- Email: [email protected]
Nonlinear model of a distribution transformer appropriate for evaluating the effects of unbalanced loads
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015 www.elsevier.com/locate/jmmm
Nonlinear model of a distribution transformer appropriate for evaluating the effects of unbalanced loads Matej Toman, Gorazd Sˇtumberger, Bojan Sˇtumberger, Drago Dolinar Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova ulica 17, 2000 Maribor, Slovenia Available online 18 April 2008
Abstract Power packages for calculation of power system transients are often used when studying and designing electromagnetic power systems. An accurate model of a distribution transformer is needed in order to obtain realistic values from these calculations. This transformer model must be derived in such a way that it is applicable when calculating those operating conditions appearing in practice. Operation conditions where transformers are loaded with nonlinear and unbalanced loads are especially challenging. The purpose of this work is to derive a three-phase transformer model that is appropriate for evaluating the effects of nonlinear and unbalanced loads. A lumped parameter model instead of a finite element (FE) model is considered in order to ensure that the model can be used in power packages for the calculation of power system transients. The transformer model is obtained by coupling electric and magnetic equivalent circuits. The magnetic equivalent circuit contains only three nonlinear reluctances, which represent nonlinear behaviour of the transformer. They are calculated by the inverse Jiles–Atherton (J–A) hysteresis model, while parameters of hysteresis are identified using differential evolution (DE). This considerably improves the accuracy of the derived transformer model. Although the obtained transformer model is simple, the simulation results show good agreement between measured and calculated results. r 2008 Elsevier B.V. All rights reserved. PACS: 02.60.Cb; 75.60.d; 84.70.+p; 91.25.ga Keywords: Transformer model; Inverse Jiles–Atherton hysteresis model; Higher order harmonics
1. Introduction Distribution transformers are essential components over a wide variety of power system applications. In these applications, transformers are used to transfer electric energy from one electrical system to another. When energy is transferred between electrical systems, power losses arise in the transformer. These losses can increase due to nonlinear loads. The majority of today’s nonlinear loads are power electronic devices which, when operating, produce higher-order harmonics in voltages and currents. They are usually connected to the transformer in such a way that they form unbalanced loads. In electrical engineering, digital simulations are an essential part at the design stage of power systems. In these simulations, an adequate digital model of a distribuCorresponding author. Tel.: +386 2 220 7073; fax: +386 2 220 7272.
E-mail address: [email protected] (M. Toman). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.04.089
tion transformer is a matter of great importance. It is also important that the transformer model is appropriate for use in simulations of those abnormal operating conditions, which may arise due to nonlinear and unbalanced loads. The purpose of this work is to derive a three-phase transformer model that can be used in power packages for the calculation of power system transients when nonlinear and unbalanced loads are present. Two different approaches can be applied at the stage of transformer modelling. The first one uses a lumped parameter model, while the second one uses finite element (FE) model. When a transformer is modelled by the FE, the magnetically nonlinear behaviour of the transformer iron core is normally accounted for by the magnetizing curve [1]. This method neglects the hysteresis phenomenon in the iron core. This is a more or less standard solution in commercial FE programmes, since inclusion of hysteresis into FE programmes is quite a demanding task. When a FE transformer model with magnetizing curve is used in
ARTICLE IN PRESS M. Toman et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015
