# HBFEM Model of HVDC Transformer

Consider a three-phase HVDC transformer model with a three-phase voltage-driven source connected to the magnetic system, which is always coupled to the external

Figure 3.10 HVDC power transmission system

Figure 3.11 B-H curve with hysteresis characteristics and a DC biased condition. (a) DC-biased hysteresis loop; (b) Magnetizing current

Figure 3.12 The block diagram of the three phase HVDC transformer including neutral points

circuits [10-13]. The current in the input circuits will be unknown, but saturation of the current waveform occurs because of the nonlinear characteristic of the magnetic core. For a three-phase transformer connected in wye-wye, as shown in Figure 3.12, a computer simulation model with a neutral *NN* and external circuits for both primary and secondary windings is obtained using the HBFEM technique.

According to the Galerkin procedure, system matrix equations of HBFEM for the HVDC three phase Y/Y connection transformer can be obtained through Faraday’s and Kirchhoff’s laws for the external circuit, as described below [15]:

where [Gk] is obtained from a single element - that is, *[G ^{e}]* = Д

^{е}/3.

*[Z*and

_{in}], [Z_{out}]*S*are external circuit impedances and cross-sectional areas of windings respectively, [T] is unit matrix,

_{in}, S_{out}*V*is the voltage of primary or secondary neutral points when it is not grounded, [C

_{NN}_{in}] and

*[C*are geometric coefficients related to transformer windings, and current density

_{out}]*J*can be presented as:

*[H]* is the single-element matrix, and the detailed definitions can be expressed from:

where matrix *D* and *N* are:

In the DC-biased case, the *D* matrix is different from a normal *D* matrix, due to the DC component *v _{0}* being involved in magnetizing, as shown in equation (3-98):

The *N* matrix is also different from the normal *N* matrix; an additional harmonic 0 is considered in the *N* matrix as presented below:

The system matrix equations of the HBFEM model in a compact form can be obtained as:

where the input and output voltages can be defined as: and