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Utilizing Preisach models of hysteresis in the computation of three-phase transformer inrush currents
Electric Power Systems Research 65 (2003) 233 /238 www.elsevier.com/locate/epsr
Utilizing Preisach models of hysteresis in the computation of threephase transformer inrush currents A.A. Adly, H.H. Hanafy *, S.E. Abu-Shady Electrical Power and Machines Department, Faculty of Engineering, Cairo University, Giza 12211, Egypt Received 7 July 2002; received in revised form 21 November 2002; accepted 4 December 2002
Abstract A new approach through which transformer magnetizing inrush currents associated with the energization of three-phase transformers may be computed while accurately taking the effect of hysteresis into consideration. Transformer operating conditions such as switching-on angle and residual flux are also included. The simulation results are compared with those obtained experimentally in a laboratory model transformer. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Magnetization; Inrush current; Hysteresis
1. Introduction When an unloaded transformer is switched on to a supply, the initial magnetizing current is generally much larger than the magnetizing current at steady-state conditions and often much larger than the rated current of the transformer. This phenomenon is known as magnetizing inrush current. Measurements of magnetizing inrush current offer important data for a power system operation and protection because this current causes more serious problems than the energizing currents of the transformer. The concept of magnetizing inrush current is fairly old, numerous researches were carried out [1 /4] to derive the mathematical formulations and explain this phenomenon. In the majority of these researches the simulation of magnetizing inrush currents under various operating conditions were developed. Magnetizing inrush phenomenon in three-phase transformers is more complicated than that in singlephase transformers. Due to the phase shift between the three-phases and the different levels of the residual fluxes in the core.
The magnetizing inrush current has a close relationship with nonlinearity and hysteresis of the iron core. Therefore the purpose of this paper is to present a new method to simulate the magnetizing inrush current taking the effect of hysteresis and different operating conditions such as residual flux and switching-on angle into consideration. In this paper, experiment and simulation are carried out to examine the proposed method.
* Corresponding author. E-mail address: [email protected] (H.H. Hanafy). 0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00015-4
Fig. 1. The equivalent circuit of unloaded three-phases.
234
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
2. The mathematical model Fig. 1 shows the equivalent circuit of unloaded threephase (star connected & isolated neutral) transformer. The voltage equations of the primary windings can be written as: d(ia (t) ib (t))
va (t) vb (t) R(ia (t) ib (t))Ls N
dFa (t) dt
dt
(2) (3) (4) (5)
(6)
where: A , cross-sectional area of the core; B (t), instantaneous flux density in the phase limb, then Eq. (1) and Eq. (2) can be re-written as: d(ia (t) ib (t)) dt dBa (t) dBb (t) NA dt dt
va (t) vb (t) R(ia (t) ib (t))Ls
dt
dt
(8)
Thus,
dBa (t) dHa dBb (t) dHb NA dHa dHb dt dt
d(ia (t) ib (t))
dt
N 2 A dBa (t) dia (t) dBb (t) dib (t) P dHa (t) dt dHb (t) dt d(i (t) ib (t)) vc (t) vb (t) R(ic (t) ib (t))Ls c 2 N A dBc (t) dic (t) dBb (t) dib (t) dt P dHc (t) dt dHb (t) dt
(12)
(13)
For very small time intervals the current and time differentials in Eqs. (12) and (13) may be, with considerable accuracy, replaced by the following difference terms: ia (t) iao Dia (t) ic (t) ico Dic (t) and t to Dt
(14)
Substituting Eqs. (3) and (14) in Eqs. (12) and (13) and making use of Preisach model’s coupled to integral equations, as will be discussed in the next section, to dB (t) dB (t) dB (t) predict the terms ð a ; b ; c Þ (i.e. instantadHa (t) dHb (t) dHc (t) neous permeabilities), we can compute Dia(t), Dic(t ) and, consequently, Dib(t).