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calculations, the differences between measured and calculated results are noticeable. The differences increase with increasing differences between the slopes of the magnetizing curve, and hysteresis at the given operating point. Since the hysteresis is not included in the transformer model, calculations give us no information about hysteresis losses that may arise. Therefore, in the opinion of the authors, hysteresis should be included in any transformer model used for operational analysis under nonlinear and unbalanced loads. Furthermore, a FE transformer model is too complex and requires too much computational effort to be appropriate for calculating power system transients. A magnetically nonlinear lumped parameter transformer dynamic model is proposed in order to investigate the behaviour of the transformer under unbalanced and nonlinear loads. This model is obtained by coupling electric and magnetic equivalent circuits. The magnetic equivalent circuit consists of lumped parameter elements where each element corresponds to a specific area of the iron core. For each area, any geometrical and magnetical nonlinear behaviour of the iron core is accounted for by a corresponding element in the magnetic equivalent circuit. In order to achieve better agreement between the measured and calculated results, the magnetically nonlinear behaviour of the iron core is taken into account by the inverse Jiles–Atherton (J–A) hysteresis model. The parameters of the inverse J–A hysteresis model are determined using a stochastic search algorithm called differential evolution (DE). The obtained transformer model is confirmed though a comparison of measured and calculated voltages and currents. 2. Nonlinear dynamic transformer model of a three-phase transformer The transformer dynamic model is derived for a threephase three-limb transformer whose primary and secondary windings are Y-connected with a neutral point and placed around three limbs, as shown in Fig. 1. The voltage equations of the presented transformer are given in matrix form by Eq. (1): u ¼ Ri þ Ls
d d i þ N U. dt dt
(1)
In Eq. (1), u ¼ ½ upa upb upc usa usb usc T is the voltage vector of the primary and secondary voltages, i ¼ ½ ipa ipb ipc isa isb isc T is the current vector of the primary and secondary currents, R ¼ diag ½ Rp Rp Rp Rs Rs Rs is the diagonal resistance matrix of the primary and secondary windings, Ls ¼ diag ½ Lsp Lsp Lsp Lss Lss Lss is the diagonal leakage inductance matrix of the primary and secondary windings, U ¼ ½ fa fb fc T is the flux linkage vector of the magnetic fluxes in the three limbs and N ¼ ½ N p I N s I T is the diagonal matrix of the primary and secondary number of turns, where IAR3 is the identity matrix. The magnetic flux linkages are nonlinearly dependent on magnetomotive force (MMF). The flux paths close through the transformer’s iron core with different lengths. The magnetic conditions of the transformer are expressed by Eq. (2), based on the magnetic equivalent circuit shown in Fig. 2: ipa N p þ isa N s ipb N p isb N s ¼ Rma ðfa Þfa Rmb ðfb Þfb , ipb N p þ isb N s ipc N p isc N s ¼ Rmb ðfb Þfb Rmc ðfc Þfc , fa þ fb þ fc ¼ 0,
Rma(fa), Rmb(fb) and Rmc(fc) are the reluctances of individual magnetic circuit branches and represent nonlinear behaviour of the transformer. Eq. (2) can be rewritten in matrix form Eq. (3), where fc is expressed as a linear combination of fa and fb: " # fa Nm i ¼ Rm ; fb " # 0 N s N s 0 N p N p , Nm ¼ 0 N p N p 0 N s N s " # Rmb ðfb Þ Rma ðfa Þ . (3) Rm ¼ Rmc ðfc Þ Rmb ðfb Þ þ Rmc ðfc Þ Flux linkages in individual magnetic circuit branches are expressed from Eq. (3) and inserted into Eq. (1), where fc is also expressed by fa and fb. The obtained matrix equations are given by the time derivative of the current vector in Eq. (4): d 1 i ¼ ðLs þ NR1 m Nm Þ ðu RiÞ. dt
Fig. 1. Three-phase three-limb transformer schematic.
(2)
Fig. 2. Magnetic equivalent circuit of the transformer.
(4)
ARTICLE IN PRESS M. Toman et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015
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The nonlinear behaviour of the transformer is taken into account using variable reluctances Rma(fa), Rmb(fb) and Rmc(fc) in Eq. (4), which are dependent on the magnetic flux. Reluctances are calculated using an inverse J–A hysteresis model Eqs. (5)–(8), which is presented in Refs. [2,3]:
such a way as to achieve best possible agreement between calculated and measured B–H hysteresis. The B–H hysteresis of the used three-limb transformer is measured, as explained in Ref. [5]. The three-phase nonlinear transformer model is given by Eqs. (1), (4) and (5)–(10). They are used to develop a transformer model in the Matlab/Simulink programme package.
