# Manual Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems

In fact, the specific losses are customarily given for unit weight but multiplying them by the specific weight yields the losses per unit volume. Usually, the specific losses are measured for one single frequency f0 e. In order to approximately take account of the dependence of the specific losses on frequency, the following algorithm is adopted. It is assumed that, neglecting excess losses,. Similarly to the current density, the magnetic flux density is also obtained in the form of a Fourier series when using the harmonic balance technique:.

In lack of any better assumption, the specific losses are simply computed for each harmonic from 64 and then added: N f. As an example, the autotransformer analyzed in  is presented here.

A Video On The Finite Element Method.

Its name plate data are given in Table 1. The FEM model used has been improved in comparison to , it consists of , finite elements. The problem has been solved using the Tformulation, resulting in 2,, degrees of freedom for the potentials. The model is shown in Fig. Two short circuit computations have been carried out with the winding currents taken to be sinusoidal and the magnetization current neglected. The losses have been computed as described above.

The computed losses in the two cases are summarized in Tables 2 and 3, given as a percentage of the total measured short circuit losses.

## A Harmonic Balance Methodology for Circuits with Fractional and Nonlinear Elements | SpringerLink

These results indicate that in parts of the transformer where significant saturation is present, like in the first laminates of the core exposed to stray magnetic fields see Fig. Geomagnetically induced currents GIC are direct currents that enter and leave the directly earthed neutrals of highvoltage star connected windings, causing a direct current DC bias in the magnetizing current of the transformer . The frequency of GIC ranges typically from 0. FEM model of the analyzed single-phase autotransformer.

The model comprises one half of the transformer.

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The tank is shown transparent, the core is yellow, the clamping plates and the tie bars are shown green. The windings and the tank shieldings are red. Losses in percentage of total measured losses of autotransformer analyzed. All quantities are sinusoidal. In recent years, high-voltage direct current HVDC transmission is widely used for intercontinental distribution of electric power. The large DC potential difference between two converting plants also generates a DC that flows into the windings of the power transformer . The core of the transformer is saturated during the half cycle in which the bias current is in the same direction as the magnetizing current, causing undesirable effects like increased noise, additional core losses as well as eddy current losses due to the higher leakage flux.

Here, we focus on solving a nonlinear steady-state power transformer problem under DC bias in the discrete Fourier domain. The Tformulation is used with the voltage in the winding directly used as the excitation, i. The geometry of a single-phase power transformer is shown in Fig. The model comprises 54, second-order hexahedral finite elements. The tie bar is made of massive steel same material as the tank in the model of Fig.

Both ferromagnetic materials are nonlinear, the corresponding B-H curves are shown in Fig.

The single winding is driven by a given sinusoidal voltage of 60 Hz. The current of the winding has a known DC bias. To validate the fixed-point method, the problem has been first solved by the method in . This method is capable of predicting the waveform of the magnetizing current based on three-dimensional static finite element analyses of the transformer. With the aid of a flux-current curve, the waveform of the magnetizing current with the prescribed DC value is predicted, the computed waveforms have been shown to agree well with measured ones.

The waveform of the winding current at a DC component of 45 A obtained by the present method is in good agreement with the result of  as shown in Fig. It has been shown that the use of FEM in conjunction with the continuous or discrete harmonic balance method is capable of providing the steady-state solution to large, complex real-world eddy current problems with the nonlinearity of. The method developed allows for the decoupled computation of the harmonics and is hence massively parallel.

### ISBN 10: 1118975766

Albanese, E. Coccorese, R. Martone, G. Miano, G. Biro, Edge element formulations of eddy current problems, Comput. Methods Appl. Biro, S.

Biro, U. Baumgartner, G. Koczka, G. Leber, B. Baumgartner, K.

Preis, G. Leber, Finite element method for nonlinear eddy current problems in power transformers, in: Proceedings of the International Colloquium Transformer Research and Asset Management, Dubrovnik, Croatia, May , , Session I, Numerical modelling, 10 pages. Biro, Y. Chen, G. Leber, K. Preis, B.

## Seminar in OMN TEAM

Wagner, Steady-state analysis of power transformers under DC bias by the finite element method with the fixed point technique, in: 18th International Conference on the Computation of Electromagnetic Fields, Sydney, Australia, July , , Numerical techniques, Paper , 2 pages. Biro, G. Koczka, K. Biro, K. Buchgraber, I.

Ticar, Voltage driven coils in finite element formulations using a current vector and a magnetic scalar potential, IEEE Trans. Hara, T. Naito, J. Koczka, S. Ausserhofer, O. Koczka, O. Lu, Y.