496 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 Finite Element Based Transformer Operational Model for Dynamic Simulations O. A. Mohammed 1, Z. Liu 1, S. Liu 1, N. Y. Abed 1, and L. J. Petersen 2 1 Florida International University, U.S.A. 2 Office of Naval Research, U.S.A. Abstract The transformer model proposed in this paper considers the effects of the excitation levels as well as the periodic fluctuations of ac excitation on the winding self and mutual inductances during dynamic operation. The inductance profile is obtained from sequential FE solutions covering a complete ac cycle at various excitation levels. The values are then used by table-look-up technique. The technical details, the creation of the inductance tables as well as the Simulink implementation, are explained. Simulation results show that the established model is capable of restoring the nonlinear magnetization phenomena of transformer iron core. The significance of this model is due to its accuracy and its applicability for dynamic simulation of interconnected components in a power system. Introduction Accurate electromagnetic transient studies, such as, harmonic load flow require accurate modeling of network elements and their components. The modeling of iron-core transformer plays an important role in the dynamic simulation of power system transients such as inrush currents, short circuits, and fault conditions. The key point of iron core transformer modeling is the representation of nonlinear magnetization. Two commonly used methods are the piece-wise linear curve and the simple saturated reluctance function [1-3]. These approaches consider the effects of average excitation level on the flux/inductance but ignore the fluctuating effects of the ac excitation itself. For cases requiring high-precision modeling, this is not adequate. Reference [4] developed an FE based method for determining the saturated transformer inductances utilizing energy perturbation. Reference [5] studied the transformer inductance variations with respect to the average excitation level and the periodic fluctuations of ac excitation. The 2D profiles are used to describe each inductance. Making use of such an inductance definition, we proposed our new transformer model. As an example, a 187.8kW, 288/232V three-phase power transformer is studied. The transformer equation, inductance calculation and 2D inductance table establishment, Simulink implementation in addition to simulation results are presented. Transformer s Equation and Inductances Calculation A. Basic equation The voltage and flux linkage equations of the three-phase transformer are as follows: [ ] [ ][ ] uabc R1 0 iabc + d [ ] ψabc (1) u ABC 0 R 2 BC dt ψ ABC [ ] [ ] [ ] ψabc L1 M 12 iabc (2) ψ ABC L 2 BC M 21 Where, u abc, bc, R 1, L 1, and, ψ abc are the voltage, current, resistance, self inductance, and the flux linkage of the primary winding. u ABC, BC, R 2, L 2, and ψ ABC are the corresponding parameters of the secondary winding. M 12 and M 21 are the mutual inductances between the primary and secondary windings. The inductancesl 1, L 2, M 12, and M 21 are considered as magnetization status dependent so as to accurately represent the nonlinear magnetization property of iron core. They are determined in terms of the maximum value and the phase angle of ac excitation during a complete electrical cycle. B. Inductance calculation and inductance table Inductances are evaluated using the energy perturbation mehtod [4]. While performing the energy perturbation algorithm, the magnetizing currents of the primary winding are assigned to bc ; zero currents are assigned
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 497 Figure 1: waveform of rated magnetizing current Figure 2: inductance profile of Laa to BC. The energy of the transformer is calculated based on the nonlinear FE magnetic field analysis of the transformer. The magnetizing currents of the primary winding is determined through circuit-fe diect coupling, while the primary winding is fed with sinusodal voltage source and the secondary winding is open-circuited. Fig. 1 shows the obtained rated excitation current waveform of the 187.8kW, 288/232V three-phase power transformer. The excitation level is represented by the magnitude of ψ m. The determination of ψ m is as follows: ψ m (ψα 2 + ψ2 β ) (3) Where, ψ α (u α R 1 i α )dt, ψ β (u β R 1 i β )dt. u α, u β, i α, and i β are obtained by transferring the sinusoidal voltage and magnetizing current of the primary winding from a b c coordinate system to α β coordinate system. While building the 2D inductance table, the excitation level is adjusted by changing the magnitude of the sinusoidal voltage source. The phase angle of the ac excitation during a complete ac cycle is identified by θ. It is calculated uding the formulation below: θ tg 1 (ψ β /ψ α ) (4) Table 1: 2-dimentional inductance table θ 1 0 2 0 3 0 358 0 359 0 360 0 ψ m 25% 0.0426 0.0425 0.0423 0.0427 0.0428 0.0431 50% 0.0430 0.0428 0.0426 0.0423 0.0425 0.0426 100% 0.0424 0.0420 0.0417 0.0417 0.0421 0.0425 150% 0.0412 0.0410 0.0405 0.0416 0.0417 0.0419 Using the primary winding self inductance L aa as an example, Table I gives the structure of the 2D inductance table and Fig.2 shows the inductance profile. Simulink Implementation and Simulation Results In our previous work, two procedures were proposed to build the machine model in Simulink; equationbased and circuit component-based [6-7]. For the implementation of the transformer equation (1), the circuit component-based model is adopted to allow arbitrary connection (Wye or Delta) of the transformer three phase winding. In the circuit component-based model in reference [7], an adjustable inductance component was developed to represent the rotor position dependent self inductances. Here, a new procedure is proposed to represent the magnetization status dependent self inductances. A constant inductance term is separated from each varying self inductance, as shown in Fig.3. The constant term is used to apply the initial condition of inductance. The varying term is used to reflect the inductance variation with the iron core magnetization status. Equation (5) gives the flux equation rewritten in terms of the separated inductances. Where L aa, L bb, are the constant inductance terms.