dM j ð1 cÞðdM irr =dBe Þ þ ðc=m0 ÞðdM an =dH e Þ , ¼ dBj 1 þ cð1 aÞðdM an =dH e Þ þ m0 ð1 cÞð1 aÞðdM irr =dBe Þ
4. Results
3. Inverse Jiles–Atherton hysteresis model
(5) dM an Ms ¼ dH e a
"
2 # a 2 He , 1 coth þ He a
dM irr ðM an M irr Þg , ¼ m0 kd dBe ( g¼
1;
ðM an M irr Þ dBe X0;
0;
ðM an M irr Þ dBe o0;
(6)
d¼
8 < þ1; : 1;
dBj dt dBj dt
X0; o0; (7)
dM j dM j dBj ¼ ; dt dBj dt
j ¼ a; b; c,
(8)
where M, Mirr and Man are the magnetization, irreversible magnetization, and the anhysteretic magnetization. m0 is the permeability of vacuum, Ms is the saturation magnetization, a is the main field parameter, a presents the anhysteretic behaviour, c is the parameter, which is proportional to the hysteresis loop width and domain flexing constant and k is the pinning parameter. In the equations of the inverse J–A hysteresis model, index j is used to represent three magnetic circuit branches, while He is denoted as He=H+aM and Be as Be=m0He. The reluctances Rmj of individual magnetic branches j, are calculated from known transformer geometry and slopes dMj/dBj, which are determined from the inverse J–A hysteresis model, as shown in Eq. (9): dH j l j 1 dM j l j ¼ , (9) Rmj ¼ m0 dBj A dBj A where A is the area of the transformer iron core and lj are the lengths of the mean flux paths. Magnetic fluxes fj through reluctances Rmj are determined at each integration step from Eq. (1). They are needed to calculate magnetic flux densities Bj (Eq. (10)), which are used in the inverse J–A hysteresis model to determine reluctances in the next integration step. Five parameters define the shape of B–H hysteresis calculated by the inverse J–A hysteresis model. These parameters are Ms, a, a, c and k. They are determined using DE according to Ref. [4] in: dBj 1 dfj ¼ ; A dt dt
j ¼ a; b; c,
(10)
The obtained transformer model was verified by comparing the calculated voltages and currents with the measured ones. Measurements were performed on a 0.4 kV, 3.5 kV A laboratory three-phase three-limb transformer, with data shown in Table 1. During all measurements, the primary and secondary windings of the transformer as well as balanced and unbalanced loads were Y-connected. In each test, the neutral points of the secondary winding and the load were connected. First, the transformer voltages and currents were measured when the transformer was supplied from the grid and loaded with balanced resistive Table 1 Transformer data Parameter
Value
Rp (O) Rs (O) Np Ns Lsp (H) Lss (H) la, lb, lc (m) A (m2)
0.914 0.898 278 278 0.0265 0.0250 480 103, 240 103, 480 103 5.52 103
Fig. 3. Voltages applied to the laboratory power transformer.
Fig. 4. Measured and calculated secondary voltages at balanced load.
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load. The measured primary voltages upa meas, upb meas and upc meas, shown in Fig. 3, were then used in order to supply the nonlinear transformer model. The model’s calculated secondary voltages usa sim, usb sim and usc sim are shown in Fig. 4. For comparison, they are presented together with the measured secondary voltages usa meas, usb meas and usc meas. The measured primary currents ipa meas, ipb meas, ipc meas and calculated primary currents ipa sim,
Fig. 5. Measured and calculated primary currents at balanced load.
Fig. 6. Measured ipa meas, ipb mea, ipc meas and calculated ipa ipc sim primary currents at unbalanced and nonlinear load.
