Bjørn Gustavsen

• PhD
• Chief Scientist at SINTEF

The so-called non-linear vector fitting (NL-VF) technique was recently introduced as a new method for rational function approximation in the frequency domain (FD). While the classical vector fitting (VF) fits a rational function to a given frequency response, NL-VF calculates the rational function approximation based on a set of input/output respon...

In this work we address the modeling of multi-port subsystems with unknown inner dynamics by utilizing vector fitting to identify state-space models for performing eigenvalue-based analysis. Vector fitting is used to characterize frequency responses obtained from frequency domain analysis, for example, from Fourier transformation of time domain dat...

Variable speed drives (VSDs) cause motor overvoltage stresses to result from fast-front voltage waveforms emerging from the converter, in particular when filters are not installed at the converter output terminals. This work investigates modeling procedures for predicting the overvoltages on a 970 kW 6.6 kV induction motor that is fed from 41-m sin...

Transformer white-box models are used by the manufacturers to calculate internal winding voltages during the lightning impulse test. The model can also be applied in general network studies, but the model accuracy should then be improved, considering the many different voltage waveforms and frequencies that can exist in the system. Accuracy improve...

A method is introduced for impedance and loss calculation of three-core power cables with steel armoring, based on 2D finite element method (FEM) computations. The pitching effect of the cores and armor wires is taken into account by 1) enforcing identical net current on the wires, and by 2) introducing a fictitious non-conductive material between...

The prediction of voltage stresses in transformer and machine windings requires the ability to calculate pulse propagation effects on the feeding cable with sufficient accuracy. The use of commonly available cable models in electromagnetic transient (EMT) programs can lead to voltage wave fronts with too weak damping at very high frequencies. This...

During lightning or switching in power system substation, particularly in the Gas Insulated Substations, very fast transient oscillations may take place. Transformers can be exposed to severe transient overvoltages, which may propagate inside windings and damage the transformer. Substation grounding impedance in this case cannot be considered as ne...

A novel measurement-based method for high-frequency modeling of shielded cables is introduced for use in very fast transient simulation studies. The method requires measurements on one cable end only, thereby being applicable to installed cable systems. Three frequency sweep voltage transfer measurements are performed at one cable end using a vecto...

This paper presents an alternative vector fitting (VF)-based approach to generate a rational model of nonlinear system behaviors. The proposed approach, named here as nonlinear VF (NL-VF), has its fundament in the previously proposed time-domain VF (TD-VF). TD-VF generates rational models using TD input/output responses and numerical convolution. U...

Lightning strokes is a main source of transformer failures in distribution systems. The discharge of a lightning current through a distribution transformer's high-voltage (HV) side arresters leads to a ground potential rise of the transformer tank. In the case of a wye-wye connected transformer with ungrounded HV neutral point, the ground potential...

Abstract—Measurement-based wide-band black-box transformer models are well suited for simulation studies of high-frequency transients in power systems. The inclusion of neutral points in such models can be difficult when the model is required to simulate voltage transfer to windings with high-impedance loads. The difficulty arises because it is the...

Rational models can be a cause of unstable time domain simulations if they are non-passive. One commonly applied method for ensuring model passivity is to combine a passivity assessment step with a passivity enforcement step in an iterative loop where the model's residue matrices are updated in each pass. This paper shows a new variant of such sche...

The automotive power system is being increasingly expanded by adding high-dynamic power electronics. These components can potentially cause malfunction or failure of safety-relevant low-voltage components. The susceptibility to disturbances is often reduced by usage of oversized passive input electronics. This paper introduces an alternative means...

This paper presents a novel approach, based on the theory of hyperplanes, for mode identification of linear systems. The proposed approach can operate on either a set of ordinary differential equations (converted to diagonal form, if needed) or a set of partial fractions derived from a synthesized transfer function of the system under analysis. For...

