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This paper proposes applying the two-stage diagonally implicit Runge-Kutta (2S-DIRK) method of numerical integration to the calculation of electromagnetic transients (EMTs) in a power system. The accuracy and the numerical stability of 2S-DIRK are almost the same as those of the trapezoidal method, while 2S-DIRK does not produce sustained numerical oscillation due to a sudden change of an inductor current or a capacitor voltage unlike the trapezoidal method. First, this paper reviews the 2S-DIRK integration scheme and derives the 2S-DIRK formulas of inductors and capacitors for both linear and nonlinear cases. Then, analytical comparisons of 2S-DIRK with the trapezoidal, backward Euler, and Gear-Shichman methods are carried out, and numerical examples which verify the analytical comparisons are shown. Finally, 2S-DIRK is compared with critical damping adjustment (CDA) implemented in Electromagnetic Transients Program (EMTP) for some simulation cases. Index Terms--Electromagnetic transient analysis, integration (mathematics), numerical stability, power electronics, power semiconductor switches, power system simulation, power system transients.

This is an introductory paper on z-transform-based methods for electromagnetic transient simulations of power systems. Since the theory of the z-transform was originally developed for the analysis of time series data defined at equidistant time steps, simulation models developed using z-transforms can readily be used in electromagnetic transient simulations based on the same time step. First, this paper briefly introduces the basics of the z-transform, and then applications to electromagnetic transient simulations of power systems are reviewed. Index Terms--Electromagnetic transient analysis, equivalent circuits, power systems, signal processing, switching transients, time series, z transforms.

This paper proposes a simple closed-form formula of the ground-return impedance of horizontal parallel wires above a lossy ground plane. It is derived by a double logarithmic approximation of the integral term with a semi-infinite interval in Carson's ground-return impedance formula. The proposed double logarithmic approximation gives better accuracy compared to a similar approximation proposed by Pizarro and Eriksson, and this is achieved by an advanced optimization technique used for determining the coefficients in the double logarithmic approximation and also by incorporating an additional variable which minimizes errors. In this paper, it is shown, using practical examples, that the derived formula gives more accurate results compared with other existing approximate formulas. It is also shown that a graphical interpretation of the double logarithmic approximation is to replace a homogeneous lossy ground with a pair of perfectly conducting return planes located at different complex depths. Index Terms--Approximation methods, electromagnetic transient analysis, ground-wave propagation, impedance, lossy circuits, power distribution lines, power transmission lines, skin effect, surges.

This paper presents an algorithm for identifying a multiphase network equivalent for electromagnetic (EM) transient calculations. Conventional rational fitting methods intrinsically get into ill condition because the [s.sup.n] terms in the rational function take a wide range of values which cannot be accurately treated within machine arithmetic accuracy. The proposed algorithm averts the ill conditioning by partitioning the given frequency response into sections along the frequency axis. Rational fitting is applied to each section of the frequency response to identify the poles, and then the corresponding residue matrices are obtained by a standard least-squares procedure using the entire frequency response. Adaptive weighting, column scaling, and iteration step adjustment are utilized to facilitate the rational fitting. The proposed identification algorithm is applied to obtain a reduced-order equivalent of a 500-kV power network, and the equivalent is used in a switching transient simulation to demonstrate its performance. Index Terms--Electromagnetic (EM) transient analysis, equivalent circuits, frequency response, interconnected power systems, multi-input multi-output (MIMO) systems, power system modeling, power system simulation, realization theory, switching transients

This paper presents the result of a study on the error in propagation velocity introduced by the staircase approximation of a thin wire in the finite difference time domain (FDTD) surge simulation. The FDTD method directly solves Maxwell's equations by discretizing the space of interest into cubic cells. Thus, it is suitable for solving very-fast surge phenomena which cannot be dealt with by conventional techniques based on the lumped- and distributed-parameter circuit theories. However, FDTD has a limitation that the shape of a conductive object must be modeled by a combination of sides of cells with forced zero electric fields. This indicates that a thin wire, one of the most important components in the surge simulation, results in a staircase approximation, if the wire is not parallel to any of the coordinate axes used for the discretization. A staircase approximation gives a slower propagation velocity due to the zigzag path which is longer than the actual length of the wire. For precise simulations, the error in propagation velocity has to be clarified quantitatively. In this paper, extensive simulations are carried out to obtain the velocity versus inclination characteristic, and it is deduced that the maximum error in propagation velocity is less than 14%. Index Terms--Electromagnetic transient analysis, finite difference time domain (FDTD) methods, Maxwell equations, simulation, surges, wire.

This paper proposes an algorithm for obtaining the periodic steady-state solution of a multiphase network including nonlinear, switching, and frequency dependent elements. Unlike existing methods which deal with nonlinear and switching elements in the time domain, the approach presented is entirely in the harmonic domain. The method will be used for the harmonic analysis of power systems and for steady-state initialization in electromagnetic transient analysis. The algorithm takes rigorously into account the inter-harmonic couplings in the Jacobian matrix of the proposed Newton-Raphson iteration process so that a quadratic convergence rate is achieved. Linear, nonlinear, switching, and frequency dependent elements are modeled in a modular approach, and any network topology can be handled by extending the Modified Nodal Equations approach to the harmonic domain case. First the algorithm is described and then applied to a test ease to demonstrate its computational performance. Index Terms--Multiport circuits, nonlinear circuits, periodic functions, power system harmonics, switching circuits.