sim,
ipb
sim,
ipb sim, ipc sim are also presented. They are shown in Fig. 5. The measured and calculated secondary currents are omitted in this paper, since they have the same waveforms as voltages shown in Fig. 4. Next, nonlinear and unbalanced load is considered. In order to form nonlinear and unbalanced load, a different resistive loads were used in phases a, b and c and a diode rectifier was included into phase a. For the given case, primary and secondary currents were measured. They are shown in Figs. 6 and 7. In order to perform simulations, a dynamic model of a diode rectifier was derived and included in the transformer model. Results of the calculated primary and secondary currents are added to Figs. 6 and 7. The presented results show very good agreement between the measured and calculated currents and voltages. This means that the calculated magnetic results can give us accurate insight into the nonlinear magnetic behaviour of each magnetic branch. Fig. 8 shows the hysteresis of an individual magnetic circuit branch over a time span covering two periods for the case of unbalanced and nonlinear load. All simulations were performed on a computer with Intel Pentium M 1.86 GHz processor and 1 GB of RAM. During simulations in the Matlab/Simulink programme package, a fixed step size of 10 ms was used for the case of balanced resistive load and 1 ms for the case of unbalanced and nonlinear load. The computational times were 12 and 100 s for the cases of balanced resistive load and unbalanced and nonlinear load, respectively. 5. Conclusion
Fig. 7. Measured isa meas, isb mea, isc meas and calculated isa sim, isb sim, isc sim secondary currents at unbalanced and nonlinear load.
This paper describes the development of a three-phase three-limb transformer model, which is appropriate for analysis of transformer operation under nonlinear and unbalanced loads. Just like FE models, the developed model can give us insight into a specific iron core area if the iron core is discretizated in more detail. Moreover, it can be easily incorporated into power packages for calculating power system transients. This gives the derived model an advantage, since FE models are too complex and timeconsuming for power system transient calculations.
Fig. 8. Calculated hysteresis in magnetic circuit branches a, b and c.
ARTICLE IN PRESS M. Toman et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015
References [1] O.A. Mohammed, N.Y. Abed, S. Liu, IEEE Trans. Magn. 42 (2006) 976. [2] J.P.A. Bastos, N. Sadowski, Electromagnetic Modeling by Finite Element Methods, CRC, New York, 2003.
e1015
[3] J.V. Leite, N. Sadowski, P. Kuo-Peng, N.J. Batistela, J.P.A. Bastos, A.A. de Espı´ ndola, IEEE Trans. Magn. 40 (2004) 1769. [4] M. Toman, G. Sˇtumberger, D. Dolinar, Compumag 16 (2007) 797. [5] M. Hadzˇiselimovic´, D. Miljavec, I. Zagradisˇ nik, ICEM 16 (2004).
Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015 www.elsevier.com/locate/jmmm
Nonlinear model of a distribution transformer appropriate for evaluating the effects of unbalanced loads Matej Toman, Gorazd Sˇtumberger, Bojan Sˇtumberger, Drago Dolinar Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova ulica 17, 2000 Maribor, Slovenia Available online 18 April 2008
Abstract Power packages for calculation of power system transients are often used when studying and designing electromagnetic power systems. An accurate model of a distribution transformer is needed in order to obtain realistic values from these calculations. This transformer model must be derived in such a way that it is applicable when calculating those operating conditions appearing in practice. Operation conditions where transformers are loaded with nonlinear and unbalanced loads are especially challenging. The purpose of this work is to derive a three-phase transformer model that is appropriate for evaluating the effects of nonlinear and unbalanced loads. A lumped parameter model instead of a finite element (FE) model is considered in order to ensure that the model can be used in power packages for the calculation of power system transients. The transformer model is obtained by coupling electric and magnetic equivalent circuits. The magnetic equivalent circuit contains only three nonlinear reluctances, which represent nonlinear behaviour of the transformer. They are calculated by the inverse Jiles–Atherton (J–A) hysteresis model, while parameters of hysteresis are identified using differential evolution (DE). This considerably improves the accuracy of the derived transformer model. Although the obtained transformer model is simple, the simulation results show good agreement between measured and calculated results. r 2008 Elsevier B.V. All rights reserved. PACS: 02.60.Cb; 75.60.d; 84.70.+p; 91.25.ga Keywords: Transformer model; Inverse Jiles–Atherton hysteresis model; Higher order harmonics
1. Introduction Distribution transformers are essential components over a wide variety of power system applications. In these applications, transformers are used to transfer electric energy from one electrical system to another. When energy is transferred between electrical systems, power losses arise in the transformer. These losses can increase due to nonlinear loads. The majority of today’s nonlinear loads are power electronic devices which, when operating, produce higher-order harmonics in voltages and currents. They are usually connected to the transformer in such a way that they form unbalanced loads. In electrical engineering, digital simulations are an essential part at the design stage of power systems. In these simulations, an adequate digital model of a distribuCorresponding author. Tel.: +386 2 220 7073; fax: +386 2 220 7272.
