PD). We calculate a correction to the rational approximation of Y which enforces the PD-criterion to be satisfied. The correction is minimal with respect to the fitting error. The method is based on linearization and constrained minimization by quadratic programming. Examples show that models not satisfying the PD-criterion can lead to an unstable simulation, even though the rational approximation has stable poles only. Enforcement of the PD-criterion is demonstrated to give a stable result.

/pdf/enforcing-passivity-for-admittance-matrices-approximated-by-4l3sscgmbe.pdf

Enforcing Passivity for Admittance Matrices Approximated by Rational Functions

This paper presents a transmission line model for the simulation of electromagnetic transients in power systems. The model can be applied to both overhead lines and cables, even in the presence of a strongly frequency dependent transformation matrix and widely different modal time delays. This has been achieved through a phase domain formulation where the modal characteristics have been utilized in the approximation for the propagation matrix. High computational efficiency is achieved by grouping modes with nearly equal velocities and by columnwise realization of the matrices for propagation and characteristic admittance.

A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables

A model to simulate the high-frequency behavior of a power transformer is presented This model is based on the frequency characteristics of the transformer admittance matrix between its terminals over a given range of frequencies The transformer admittance characteristics can be obtained from measurements or from detailed internal models based on the physical layout of the transformer The elements of the nodal admittance matrix are approximated with rational functions consisting of real as well as complex conjugate poles and zeroes These approximations are realized in the form of an RLC network in a format suitable for direct use with the ElectroMagnetics Transient Program (EMTP) The high-frequency transformer model can be used as a stand-alone linear model or as an add-on module of a more comprehensive model where iron core nonlinearities are represented in detail >

A high frequency transformer model for the EMTP

A method is developed to reduce large power systems to single- and multiport frequency-dependent equivalents. These equivalents consist of simple RLC modules that faithfully reproduce the frequency characteristics of the network. The method is implemented in the EletroMagnetics Transient Program (EMTP) and has been tested at Ontario Hydro. The implementation involves a preprocessor program, the Frequency Dependent Equivalent (FDNE), to generate the model and a EMTP time step loop module to calculate the transient response. The use of the FDNE results in major reductions in computer time and is especially beneficial for multicase statistical EMTP studies. An example showing the accuracy and efficiency of the FDNE when used to reduce a large 500 kV network is presented. >

Multi-port frequency dependent network equivalents for the EMTP

This paper describes studies carried out to investigate the failure of one phase of a 990 MVA generator step-up transformer bank at one of Ontario Hydro's major generating stations. A high accuracy frequency dependent transformer model was used to determine the behaviour of the transformer at the terminals under transient conditions. With this information, and with measured transfer functions between the terminals and various points inside the windings, it was possible to create internal voltage distribution maps. This information was then used to evaluate the insulation stresses within the winding, and to ascertain that the use of additional capacitance at the HV terminals to reduce the magnitude and frequency of incoming transients would be adequate to reduce the disk-to-disk stresses near the line end of the HV winding. The values of the HV terminal capacitances were chosen so that no undue stresses in other portions of the winding would take place.

Analysis of internal winding stresses in EHV generator step-up transformer failures

Application of metal oxide surge arresters in power systems has been traditionally linked to electrical equipment protection. The industry has noted a very significant increase in the application of metal oxide arresters on transmission lines in an effort to reduce lightning initiated flashovers. This paper describes a cost-effective installation of surge arresters on a partially shielded transmission line. It compares several options with respect to cost and gains in terms of lightning performance improvement. This paper also presents a new approach to calculate risk of failure of transmission line surge arresters (TLSAs) due to lightning strokes to towers, shield/phase conductors, and illustrates the method with an example for a partially shielded line. Results show that the risk of failure is very small. The installation of TLSAs completed in 1997 demonstrated that it is possible to afford adequate lightning protection levels by selectively applying surge arresters only to the towers most sensitive to backflashover and shielding failures.

A.S. Morched

Papers

A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables

A high frequency transformer model for the EMTP

Multi-port frequency dependent network equivalents for the EMTP

Analysis of internal winding stresses in EHV generator step-up transformer failures

Transmission line arrester energy, cost, and risk of failure analysis for partially shielded transmission lines