3. Computing core magnetization and instantaneous permeabilities using the volume integral equations coupled to the Preisach model
dt
va (t) vb (t) R(ia (t) ib (t))Ls
d(ia (t) ib (t)) dt
(7)
d(ic (t) ib (t))
dBb (t)
(11)
;
va (t) vb (t) R(ia (t) ib (t))Ls
F(t) AB(t)
dBc (t)
P
where, P , mean length of the phase limb. Substituting Eq. (11) in Eqs. (9) and (10), we obtain:
where: Vml, the peak of line voltage; v, supply frequency, a , switching phase angle of phase (a); R , total primary circuit resistance per phase, Ls, primary circuit leakage inductance per phase; N , primary turns per phase; ia(t ), magnetizing current of phase (a); ib(t), magnetizing current of phase (b); ic(t), magnetizing current of phase (c); Fa(t), instantaneous flux linkage of phase (a); Fb(t), instantaneous flux linkage of phase (b); Fc(t), instantaneous flux linkage of phase (c). Substituting F(t) of each phase as
NA
Ni(t)
(1)
dt
d(i (t) ib (t)) vc (t) vb (t) R(ic (t) ib (t))Ls c dt dFc (t) dFb (t) N dt dt ib (t) (ia (t)ie (t)) va (t) vb (t) Vml sin(vtap=6) vc (t) vb (t) Vml sin(vta p=6)
(10)
The magnetic field acting on a phase limb may be roughly correlated to the current of the same phase by the expression: H(t):
dFb (t)
vc (t) vb (t) R(ic (t) ib (t))Ls
d(i (t) ib (t)) vc (t) vb (t) R(ic (t) ib (t))Ls c dt dBc (t) dHc dBb (t) dHb NA dHc dHb dt dt
(9)
Various numerical approaches, such as finite-element and finite-difference techniques, may be utilized while dealing with 3D electromagnetic field analysis. It turns out that the integral equations approach may be regarded among the most efficient techniques. This is because, unlike for the case of finite-elements and finitedifferences, discretization is confined to the ferromagnetic zones and not the whole problem universe. As previously presented in [5 /7], 3D iterative integral equations formulation may be given by:
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
Fig. 2. Transformer core discretization into eight sub-volumes.
¯n H¯ n1 i;t H i;t Nv 1 X t H¯ ci;t 9i 4p j 1
gg
¯ nj;t × dS¯j M Sj
rj;i
H¯ ni;t ; (15)
where Sj is the area of the discretized sub-volume ‘j’, n denotes the iteration step, H¯ c is the field at the center of a sub-volume due to the transformer coils and may be computed using Biot-Savart law, H¯ is the total field at a ¯ represents the sub-volume magsub-volume center, M netization and is updated at every iteration step using the Preisach hysteresis model [8], Nv is the total number of discretized sub-volumes, dSj is a surface element corresponding to sub-volume ‘j ’, while t is a constant depending on the core properties. As discussed and shown in Ref. [5], the convergence of the iterative approach, for any time step, is guaranteed by choosing t according to: 00 t0 xmax
2 ; where; xmax 2
dM dH
j
;
(16) (17)
max
Hence, magnetic B /H characteristics eventually play a role in expected solution. Moreover, working flux density also plays a role in expected field solution for any problem involving magnetic regions. Once, for any time instant the coil currents are known, together with magnetization values for every
235
magnetic sub-volume, total field applied at any point in space may be computed. In this work, the transformer core has been assumed to be made up of eight different sub-volumes as shown in Fig. 2. (i.e. Nv /8). Within each volume, only magnetization along its axial length is assumed to exist due to shape-introduced anisotropy. Given that transformer cores are made up of thin laminated sections, eddy current related distortions on current waveforms may be neglected. Thus, quasi-static computations may be used in order to compute field distribution outside the core as well as instantaneous permeability from the point of view of the transformer primary winding. It should be pointed out here that since Preisach model is involved, the computed permeability becomes dependent on instantaneous current as well as magnetization history. Consequently, different remanent magnetizations are quite expected to lead to different computational results.