E-mail address: [email protected] (M. Toman). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.04.089
tion transformer is a matter of great importance. It is also important that the transformer model is appropriate for use in simulations of those abnormal operating conditions, which may arise due to nonlinear and unbalanced loads. The purpose of this work is to derive a three-phase transformer model that can be used in power packages for the calculation of power system transients when nonlinear and unbalanced loads are present. Two different approaches can be applied at the stage of transformer modelling. The first one uses a lumped parameter model, while the second one uses finite element (FE) model. When a transformer is modelled by the FE, the magnetically nonlinear behaviour of the transformer iron core is normally accounted for by the magnetizing curve [1]. This method neglects the hysteresis phenomenon in the iron core. This is a more or less standard solution in commercial FE programmes, since inclusion of hysteresis into FE programmes is quite a demanding task. When a FE transformer model with magnetizing curve is used in
ARTICLE IN PRESS M. Toman et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015
e1012
calculations, the differences between measured and calculated results are noticeable. The differences increase with increasing differences between the slopes of the magnetizing curve, and hysteresis at the given operating point. Since the hysteresis is not included in the transformer model, calculations give us no information about hysteresis losses that may arise. Therefore, in the opinion of the authors, hysteresis should be included in any transformer model used for operational analysis under nonlinear and unbalanced loads. Furthermore, a FE transformer model is too complex and requires too much computational effort to be appropriate for calculating power system transients. A magnetically nonlinear lumped parameter transformer dynamic model is proposed in order to investigate the behaviour of the transformer under unbalanced and nonlinear loads. This model is obtained by coupling electric and magnetic equivalent circuits. The magnetic equivalent circuit consists of lumped parameter elements where each element corresponds to a specific area of the iron core. For each area, any geometrical and magnetical nonlinear behaviour of the iron core is accounted for by a corresponding element in the magnetic equivalent circuit. In order to achieve better agreement between the measured and calculated results, the magnetically nonlinear behaviour of the iron core is taken into account by the inverse Jiles–Atherton (J–A) hysteresis model. The parameters of the inverse J–A hysteresis model are determined using a stochastic search algorithm called differential evolution (DE). The obtained transformer model is confirmed though a comparison of measured and calculated voltages and currents. 2. Nonlinear dynamic transformer model of a three-phase transformer The transformer dynamic model is derived for a threephase three-limb transformer whose primary and secondary windings are Y-connected with a neutral point and placed around three limbs, as shown in Fig. 1. The voltage equations of the presented transformer are given in matrix form by Eq. (1): u ¼ Ri þ Ls
d d i þ N U. dt dt
(1)
In Eq. (1), u ¼ ½ upa upb upc usa usb usc T is the voltage vector of the primary and secondary voltages, i ¼ ½ ipa ipb ipc isa isb isc T is the current vector of the primary and secondary currents, R ¼ diag ½ Rp Rp Rp Rs Rs Rs is the diagonal resistance matrix of the primary and secondary windings, Ls ¼ diag ½ Lsp Lsp Lsp Lss Lss Lss is the diagonal leakage inductance matrix of the primary and secondary windings, U ¼ ½ fa fb fc T is the flux linkage vector of the magnetic fluxes in the three limbs and N ¼ ½ N p I N s I T is the diagonal matrix of the primary and secondary number of turns, where IAR3 is the identity matrix. The magnetic flux linkages are nonlinearly dependent on magnetomotive force (MMF). The flux paths close through the transformer’s iron core with different lengths. The magnetic conditions of the transformer are expressed by Eq. (2), based on the magnetic equivalent circuit shown in Fig. 2: ipa N p þ isa N s ipb N p isb N s ¼ Rma ðfa Þfa Rmb ðfb Þfb , ipb N p þ isb N s ipc N p isc N s ¼ Rmb ðfb Þfb Rmc ðfc Þfc , fa þ fb þ fc ¼ 0,
Rma(fa), Rmb(fb) and Rmc(fc) are the reluctances of individual magnetic circuit branches and represent nonlinear behaviour of the transformer. Eq. (2) can be rewritten in matrix form Eq. (3), where fc is expressed as a linear combination of fa and fb: " # fa Nm i ¼ Rm ; fb " # 0 N s N s 0 N p N p , Nm ¼ 0 N p N p 0 N s N s " # Rmb ðfb Þ Rma ðfa Þ . (3) Rm ¼ Rmc ðfc Þ Rmb ðfb Þ þ Rmc ðfc Þ Flux linkages in individual magnetic circuit branches are expressed from Eq. (3) and inserted into Eq. (1), where fc is also expressed by fa and fb. The obtained matrix equations are given by the time derivative of the current vector in Eq. (4): d 1 i ¼ ðLs þ NR1 m Nm Þ ðu RiÞ. dt
Fig. 1. Three-phase three-limb transformer schematic.