Lightning strokes is a main source of transformer failures in distribution systems. The discharge of a lightning current through a distribution transformer's high-voltage (HV) side arresters leads to a ground potential rise of the transformer tank. In the case of a wye-wye connected transformer with ungrounded HV neutral point, the ground potential...

Passivity is a fundamental requirement for rational models to guarantee stable behavior when included in general time domain simulations. This paper introduces a new variant of one passivity enforcement scheme that is based on residue matrix perturbation. The constrained least squares problem associated with the passivity condition is transformed i...

Research tests were performed on a shell-form autotransformer with a more rigorous impulse test methodology than the traditional recurrent surge oscillograph method. These tests included non-standard terminal connections with open terminals, differing from those tests defined by the international standards related with impulse testing of power tran...

A wide-band, frequency-dependent five-terminal model is developed that represents one HVDC transformer unit in the French-English IFA2000 HVDC interconnection. Three such interconnected 1-ph units constitute one 3-ph transformer bank needed in 12-pulse conversion. The model is obtained via admittance frequency sweep measurements on the transformer'...

Research tests were performed on a shell-form autotransformer with a more rigorous impulse test methodology than the traditional recurrent surge oscillograph method. These tests included non-standard terminal connections with open terminals, differing from those tests defined by the international standards related with impulse testing of power tran...

White-box detailed transformer models are used by manufacturers for predicting internal overvoltages in transformer windings during the lightning impulse test. One such model is the d-factor model which is based on a lumped-parameter description based on winding discretization with inclusion of losses via an empirical, frequency-dependent damping f...

This study presents the impact of wind power plant components modelling on harmonics propagation and harmonic small‐signal stability studies. Different types of cable and transformer modelling techniques are taken into consideration, e.g. state‐of‐the‐art standard models and latest academic developments. The models are compared based on system‐leve...

Offshore umbilical cables often include both power cables and elements such as metallic tubes and signal cables. It was concluded from a recent failure of stainless tubes that AC corrosion was the failure cause. This paper describes how AC corrosion can lead to tube failure if a puncture exists in the insulating tube coating. The variation in power...

Transformer manufacturers make routine use of white-box detailed transformer models for ensuring that the transformer will pass the lightning impulse test. For use in general simulation studies, the model should additionally be multi-phase and properly reproduce the transformer's input impedance characteristics and voltage ratios at 50/60 Hz. One c...

The transformer manufacturers make use of electromagnetic transient calculations by detailed white-box models to ensure that the transformer will withstand the lightning impulse test during the factory acceptance test. In principle, such models could be used in general electromagnetic transient simulation of overvoltages in the system. One of the o...

The transformer manufacturers make use of electromagnetic transient calculations by detailed white-box models to ensure that the transformer will withstand the lightning impulse test during the factory acceptance test. In principle, such models could be used in general electromagnetic transient simulation of overvoltages in the system. One of the o...

Transformer terminal equivalents obtained via admittance measurements are suitable for simulating high-frequency transient interaction between the transformer and the network. This paper augments the terminal equivalent approach with a measurement-based voltage transfer function model which permits calculation of voltages at internal points in the...

The authors present a methodology to improve the rational modelling of non-uniform lines (NuLs). First, they formulate a segmented chain matrix representation of the NuL, which is converted into a nodal admittance formulation with respect to the two line ends. The admittance matrix is fitted with rational functions while utilising the so-called cha...

A stable and passive six-terminal model is developed for a 45-MVA generator step-up transformer basedon frequency sweep measurements in the range 5 Hz–10 MHz and curve fitting with rational functions.The modeling employs separate treatment of the zero sequence system in combination with a new variantof the fast residue perturbation method for passi...

Cable-series impedance modeling is widely applied in electromagnetic transients simulations and power-loss calculations. The calculations are usually performed in a 2-D frame using the finite-element method or alternative approaches, such as MoM-SO, thereby losing the 3-D effects imposed by twisting wire screens and armors. One of the implications...