4. Computational and experimental results A 3.5 kVA, 175/380 V, Y /Y three-phase core type transformer is used for on-site measurements of magnetizing inrush currents in the laboratory. As a first step, core B /H characteristic have been inferred by energizing one of the primary coils and displaying the time integration of induced emf in an auxiliary coil wound around the same limb, versus energization current using a dual-channel oscilloscope. Knowing the limiting B /H characteristic a normal distribution Preisach distribution density function has been assumed (refer to Ref. [8]). This is a necessary step to deduce the, so-called, first-order reversal curves required to facilitate the numerical implementation of Preisach models. For the core under consideration, inferred B /H first-order reversal curves are shown in Fig. 3.
Fig. 3. Inferred B /H first-order reversal curves for the core under consideration and based on experimental acquisition of the limiting curve.
236
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
Fig. 4. Computed results for (a /0.08) and (lar /0.3, lbr / /0.2 and lcr / /0.1).
Fig. 6. Computed results for (a /210.08) and (lar /0.0, lbr /0.0 and lcr /0.0).
Fig. 5. (a) Waveform of supply voltage of phase A, (b) Measured results for (a /0.08) and (lar /0.3, lbr / /0.2 and lcr / /0.1) [2.5 A/ div and 10 ms/div.].
Fig. 7. (a) Waveform of supply voltage of phase A, (b) Measured results for (a /210.08) and (lar /0.0, lbr /0.0 and lcr /0.0) [2.0 A/ div and 10 ms/div.].
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
237
Different residual flux in the transformer iron core is achieved by using proper values of DC excitation after initially demagnetizing the core. Demagnetization has been accomplished by following the steps below: A) Apply and increase an AC voltage supply to the primary windings until the core reaches saturation. B) Slowly decrease the supply voltage down to zero in order to demagnetize the core. C) To ensure that the experiment is exactly under control with predetermined conditions, this demagnetizing process must be carried out before each test. Using the B /H curve as a guide, temporarily apply the appropriate DC current that would lead to the required residual magnetization.
Fig. 8. Computed results for (a /180.08) and (lar / /0.1, lbr /0.2 and lcr / /0.1).
Table 1 Simulated and measured current peaks for (a /0.08) and per unit residual fluxes of (lar /0.3, lbr / /0.2 and lcr / /0.1) Cycle number
1
2
3
4
5
6
7
8
9
10
Ia
Simu. Meas.
9.15 8.8
6.8 6.2
5.4 4.8
4.4 3.9
3.8 3.2
3.45 3.0
3.0 2.8
2.72 2.5
2.5 2.4
2.3 2.1
Ib
Simu. Meas.
/4.0 /4.0
/3.0 /2.8
/2.4 /2.3
/2.0 /2.0
/1.77 /1.8
/1.56 /1.5
/1.4 /1.3
/1.26 /1.15
/1.1 /1.1
/1.0 /1.0
Ic
Simu. Meas.
/5.0 /5.0
/3.6 /3.5
/2.8 /2.8
/2.32 /2.3
/2.0 /1.8
/1.7 /1.6
/1.5 /1.5
/1.33 /1.3
/1.2 /1.1
/1.0 /1.0
Table 2 Simulated and measured current peaks for (a /210.08) and per unit residual fluxes of (lar /0.0, lbr /0.0 and lcr /0.0) Cycle number
1
2
3
4
5
6
7
8
9
10
Ia
Simu. Meas.
/8.3 /8.0
/4.73 /4.2
/3.28 /3.0
/2.5 /2.4
/2.0 /1.6
/1.68 /1.5
/1.44 /1.3
/1.27 /1.2
/1.14 /1.1
/1.0 /0.95
Ib
Simu. Meas.
/4.74 /4.6
/2.72 /2.8
/1.92 /2.0
/1.5 /1.6
/1.22 /1.3
/1.0 /1.1
/0.88 /0.85
/0.78 /0.8
/0.7 /0.7
/0.64 /0.6
Ic
Simu. Meas.