(2)
Fig. 2. Magnetic equivalent circuit of the transformer.
(4)
ARTICLE IN PRESS M. Toman et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015
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The nonlinear behaviour of the transformer is taken into account using variable reluctances Rma(fa), Rmb(fb) and Rmc(fc) in Eq. (4), which are dependent on the magnetic flux. Reluctances are calculated using an inverse J–A hysteresis model Eqs. (5)–(8), which is presented in Refs. [2,3]:
such a way as to achieve best possible agreement between calculated and measured B–H hysteresis. The B–H hysteresis of the used three-limb transformer is measured, as explained in Ref. [5]. The three-phase nonlinear transformer model is given by Eqs. (1), (4) and (5)–(10). They are used to develop a transformer model in the Matlab/Simulink programme package.
dM j ð1 cÞðdM irr =dBe Þ þ ðc=m0 ÞðdM an =dH e Þ , ¼ dBj 1 þ cð1 aÞðdM an =dH e Þ þ m0 ð1 cÞð1 aÞðdM irr =dBe Þ
4. Results
3. Inverse Jiles–Atherton hysteresis model
(5) dM an Ms ¼ dH e a
"
2 # a 2 He , 1 coth þ He a
dM irr ðM an M irr Þg , ¼ m0 kd dBe ( g¼
1;
ðM an M irr Þ dBe X0;
0;
ðM an M irr Þ dBe o0;
(6)
d¼
8 < þ1; : 1;
dBj dt dBj dt
X0; o0; (7)
dM j dM j dBj ¼ ; dt dBj dt
j ¼ a; b; c,
(8)
where M, Mirr and Man are the magnetization, irreversible magnetization, and the anhysteretic magnetization. m0 is the permeability of vacuum, Ms is the saturation magnetization, a is the main field parameter, a presents the anhysteretic behaviour, c is the parameter, which is proportional to the hysteresis loop width and domain flexing constant and k is the pinning parameter. In the equations of the inverse J–A hysteresis model, index j is used to represent three magnetic circuit branches, while He is denoted as He=H+aM and Be as Be=m0He. The reluctances Rmj of individual magnetic branches j, are calculated from known transformer geometry and slopes dMj/dBj, which are determined from the inverse J–A hysteresis model, as shown in Eq. (9): dH j l j 1 dM j l j ¼ , (9) Rmj ¼ m0 dBj A dBj A where A is the area of the transformer iron core and lj are the lengths of the mean flux paths. Magnetic fluxes fj through reluctances Rmj are determined at each integration step from Eq. (1). They are needed to calculate magnetic flux densities Bj (Eq. (10)), which are used in the inverse J–A hysteresis model to determine reluctances in the next integration step. Five parameters define the shape of B–H hysteresis calculated by the inverse J–A hysteresis model. These parameters are Ms, a, a, c and k. They are determined using DE according to Ref. [4] in: dBj 1 dfj ¼ ; A dt dt
j ¼ a; b; c,
(10)
The obtained transformer model was verified by comparing the calculated voltages and currents with the measured ones. Measurements were performed on a 0.4 kV, 3.5 kV A laboratory three-phase three-limb transformer, with data shown in Table 1. During all measurements, the primary and secondary windings of the transformer as well as balanced and unbalanced loads were Y-connected. In each test, the neutral points of the secondary winding and the load were connected. First, the transformer voltages and currents were measured when the transformer was supplied from the grid and loaded with balanced resistive Table 1 Transformer data Parameter
Value
Rp (O) Rs (O) Np Ns Lsp (H) Lss (H) la, lb, lc (m) A (m2)
0.914 0.898 278 278 0.0265 0.0250 480 103, 240 103, 480 103 5.52 103
Fig. 3. Voltages applied to the laboratory power transformer.