Transformer terminal equivalents obtained via admittance measurements are suitable for simulating high-frequency transient interaction between the transformer and the network. This paper augments the terminal equivalent approach with a measurement-based voltage transfer function model which permits calculation of voltages at internal points in the...

Frequency-dependent transmission line modeling by the traveling wave method requires to approximate the propagation function with a delayed rational function. Some approaches are based on modal decomposition where scalar functions are fitted with a rational model plus a single time delay. The delay is calculated from the modal velocity and the mini...

Admittance frequency-sweep measurements is an accepted procedure for characterizing transformer terminal behavior for the purpose of frequency-dependent black-box modeling. The errors introduced by the measurement cables is with one existing practice mitigated by removal of the associated shunt capacitance effect. In this paper, it is shown that th...

In this paper a new method for power system stability analysis is introduced. The method is based on injection of a small voltage or current in an arbitrary point of the system. The apparent impedance is defined as the ratio between the voltage and current in the injection point. It is shown that the apparent impedance can be used to estimate the e...

The Universal Line Model (ULM) is widely used for the simulation of electromagnetic transients in electrical power transmission systems. This paper proposes an improvement in the computational efficiency of the ULM by introducing a methodology for rational approximation of smooth frequency domain responses using real poles only through Vector Fitti...

Connection of offshore wind farms to shore requires the use of submarine cables. In the case of long HVAC connections, the capacitive charging currents limit the transfer capability and lead to high losses. This paper shows that the losses can be substantially reduced by continuously adjusting the cable operating voltage according to the instantane...

Connection of offshore wind farms to shore requires the use of submarine cables. In the case of long HVAC connections, the capacitive charging currents limit the transfer capability and lead to high losses. This paper shows that the losses can be substantially reduced by continuously adjusting the cable operating voltage according to the instantane...

Offers an overview of state of the art passive macromodeling techniques with an emphasis on black-box approaches This book offers coverage of developments in linear macromodeling, with a focus on effective, proven methods. After starting with a definition of the fundamental properties that must characterize models of physical systems, the authors...

The main idea of black-box macromodeling is to approximate the dynamic behavior of complex systems in terms of lowcomplexity models or equivalent circuits. Such compact models can be derived through robust numerical algorithms, such as the Vector Fitting scheme, starting from frequency-or time-domain responses of the system, and without any specifi...

A large number of electromagnetic transient studies have so far reported findings related to the overvoltage behavior of systems within a broad frequency range. However, in most cases, actual grounding effects have been either not taken into account or have been just partially considered. Although an accurate methodology to study grounding effects...

This paper presents a new approach to wide-band modeling of tower footing grounding impedances. Appropriate representation of transmission tower footing impedances is essential in electromagnetic transient simulations for proper evaluation of overvoltages and flashovers in the system during fast-front transient conditions such as lightning strikes....

This chapter introduces the vector fitting (VF) method for fitting a rational function (model) to a set of frequency-domain or time-domain tabulated data. It starts from the linearized rational fitting formulation obtained by weighting, known as Levy's approach. The chapter explains the Sanathanan-Koerner (SK) iteration, which compensates the undes...

The concept of passivity is based on the black-box input-output characterization of the system. Passivity implies that the cumulative net energy entering the system is nonnegative at all times. This condition leads to a precise characterization of passivity in terms of the system transfer function, which must be positive real for immittance represe...

Lumped linear time-invariant (LTI) systems are described by ordinary differential equations (ODEs) expressed in state-space or descriptor forms. This chapter considers distributed LTI systems. Two simple examples show how distributed systems can be interpreted as lumped systems having an infinite number of state variables. Distributed systems natur...

This chapter considers lumped linear time-invariant (LTI) systems for which one knows the full-size state-space, descriptor, or modified nodal analysis (MNA) equations. These systems are often obtained from first-principle formulations based on circuit or discretized field equations. The number of variables occurring in these descriptions is usuall...