11.4 11.2
6.34 6.6
4.3 4.8
3.2 3.2
2.53 2.8
2.0 2.1
1.76 1.8
1.52 1.55
1.34 1.3
1.2 1.2
Table 3 Simulated and measured current peaks for (a /180.08) and per unit residual fluxes of (lar / /0.1, lbr /0.2 and lcr / /0.1) Cycle number
1
2
3
4
5
6
7
8
9
10
Ia
Simu. Meas.
/11.0 /8.8
/6.28 /5.84
/4.39 /4.24
/3.32 /3.2
/2.72 /2.5
/2.32 /2.24
/2.0 /2.0
/1.8 /1.6
/1.63 /1.6
/1.5 /1.45
Ib
Simu. Meas.
4.8 4.0
2.77 2.56
1.93 1.84
1.48 1.36
1.2 1.2
1.0 0.98
0.88 0.86
0.78 0.75
0.7 0.7
0.65 0.64
Ic
Simu. Meas.
6.6 5.4
3.84 3.6
2.66 2.64
2.0 2.2
1.6 1.7
1.32 1.4
1.11 1.1
096 1.0
0.84 0.9
0.75 0.8
238
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
residual fluxes (lar /0.3, lbr / /0.2 and lcr / /0.1). Table 1 contains the peaks of the measured and simulated waves under these conditions. Table 2 gives a comparison between the peaks of calculated and measured results of the inrush currents those are shown in Figs. 6 and 7 for (a /210.08) and (lar /0.0, lbr /0.0 and lcr /0.0). Finally Figs. 8 and 9 show the simulated and measured waveforms for the inrush currents at (a / 180.08) and (lar / /0.1, lbr /0.2 and lcr / /0.1). A comparison between the peaks of the measured and simulated waves under these conditions is given in Table 3.
5. Conclusion This paper presents an accurate method for predicting the magnetizing inrush current taking the hysteresis of the core into consideration. The simulation results by the new method are compared with the experimental results. A good agreement between simulation and experimental results was found under different operating conditions.
References
Fig. 9. (a) Waveform of supply voltage of phase A, (b) Measured results for (a /180.08) and (lar / /0.1, lbr /0.2 and lcr / /0.1) [2.0 A/div and 10 ms/div.].
The inrush current waveforms are recorded by using digital oscilloscope through current transducers. The switch-on angles of the transformer primary voltage are taken into a random process and could be determined by noticing the voltage waveforms on the digital oscilloscope. Under different values of residual flux and switchingon angle the inrush currents are measured and calculated by the proposed method. Figs. 4 and 5 show the simulated and a typical oscillogram for the inrush currents at switching-on angle (a /0.08) and per unit
[1] L.A. Finzi, W.H. Mutschler, The inrush of magnetizing current in single-phase transformer, AIEE Trans. 70 (1951) 1436 /1438. [2] R. Yacamini, A. Abu Nasser, Numerical calculation of inrush current in single-phase transformers, IEE Proc. 128 (6) (1981) 327 / 334. [3] S. Akpinar, M. Coulson, Calculation of transient current in transformers, IEE Proc. 129 (1) (1982) 30 /34. [4] C.E. Lin, C.L. Cheng, J.C. Yen, Investigation of inrush current in transformers, IEEE Trans. On power delivery 8 (1) (1993) 246 / 254. [5] I.D. Mayergoyz, Iterative methods for calculations of static fields in inhomogeneous, anisotropic and nonlinear media, Naukova Dumka, 1979. [6] G. Friedman, I.D. Mayergoyz, Computation of magnetic field in media with hysteresis, IEEE Trans. Magn. 25 (1989) 3934 /3936. [7] A.A. Adly, I.D. Mayergoyz, R.D. Gomez, E.R. Burke, Computation of magnetic fields in hysteretic media, IEEE Trans. Magn. 29 (1993) 2380 /2382. [8] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer Verlag, New York, 1991.