Fig. 4. Measured and calculated secondary voltages at balanced load.
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M. Toman et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015
load. The measured primary voltages upa meas, upb meas and upc meas, shown in Fig. 3, were then used in order to supply the nonlinear transformer model. The model’s calculated secondary voltages usa sim, usb sim and usc sim are shown in Fig. 4. For comparison, they are presented together with the measured secondary voltages usa meas, usb meas and usc meas. The measured primary currents ipa meas, ipb meas, ipc meas and calculated primary currents ipa sim,
Fig. 5. Measured and calculated primary currents at balanced load.
Fig. 6. Measured ipa meas, ipb mea, ipc meas and calculated ipa ipc sim primary currents at unbalanced and nonlinear load.
sim,
ipb
sim,
ipb sim, ipc sim are also presented. They are shown in Fig. 5. The measured and calculated secondary currents are omitted in this paper, since they have the same waveforms as voltages shown in Fig. 4. Next, nonlinear and unbalanced load is considered. In order to form nonlinear and unbalanced load, a different resistive loads were used in phases a, b and c and a diode rectifier was included into phase a. For the given case, primary and secondary currents were measured. They are shown in Figs. 6 and 7. In order to perform simulations, a dynamic model of a diode rectifier was derived and included in the transformer model. Results of the calculated primary and secondary currents are added to Figs. 6 and 7. The presented results show very good agreement between the measured and calculated currents and voltages. This means that the calculated magnetic results can give us accurate insight into the nonlinear magnetic behaviour of each magnetic branch. Fig. 8 shows the hysteresis of an individual magnetic circuit branch over a time span covering two periods for the case of unbalanced and nonlinear load. All simulations were performed on a computer with Intel Pentium M 1.86 GHz processor and 1 GB of RAM. During simulations in the Matlab/Simulink programme package, a fixed step size of 10 ms was used for the case of balanced resistive load and 1 ms for the case of unbalanced and nonlinear load. The computational times were 12 and 100 s for the cases of balanced resistive load and unbalanced and nonlinear load, respectively. 5. Conclusion
Fig. 7. Measured isa meas, isb mea, isc meas and calculated isa sim, isb sim, isc sim secondary currents at unbalanced and nonlinear load.
This paper describes the development of a three-phase three-limb transformer model, which is appropriate for analysis of transformer operation under nonlinear and unbalanced loads. Just like FE models, the developed model can give us insight into a specific iron core area if the iron core is discretizated in more detail. Moreover, it can be easily incorporated into power packages for calculating power system transients. This gives the derived model an advantage, since FE models are too complex and timeconsuming for power system transient calculations.
Fig. 8. Calculated hysteresis in magnetic circuit branches a, b and c.
ARTICLE IN PRESS M. Toman et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e1011–e1015
References [1] O.A. Mohammed, N.Y. Abed, S. Liu, IEEE Trans. Magn. 42 (2006) 976. [2] J.P.A. Bastos, N. Sadowski, Electromagnetic Modeling by Finite Element Methods, CRC, New York, 2003.
e1015
[3] J.V. Leite, N. Sadowski, P. Kuo-Peng, N.J. Batistela, J.P.A. Bastos, A.A. de Espı´ ndola, IEEE Trans. Magn. 40 (2004) 1769. [4] M. Toman, G. Sˇtumberger, D. Dolinar, Compumag 16 (2007) 797. [5] M. Hadzˇiselimovic´, D. Miljavec, I. Zagradisˇ nik, ICEM 16 (2004).