In electromagnetic transients studies, only a limited part of a transmission system can be modelled in detail. The remaining parts of the system have to be reduced to equivalents. Simple equivalents which reproduce the network's response at few frequency points can introduce significant errors into the calculated transients. A method to generate network equivalents from the frequency response of its admittance has been developed. The equivalents are in the form of lumped parameter circuits, have a simple structure and reproduce the network's behaviour over a wide frequency range. The developed method was used to generate an equivalent for a sample system. Voltage transients due to an energization of a transmission line from that sample system were calculated. The results of calculations with the full system representation and the developed equivalent showed a good agreement.

Transmission Network Equivalents for Electromagnetic Transients Studies

The paper presents a mathematical corona model and its use for transmission line modelling with corona and, if desired, with frequency dependence of its parameters. The corona model has been developed from macroscopical physical laws, reflecting relations between charge, electric field intensity and voltage. Space charges, their development and their displacement are taken into account. The parameters of the model are determined so that the simulated qv curves closely fit those available from measurements. The corona model is then used as an element of the discretized transmission line model. This consists either of longitudinal inductances or of travel delay modules. Modelling of frequency dependence is optional. The line equations are integrated using the trapezoidal rule, so that Norton equivalents are obtained for the two line ends for the purpose of interfacing with the external system. Thus the model is fully compatible with an Electro-Magnetic Transients Program (EMTP).

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Corona Modelling for the Calculation of Transients on Transmission Lines

It is known that an untransposed open-ended line causes a negative sequence component in the current and the generator reacts through third and higher harmonic voltages. The currents and voltages can be greatly amplified by resonance to any of these harmonics. The phenomenon of machine-line interaction has traditionally been explained as a sequence of interactive events between the line and the machine. This paper presents a unified noniterative mathematical representation of this interaction. The analysis is formulated in the harmonic domain abc-frame to detect resonance conditions and for the assessment of their stability.

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Harmonic domain characterization of the resonant interaction between generator and transmission line

The paper presents the calculation of electromagnetic transients (EMT) on three-phase transmission lines with frequency dependent parameters and corona, the latter represented by the model of Ref. [1]. The transmission line is divided into segments, each with a small travel delay, a longitudinal block for the modelling of frequency dependence, and a transversal model of the nonlinear effect of corona. Longitudinal "shaping" is obtained by parallel R,L elements corresponding to a series of such blocks obtained by fitting the modal line impedances over a wide range of frequencies. The computations are performed in the modal domain for longitudinal phenomena and in the phase domain for the transversal branches which include corona in each phase. The travel delay effectively decouples the nodes and the calculations for the corona branches. Thus it permits the representation of the line by Norton equivalents at its ends, so that the model can be included in a general purpose Electro-Magnetic Transients Program (EMTP). The computational results presented in the paper show wave shapes similar to those observed in measurements [13] and clearly demonstrate the significance of modelling both frequency dependence and corona.

Computation of Electro-Magnetic Transients on Three-Phase Transmission Lines with Corona and Frequency Dependent Parameters

This paper presents a Newton-type methodology to calculate the transient or periodic steady state of an electrical network including nonlinear elements. The basic idea is to decompose the complete network into linear and nonlinear subnetworks. On one hand, a nodal representation of the linear network is considered. On the other hand, a Jacobian corresponding to a nonlinear element is calculated numerically via input perturbations, with the terminal voltage being the input and the device current as being the output. The latter is calculated in the time domain, via a polynomial representation, and converted back into the frequency domain by numerical Laplace transform operations. Finally, the solutions corresponding to the linear and nonlinear subnetworks are included in a Newton-type iterative scheme having a current mismatch (at the point of coupling) as its basis. Two examples involving nonlinear loads in a network are presented for illustration of the aforementioned procedures.

Frequency-Domain Computation of Steady and Dynamic States Including Nonlinear Elements

This paper considers a very general formulation of the differential equations of a dynamic system in periodic steady state. These are linearized around an operating point, in algebraic form in terms of incremental harmonic phasor components. The solution is iterative, either Newton-type with quadratic convergence in the neighbourhood of the solution, or it has linear convergence if the Jacobian is not updated at each iteration.

Steady-state analysis of nonlinear dynamic systems with periodic excitation based on linearization in harmonic space

Discusses the calculation of inductances of transmission lines above nonideal ground based on the images of the conductors. The earth is assumed of finite conductivity but homogeneous. For this case Carson's correction terms (1926) of the impedance are well known and widely used. Carson's corrections are, however, expressed in the form of truncated series so that their evaluation normally requires the use of a small computer program. The author proposes an alternative solution to the problem. Recognizing that the flux in the earth is not in phase with the return current, an equivalent return path is proposed at a complex depth. This preserves the possibility of using images for practical calculations even though a complex depth or distance has no physical significance. The author shows a very simple formula equivalent to Carson's equations, and establishes its usefulness by examining its accuracy.

Correspondence: Approximation to Carson's loss formulae

The paper is essentially an overview presenting an outline of how Electro-Magnetic Transients (EMTs) are calculated in power system networks for means of an EMT Program based on travelling wave modelling of transmission lines. This paper presents the following steps: (1) For circuits with lumped parameters, the individual branches are modelled by different equations which are discretized to provide Norton equivalents. These are assembled producing nodal equations, to be solved step-by-step. (2) Transmission lines are modelled by time lags and Norton equivalents, even if their parameters are frequency dependent or multi-phase, in which case model decompositions are used. This paper uses state equations for transmission line modelling. Inclusion of transmission line Norton equivalents in the nodal equations does not complicate the system problem due to the effective time-decoupling of the computations. Some recent developments such as Z-transforms, s-domain analysis and corona modelling are also described.

Electro-magnetic transients on overhead transmission lines — An overview

Natural frequencies of transmission lines are identified by the poles obtained from line transfer function matrices. The procedure is based on an analysis in terms of the complex frequency s which has been introduced in the expressions of line parameters and transfer functions using the concept of complex depth.

Natural frequencies of transmission lines

A multiphase transmission line is modelled by sinusoidally distributed charges on a cylindrical surface. In its inside, the field is uniform and outside the cylinder it can be obtained by a rotating equivalent dipole. Total charge, capacitance, required number of conductors and natural power are calculated for this idealized multiphase line model, with or without ground wires and shielding. The practical advantages of this model are presented in its application to actual multiphase transmission line geometries for the calculation of the ground level potential gradient and of maximum conductor surface gradients.

A. Semlyen

Papers

Steady-state analysis of nonlinear dynamic systems with periodic excitation based on linearization in harmonic space

Correspondence: Approximation to Carson's loss formulae

Electro-magnetic transients on overhead transmission lines — An overview

Natural frequencies of transmission lines

Infinite phase order modelling of multi-phase transmission lines