# HBFEM and Its Future Applications

## HBFEM Model of Three-Phase Power Transformer

### Three-Phase Transformer

A three-phase transformer-based magnetic frequency-tripling devices was introduced in Chapter 4.3, where the production of triple-frequency output from a three-phase fundamental frequency source was based on nonlinear magnetic saturation characteristics, as illustrated in Figure 6.2. However, a three-phase transformer used in electric power systems, including HVDC power system with different winding connections, including Y/Y, Y/Д, Д/Y and Д/Д, is also a nonlinear magnetic system [1]. In particular, the transformer is under the DC or quasi-DC biased condition, a significant nonlinear phenomenon is accrued, and a larger amount of harmonics are generated when the transformer is saturated.

Figure 6.1 shows the Д/Y and Д/Д winding connection used in an electric power system. The Д/Y connection is often used in a 10-12 kV/200-400 V distribution network, which is now widely used in solar PV and winding power-based renewable energy systems or a distributed energy resource (DER) low-voltage (LV) network.

By using the HBFEM approach, magnetic nonlinear characteristics, hysteresis losses, eddy current losses, and magnetic flux distribution for each harmonic component can be calculated and presented. HBFEM can provide magnetic flux distribution and eddy current losses at each harmonic [2].

*Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems,* First Edition. Junwei Lu, Xiaojun Zhao and Sotoshi Yamada.

© 2016 John Wiley & Sons Singapore Pte. Ltd. Published 2016 by John Wiley & Sons Singapore Pte. Ltd. Companion website: www.wiley.com/go/lu/HBFEM

Figure 6.1 A circuit diagram of a three-phase transformer. (a) Д/Y; (b) Д/Д

Figure 6.2 B-H curve with hysteresis