This chapter introduces some fundamental concepts on signals and systems. The emphasis is on the special class known as linear time-invariant (LTI) systems. The chapter reviews the concepts of linearity, time invariance, causality, stability, and passivity of LTI systems. It starts by providing elementary time-domain definitions. After reviewing th...

This chapter presents some closing thoughts on the concepts discussed in the various chapters of this book. The book presents several passive macromodeling schemes, with emphasis on vector fitting (VF) and passivity assessment and enforcement. It includes some relevant background material, basic definitions of signals, systems, their mathematical r...

This chapter addresses the problem of taking a given macromodel as a black box, represented by a rational transfer matrix or an associated state-space realization, and performing transient simulations. It commences with a review of basic techniques for the numerical solution of continuous-time dynamic systems. The chapter presents a simple example...

This chapter discusses black-box macromodeling methods, which are indeed the main focus of this book. It introduces least squares (LS) techniques, as applied to curve fitting. A detailed analysis is provided for linear LS fitting, which is shown to be optimal under several standpoints. The chapter shows that a direct LS approach for rational curve...

This chapter applies the macromodeling techniques developed throughout this book to a number of problems from different application areas. The main emphasis is on electrical (low frequency) and electronic (high frequency) applications. The chapter discusses applications to signal and power integrity characterization and simulation, computational el...

This chapter explains the vector fitting (VF) algorithm to the simultaneous fitting of several (scalar) elements collected in a vector of frequency responses. It introduces alternative multi-input multi-output (MIMO) model formulations and demonstrates that the common pole fitting capability of VF leads directly to compact state-space models in a n...

The full dynamic behavior of linear time-invariant (LTI) systems can only be captured by some internal description, that is, when detailed information on the system, its components, and its governing physical laws is available. The chapter starts with a simple dynamic circuit example, whose constitutive equations are derived in various forms. The e...

This chapter starts with a review of the various passivity constraints for lumped immittance and scattering systems described by state-space equations. The basic idea in the chapter is to correct a given system or macromodel by some small amount, such that the resulting perturbed system satisfies the appropriate passivity constraints. After reviewi...

This chapter introduces techniques that are suitable for modeling multi-conductor transmission lines (MTLs) in terms of their terminal behavior, including frequency-dependent effects. It starts by defining the fundamental characteristics of the line in terms of per-unit-length (PUL) parameters, wave propagation characteristics, and terminal admitta...

Rational function approximations are widely used for the modelling of frequency-dependent effects in power system components. This paper proposes to use optimization by the damped Gauss–Newton method to improve the accuracy of rational approximations calculated by three mainstream fitting techniques: Bode's asymptotic approximation, Sanathanan–Koer...

1This paper presents results from time domain transient simulations using a wide band black-box model of a 100 MVA transformer. Unlike in other studies, a modified transformer model is used which also allows investigating voltage transfers from external terminals to chosen nodes along the windings. The severity of voltage stresses on transformer's...

A 420-kV gapped-core five-legged variable shunt reactor is modeled in the frequency range 5 Hz-10 MHz based on frequency sweep measurements and curve fitting using rational functions. Comparison with time-domain measurements at reduced voltage shows that the model can accurately predict the transient behavior of the shunt reactor, both for impingin...

Black-box transformer high-frequency models can be determined via frequency sweep measurements and curve fitting based on rational functions. This procedure often leads to unacceptable model behaviors around 50/60 Hz, making it hard to properly initialize the model for high-frequency transient events. In this paper, we overcome the accuracy problem...

Switching transients and earth faults may lead to resonant overvoltages at wind turbine transformer terminals as well as inside High Voltage (HV) and Low Voltage (LV) windings. The winding design in a transformer could strongly influence the internal voltage distribution as function of frequency, and this has a great impact on risk of insulation fa...