This paper presents a methodology for the identification of a reduced-order dynamic equivalent of a nonlinear power network for the simulation of electromagnetic transients. The equivalent is deduced from the observer companion form of the state equations with periodic coefficients and includes the effects of the nonlinearity of the power network at its operating point. A previously proposed concept, the harmonic domain dynamic transfer function (HDDTF), is used to characterize the network's transient behavior, superimposed on the steady state. The HDDTF is obtained by linearization of the nonlinear state equations of the network corresponding to harmonic perturbations applied to the steady-state operating point. Then, reduced-order companion-form state equations with periodic coefficients are fitted to the HDDTF in the frequency domain using a least-squares procedure based on the SVD and QR algorithms. The fitting procedure includes sequential weighting, column scaling, and vertical partitioning to improve computational accuracy and efficiency. The SVD algorithm serves to determine an appropriate model order. A test network with nonlinear inductances is used to demonstrate the performance of the identification method as well as the time-domain simulation results. Index Terms--Dynamics, electromagnetic transient analysis, equivalent circuits, identification, large-scale systems, nonlinear circuits, periodic functions, power system harmonics, saturable cores.

This paper presents a new concept called harmonic domain dynamic transfer function (HDDTF), which characterizes the dynamics of a nonlinear, time-periodic network as seen from a port (or multiple ports), in terms of the frequency response of harmonic perturbations superimposed on its underlying periodic steady state. It pertains to the transient behavior superimposed on the steady state. The HDDTF is a transfer-function matrix H(s) relating the vectors of harmonic domain input and output endowed with s-domain properties. Because the network can contain saturable (nonlinear) elements and periodically-switching (time-periodic) power electronics components, the HDDTF may be used for the analysis of power quality problems. It may also serve for the identification of a reduced-order dynamic equivalent of a nonlinear, time-periodic network to be used in time-domain transient simulations. The HDDTF is obtained by linearization, about the periodic steady state, of the nonlinear state equations describing a given network. Following the derivation of the HDDTF, a modal analysis to characterize the HDDTF by its diagonalization is presented. Two test systems are used to produce numerical examples. Index Terms--Dynamics, electromagnetic transient analysis, large-scale systems, nonlinear circuits, periodic functions, power electronics, power quality, power system harmonics, saturable cores, transfer functions.

This paper investigates the charge versus voltage (q-v) curves of surge corona on transmission lines. Two measurement methods of the q-v curves are proposed, and field test results are shown to validate the methods. Since the new methods do not require a charge-measuring conductor, such as a corona cage, the q-v curves are measured in the real electric-field distribution. One of the methods obtains the amount of charges q by numerically integrating the digitally stored waveform of injected current, assuming that the line is short in length and open ended. Thus, it is suitable for measurement using an experimental setup. The other calculates q from voltage and current waveforms based on the traveling-wave theory, assuming that the line is long enough to expect no reflected waves, and it is applicable to existing 'real' transmission lines. Measured q-v curves are compared with simulated ones, and their physical properties are discussed in the paper. Index Terms--Charge measurement, corona, power transmission lines, space charge, surges.

This paper proposes a model of core-type distribution transformers for electromagnetic (EM) transient studies. The model accurately reproduces not only the impedance characteristics seen from each terminal of a core-type distribution transformer but also the surge-transfer characteristics between the primary and secondary sides in a wide range of frequencies. Due to the above capability, the proposed model enables the accurate evaluation of overvoltages on distribution lines including consumer-side overvoltages. In this paper, a 10-kVA transformer is modeled, and transient-simulation results agree well with laboratory-test ones. Index Terms--Electromagnetic transient analysis, power distribution, power transformers.

Simulation of very fast surge phenomena in a three-dimensional (3-D) structure requires a method based on Maxwell's equations, such as the finite difference time domain (FDTD) method or the method of moments (MoM), because circuit-equation-based methods cannot handle the phenomena. This paper presents a method of thin wire representation for the FDTD method which is suitable for the 3-D surge simulation. The thin wire representation is indispensable to simulate electromagnetic surges on wires or steel flames of which the radius is smaller than a discretized space step used in the FDTD simulation. Comparisons between calculated and laboratory test results are presented to show the accuracy of the proposed thin wire representation. The development of a general surge analysis program based on the FDTD method is also described in the present paper. Index Terms--Electromagnetic transient analysis, finite difference time domain (FDTD) methods, Maxwell equations, simulation, surges, wire.

This article presents a comparison between simulation and laboratory-test results of lightning surge propagation from a distribution line to a consumer entrance via a pole-mounted transformer. The simulation result, obtained using a transformer model previously proposed by the authors, closely reproduces the laboratory-test result. Index Terms--Electromagnetic transient analysis, lightning, power distribution lines, power transformers, surges.