Utilizing Preisach models of hysteresis in the computation of threephase transformer inrush currents A.A. Adly, H.H. Hanafy *, S.E. Abu-Shady Electrical Power and Machines Department, Faculty of Engineering, Cairo University, Giza 12211, Egypt Received 7 July 2002; received in revised form 21 November 2002; accepted 4 December 2002
Abstract A new approach through which transformer magnetizing inrush currents associated with the energization of three-phase transformers may be computed while accurately taking the effect of hysteresis into consideration. Transformer operating conditions such as switching-on angle and residual flux are also included. The simulation results are compared with those obtained experimentally in a laboratory model transformer. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Magnetization; Inrush current; Hysteresis
1. Introduction When an unloaded transformer is switched on to a supply, the initial magnetizing current is generally much larger than the magnetizing current at steady-state conditions and often much larger than the rated current of the transformer. This phenomenon is known as magnetizing inrush current. Measurements of magnetizing inrush current offer important data for a power system operation and protection because this current causes more serious problems than the energizing currents of the transformer. The concept of magnetizing inrush current is fairly old, numerous researches were carried out [1 /4] to derive the mathematical formulations and explain this phenomenon. In the majority of these researches the simulation of magnetizing inrush currents under various operating conditions were developed. Magnetizing inrush phenomenon in three-phase transformers is more complicated than that in singlephase transformers. Due to the phase shift between the three-phases and the different levels of the residual fluxes in the core.
The magnetizing inrush current has a close relationship with nonlinearity and hysteresis of the iron core. Therefore the purpose of this paper is to present a new method to simulate the magnetizing inrush current taking the effect of hysteresis and different operating conditions such as residual flux and switching-on angle into consideration. In this paper, experiment and simulation are carried out to examine the proposed method.
* Corresponding author. E-mail address: [email protected] (H.H. Hanafy). 0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00015-4
Fig. 1. The equivalent circuit of unloaded three-phases.
234
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
2. The mathematical model Fig. 1 shows the equivalent circuit of unloaded threephase (star connected & isolated neutral) transformer. The voltage equations of the primary windings can be written as: d(ia (t) ib (t))
va (t) vb (t) R(ia (t) ib (t))Ls N
dFa (t) dt
dt
(2) (3) (4) (5)
(6)
where: A , cross-sectional area of the core; B (t), instantaneous flux density in the phase limb, then Eq. (1) and Eq. (2) can be re-written as: d(ia (t) ib (t)) dt dBa (t) dBb (t) NA dt dt
va (t) vb (t) R(ia (t) ib (t))Ls
dt
dt
(8)
Thus,
dBa (t) dHa dBb (t) dHb NA dHa dHb dt dt
d(ia (t) ib (t))
dt
N 2 A dBa (t) dia (t) dBb (t) dib (t) P dHa (t) dt dHb (t) dt d(i (t) ib (t)) vc (t) vb (t) R(ic (t) ib (t))Ls c 2 N A dBc (t) dic (t) dBb (t) dib (t) dt P dHc (t) dt dHb (t) dt
(12)
(13)
For very small time intervals the current and time differentials in Eqs. (12) and (13) may be, with considerable accuracy, replaced by the following difference terms: ia (t) iao Dia (t) ic (t) ico Dic (t) and t to Dt
(14)
Substituting Eqs. (3) and (14) in Eqs. (12) and (13) and making use of Preisach model’s coupled to integral equations, as will be discussed in the next section, to dB (t) dB (t) dB (t) predict the terms ð a ; b ; c Þ (i.e. instantadHa (t) dHb (t) dHc (t) neous permeabilities), we can compute Dia(t), Dic(t ) and, consequently, Dib(t).