Wideband, black-box transformer models can be obtained via frequency sweep measurements on transformer terminals followed by rational model extraction via passive curve fitting. However, these models do not include core nonlinearity effects. A modeling approach is introduced where a wideband linear model is connected in parallel with a nonlinear 50...

Field solvers are often needed to properly characterize the behavior of grounding structures in a wide frequency range. These solvers provide accurate results but at high computational cost. Design tasks, such as design optimization and sensitivity analysis, require repeated simulations for different values of system parameters (e.g., conductor geo...

White-box transformer models are used by transformer manufacturers during the dielectric design of windings. The models are often based on constant parameters (RLCG matrices) with the high-frequency losses accounted for by a scaling of the dc resistance ( k-Factor). We show an efficient procedure for interfacing such models with Electromagnetic Tra...

The standard lightning impulse voltage wave used in transformer factory testing does not properly cover all the overvoltage stresses that a transformer can experience while in service. To properly asses the internal overvoltages that can occur in practice, we propose to interface the manufacturer's white-box transformer model with EMTP-type simulat...

High-voltage cables with paper-oil based insulation are still widely used in electric power systems. One issue with EMTP-type modeling and simulation of such cables is that the permittivity of the insulation is frequency-dependent, thereby contributing to the dispersion of traveling waves. This frequency dependency is not taken into account in curr...

Switching transients and earth fault in a wind farm collection grid are two transient phenomena which can lead to resonant overvoltages at the LV terminal of the wind turbine transformers as well as inside HV and LV windings. The aim of this paper is to analyze the potential of the resonant overvoltage for various winding designs; disc, layer and p...

This paper presents a new procedure to characterize an admittance matrix over a wide frequency band, for the purpose of black-box modeling of power transformers. Unlike previous approaches, our setup is based on a commercial sweep frequency-response analysis measurement instrument which is lightweight and considerably less expensive than dedicated...

We describe a practical modeling approach for including the frequency dependency of transformer winding resistance in Electromagnetic Transients Program-type simulation tools, based on typical curves of ${rm R}(omega)/{rm L}$ ratios. From the given resistance component, the inductance component is calculated via Kramers–Kronig dispersion relations....

The protection of offshore wind farms (OWFs) against overvoltages, especially resonant overvoltage, is of paramount importance because of poor accessibility and high repair costs. In this paper, we study how switching overvoltages at the wind turbine transformer (WTT) medium voltage (MV) side can lead to high overvoltages on the low voltage (LV) si...

Modeling of frequency-dependent components and subnetworks is often based on a terminal description by an admittance matrix in the frequency domain. One challenge in the extraction of state-space models from such data is to prevent possible error magnification when the model is to be applied in time-domain simulations. The error magnification is a...

The installation of medium voltage transformers in wind turbines motivates the wind turbine manufacture, operators and researchers to review the challenges of insulation failure owing to overvoltages. Among these overvoltages, internal resonant overvoltages are most crucial and destructive. One of the protective measures, which decrease the potenti...

We present an efficient numerical technique for calculating the series impedance matrix of systems with round conductors. The method is based on a surface admittance operator in combination with the method of moments and it accurately predicts skin and proximity effects. The application to a three-phase armored cable with wire screens demonstrates...

Switching transients and ground faults may lead to resonant overvoltages at wind turbine transformer terminals as well as inside High Voltage (HV) and Low Voltage (LV) windings. The winding design in a transformer will strongly influence the internal voltage distribution as function of frequency. This has a great impact on risk of insulation failur...

Switching transients and earth fault in a wind farm collection grid are two transient phenomena which can lead to resonant overvoltages at the LV terminal of the wind turbine transformers as well as inside HV and LV windings. The aim of this paper is to analyze the potential of resonant overvoltage for various winding designs; disc, layer and panca...

Earth fault and switching operation may lead to resonant overvoltages in Offshore Wind Farms (OWF) with higher amplitude and rate of rise (du/dt) compared to other overvoltages. Overvoltages appear on High Voltage (HV) cables and Low Voltage (LV) side of wind turbines, and can result in insulation failure of the Wind Turbine Transformers (WTTs), in...