498 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 Figure 3: inductance separation Figure 4: circuit diagram of phase a winding ψ a ψ b ψ c ψ A ψ B ψ C L aa L ab L ac M aa M ab M ac L ba L bb L bc M ba M bb M bc L ca L cb L cc M ca M cb M cc M Aa M Ab M Ac L AA L AB L AC M Ba M Bb M Bc L BA L BB L BC M Ca M cb M cc L CA L CB L CC L aa 0 0 0 0 0 0 L bb 0 0 0 0 0 0 L cc 0 0 0 0 0 0 L AA 0 0 0 0 0 0 L BB 0 0 0 0 0 0 L CC L aa L ab L ac M aa M ab M ac L ba L bb L bc M ba M bb M bc + L ca L cb L cc M ca M cb M cc M Aa M Ab M Ac L AA L AB L AC M Ba M Bb M Bc L BA L BB L BC M Ca M cb M cc L CA L CB L CC (5) For simplicity, the back EMF of phase a is given below as an example: e a dψ a dt L d aa dt + e a (6) Figure 5: Block diagram of the transformer model in Simulink
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 499 e a d dt (L aa + L ab + L ac + M aa + M ab + M ac ) (7) The circuit diagram of phase a winding is shown in Fig. 4. The controlled voltage source is used to represent the back EMF term e a. Fig.5 is the circuit diagram of the transformer model. Subsystems 1 and 2 are used to calculate the magnitude of ψ m and the phase angle θ. According to ψ m and θ, the inductances are picked up from the 2D tables stored in blocks L1, M12, L2, and M21. The flux linkage and back EMF of the primary and secondary windings are calculated using equations (1) and (5). Table 2 L aa 43.9 L ab -22.1 L ac -21.4 M aa 35.4 M ab -17.8 M ac -17.3 L ba -22.1 L bb 44.4 L bc -22.1 M ba -17.8 M bb 35.9 M bc -17.8 L ca -21.4 L cb -22.1 L cc 43.8 M ca -17.3 M cb -17.8 M cc 35.4 M Aa 35.4 M Ab -17.8 M Ac -17.3 L AA 28.6 L AB -14.4 L AC -13.9 M Ba -17.8 M Bb 35.8 M Bc -17.8 L BA -14.4 L BB 29.0 L BC -14.4 M Ca -17.3 M Cb -17.8 M Cc 35.4 L CA -13.9 L CB -14.4 L CC 28.6 Current (A) Current (A) (a) Time (s) (b) Time (s) Figure 6: Magnetizing current waveform obtained by, (a) using inductances in Table 1, (b) using inductances in Table 2 For comparison purpose, the mean values of the transformer winding inductances are calculated also, which are given in Table 2. Then, the no load experiment is performed using the inductances in Table 1 and Table 2 respectively. Fig.6 shows the magnetizing current waveform obtained from simulation. Comparison of Fig.6(a) and Fig.6(b) indicates that the proposed transformer model restores the nonlinear magnetization phenomenon of the iron core. Comparison of Fig.6(a) with Fig. 1 shows that the proposed FE based transformer model can be considered as accurate as the full FE model. In addition, the FE based transformer model supports very fast simulation speed, while the full FE model is computational cumbersome. Conclusion An accurate transformer model is proposed for dynamic simulation purposes. It uses the magnetization status dependent inductances to restore the nonlinear magnetization behavior of the transformer iron core. The inductance variations due to the excitation level and the periodic fluctuations of ac excitation are considered, which are obtained from sequential FE solutions. The definition of 2D inductance table is given and its implementation in Simulink is studied. Verification examples show the correctness and validity of the developed transformer model. Compared with the conventional transformer models, the proposed model provides an accurate description of the iron core magnetization behavior and its applicability to dynamic simulations. REFERENCES 1. leon, F. de and A. Semlyen, Complete Transformer Model for Electromagnetic Transients, IEEE Trans. Power Delivery, Vol. 9, No. 1, 231-239, 1994.
500 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 2. Chen, X. and S. Venkata, A Three-phase Three-winding Core-type Transformer Model for Low-frequency Transient Studies, IEEE Trans. on Power Delivery, Vol. 12, No. 2, 775-782, 1997. 3. Pedra, J., F. Corcoles, L. Sainz and R. Lopez, Harmonic Nonlinear Transformer Modeling, IEEE Trans. on Power Delivery, Vol. 19, No. 2, 884-890, 2004. 4. Mohammed, O. A. and N. A. Demerdash, A 3 D Finite Element Perturbational Method for Determining Saturated Values of Transformer Winding Inductances Including Experimental Verification, IEEE Trans. on Magnetics, Vol. 21, No. 5, 1877-1879, 1985. 5. Mirafzal, B. and N. A. Demerdash, Effects of Inductance Nonlinearities in a Transformer-rectifier DC Motor Drive System on the AC Side Harmonic Distortion Using a Time-stepping Coupled Finite Element-circuit Technique, proceedings of Electric Machines and Drives Conference, IEMDC 03, Vol. 3, 1755-1759. 6. Mohammed, O. A., S. Liu and Z. Liu, Physical Modeling of PM Synchronous Motors for Integrated Coupling with Machine Drives, IEEE Trans. on Magnetics, 2005. 7. Mohammed, O. A., S. Liu and Z. Liu, A Phase Variable Model of Brushless DC motor Based on Physical FE Model and Its Coupling with External Circuits, IEEE Trans. on Magnetics, 2005.