# QIC Modeling Using HBFEM Model

Figure 6.22 shows a DC-biased transformer model in GIC or HVDC events, where the internal equivalent circuit of the transformer (e.g., leakage inductance and winding resistance) and external equivalent circuits (e.g., transmission line impedance and ground resistance) should be included in the HBFEM simulation model. The DC bias transformer model can be excited as voltage source-driven or current source-driven, depending on the availability of the calculated or observed excitation data. If the induced quasi-DC voltage is obtained, then the excitation voltage source applied to the transmission line will include both DC voltage and sinusoidal voltage, with fundamental frequency (50/60 Hz). Based on the voltage source-driven HBFEM model discussed in Chapter 3.5, the unknown current magnitude can be calculated through this HBFEM model.

Transformer (A) is connected to transformer (B) through a long-distance transmission line and excited by the voltage source, including AC from the generator and DC induced by GIC. The third harmonic magnetizing current, caused by the non-linear magnetic core B-H characteristic during GIC or HVDC events, remains trapped inside the

Figure 6.22 GIC or HVDC DC-biased transformer, including internal equivalent circuit (leakage inductance and winding resistance) and external circuits (transmission line impedance)

Figure 6.23 (a) B-H curve with hysteresis characteristics and DC-biased condition; (b) excitation

current during DC bias

secondary Д winding. Third harmonic currents are zero-sequence currents, which cannot enter or leave a Д connection, but can flow within the Д.

In the DC-biased case [15], the B-H curve of the transformer with hysteresis characteristics is illustrated as Figure 6.23. The mathematical expression of H(B), as presented in Chapter 3, can be obtained from a B-H curve data table.

If the secondary windings are in the open circuit condition, the HBFEM numerical model (including internal and external impedances) will be derived as follows:

where *[H]* (= [5] + [M]) is the system coefficient matrix, [*G _{k}]* is the geometric matrix of the transformer windings,

*[Z*and

_{k},_{in}]*S*are external circuit impedances and crosssectional areas of windings, respectively, [T] is the unit matrix and

_{kin}*V*is the voltage at the neutral point which can be obtained through DC equivalent resistance Z

_{Np}_{0}=

*V*(DC equivalent resistance obtained from DC equivalent circuits including ground resistance, transmission line DC resistance and transformer winding DC resistance), and

_{Np}/J_{0}*[C*refers to geometric coefficients related to transformer windings. With the given input or excitation voltage sources {

_{kin}*V*

_{k},_{in}}:

where *V*_{0} is the quasi-DC voltage (DC component) caused by GIC.

The unknown magnetic vector potential {A_{n}} and the unknown current density {J_{kn}} in the system equation (6-72) can be obtained as follows:

where *n* is the harmonic number and *k* is the transformer winding number or phase number.

Using the HBFEM approach, the single element matrix [*H*] and the detailed definitions of each matrix can be expressed from:

where matrices *D* and *N* (including the DC component) are expressed as follows:
where the magnetic reluctance can be obtained from equation (6-78):

The HBFEM system matrix equations can be obtained as:
where *{V _{Np}* } = {3x J

_{0}xZ

_{0}}.

By substituting the following input voltages and current: the HBFEM system matrix equation in a compact form can be expressed as:

With given voltage excitation sources *V _{k}* and, by solving the above HBFEM matrix equations, the magnetic flux density B and current

*i*(including all harmonic components) can be obtained.