The universal line model is among the most accurate frequency-dependent transmission-line model available in Electromagnetic Transients Program-type simulation tools. One major drawback of this line model is that it sometimes gives unstable simulation results. The instability is related to the occurrence of close poles in the rational model of the...

Wide-band cable models for the prediction of electromagnetic transients in power systems require the accurate calculation of the cable series impedance as function of frequency. A surface current approach was recently proposed for systems of round solid conductors, with inclusion of skin and proximity effects. In this paper we extend the approach t...

Thermal modeling of power electronics modules is often used for the prediction of internal temperatures which is essential for safe operation and life time estimations. However, the models provided by manufacturers are normally quite simplified, accounting only partially for the crosscoupling effects between chips. A more accurate analysis is possi...

We present a fast and fully-automated algorithm to extract the series impedance of cables with round conductors, such as USB and HDMI cables. Our solution combines the method of moments with a surface admittance operator, and is much more efficient than a finite elements approach. Moreover, it is fully automated, owing to an adaptive scheme to auto...

This paper presents a new macro modeling methodology for solid-state power amplifiers (SSPAs) and packaged transistors used in communication systems. The model topology is based on the principle of harmonic superposition recently introduced by Agilent Technologies' X-parametersTM combined with dynamic Volterra theory. The resulting multi-harmonic b...

Frequency-dependent effects in power system components and subnetworks can be efficiently represented via rational function-based models that characterize the component port behavior as a function of frequency. The port behavior can be defined by alternative parameter sets (e.g., admittance (Y-), impedance (Z-), or scattering (S-) parameters). The...

Voltage surges arising from transient events, such as switching operations or lightning discharges, are one of the main causes of transformer winding failure. The voltage distribution along a transformer winding depends greatly on the waveshape of the voltage applied to the winding. This distribution is not uniform in the case of steep-fronted tran...

The beneficiary of Offshore Wind Farms (OWF) to land wind farms in the sense of more power productivity and land view disturbance leads to more investment and consideration. Nevertheless, the maintenance of OWFs is a significant issue due to accessibility and repair costs. Thus, the protection of OWFs electrical components should be investigated in...

The application of transformers in offshore wind farms requires more consideration in the sense of reliability and protection than onshore ones due to the higher cost of accessibility and repair. Switching of offshore wind farms and lightning strikes are the main two transient phenomena which can lead to harmful damages to Wind Turbine Transformer...

Transient and harmonic coupling effects between parallel overhead lines are normally modeled via frequency-dependent traveling-wave-type transmission-line models. Several electromagnetic transient programs rely on a line model based on a constant transformation matrix and modes (FD-line). These line models can, however, give substantial errors in t...

We benchmark two different approaches in high frequency modeling of multiconductor cables for EMI system level simulations. Our models are obtained by fitting simulated or measured data of the cable's port admittance matrix to a lumped network. The first approach is based on simulated p.u.l. parameters, whereas the second one relies on measured s-p...

Direct application of model extraction methods to tabulated admittance data often gives a model which can suffer in accuracy when applied with high-impedance terminations. The problem is relevant in situations where the admittance matrix has a large eigenvalue ratio since the small eigenvalues are likely to become corrupted. We show a modeling proc...

This paper investigates the high-frequency properties of a 3.14-MVA 35/1.0/0.4-kV wind turbine transformer commonly used for the doubly fed asynchronous generator. Frequency sweep measurements carried out for this transformer show that the voltage transfer from the 35-kV side to either of the two low-voltage taps is strongly dependent on the loadin...

Transmission-line models for electromagnetic transient simulation are usually based on the traveling-wave method. While recent developments have focused on phase-domain formulations with an emphasis on accuracy and passivity enforcement, no consideration has been given to ensure the symmetry of the model. In this paper, we introduce the symmetry pr...