3. Computing core magnetization and instantaneous permeabilities using the volume integral equations coupled to the Preisach model
dt
va (t) vb (t) R(ia (t) ib (t))Ls
d(ia (t) ib (t)) dt
(7)
d(ic (t) ib (t))
dBb (t)
(11)
;
va (t) vb (t) R(ia (t) ib (t))Ls
F(t) AB(t)
dBc (t)
P
where, P , mean length of the phase limb. Substituting Eq. (11) in Eqs. (9) and (10), we obtain:
where: Vml, the peak of line voltage; v, supply frequency, a , switching phase angle of phase (a); R , total primary circuit resistance per phase, Ls, primary circuit leakage inductance per phase; N , primary turns per phase; ia(t ), magnetizing current of phase (a); ib(t), magnetizing current of phase (b); ic(t), magnetizing current of phase (c); Fa(t), instantaneous flux linkage of phase (a); Fb(t), instantaneous flux linkage of phase (b); Fc(t), instantaneous flux linkage of phase (c). Substituting F(t) of each phase as
NA
Ni(t)
(1)
dt
d(i (t) ib (t)) vc (t) vb (t) R(ic (t) ib (t))Ls c dt dFc (t) dFb (t) N dt dt ib (t) (ia (t)ie (t)) va (t) vb (t) Vml sin(vtap=6) vc (t) vb (t) Vml sin(vta p=6)
(10)
The magnetic field acting on a phase limb may be roughly correlated to the current of the same phase by the expression: H(t):
dFb (t)
vc (t) vb (t) R(ic (t) ib (t))Ls
d(i (t) ib (t)) vc (t) vb (t) R(ic (t) ib (t))Ls c dt dBc (t) dHc dBb (t) dHb NA dHc dHb dt dt
(9)
Various numerical approaches, such as finite-element and finite-difference techniques, may be utilized while dealing with 3D electromagnetic field analysis. It turns out that the integral equations approach may be regarded among the most efficient techniques. This is because, unlike for the case of finite-elements and finitedifferences, discretization is confined to the ferromagnetic zones and not the whole problem universe. As previously presented in [5 /7], 3D iterative integral equations formulation may be given by:
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
Fig. 2. Transformer core discretization into eight sub-volumes.
¯n H¯ n1 i;t H i;t Nv 1 X t H¯ ci;t 9i 4p j 1
gg
¯ nj;t × dS¯j M Sj
rj;i
H¯ ni;t ; (15)
where Sj is the area of the discretized sub-volume ‘j’, n denotes the iteration step, H¯ c is the field at the center of a sub-volume due to the transformer coils and may be computed using Biot-Savart law, H¯ is the total field at a ¯ represents the sub-volume magsub-volume center, M netization and is updated at every iteration step using the Preisach hysteresis model [8], Nv is the total number of discretized sub-volumes, dSj is a surface element corresponding to sub-volume ‘j ’, while t is a constant depending on the core properties. As discussed and shown in Ref. [5], the convergence of the iterative approach, for any time step, is guaranteed by choosing t according to: 00 t0 xmax
2 ; where; xmax 2
dM dH
j
;
(16) (17)
max
Hence, magnetic B /H characteristics eventually play a role in expected solution. Moreover, working flux density also plays a role in expected field solution for any problem involving magnetic regions. Once, for any time instant the coil currents are known, together with magnetization values for every
235
magnetic sub-volume, total field applied at any point in space may be computed. In this work, the transformer core has been assumed to be made up of eight different sub-volumes as shown in Fig. 2. (i.e. Nv /8). Within each volume, only magnetization along its axial length is assumed to exist due to shape-introduced anisotropy. Given that transformer cores are made up of thin laminated sections, eddy current related distortions on current waveforms may be neglected. Thus, quasi-static computations may be used in order to compute field distribution outside the core as well as instantaneous permeability from the point of view of the transformer primary winding. It should be pointed out here that since Preisach model is involved, the computed permeability becomes dependent on instantaneous current as well as magnetization history. Consequently, different remanent magnetizations are quite expected to lead to different computational results.
4. Computational and experimental results A 3.5 kVA, 175/380 V, Y /Y three-phase core type transformer is used for on-site measurements of magnetizing inrush currents in the laboratory. As a first step, core B /H characteristic have been inferred by energizing one of the primary coils and displaying the time integration of induced emf in an auxiliary coil wound around the same limb, versus energization current using a dual-channel oscilloscope. Knowing the limiting B /H characteristic a normal distribution Preisach distribution density function has been assumed (refer to Ref. [8]). This is a necessary step to deduce the, so-called, first-order reversal curves required to facilitate the numerical implementation of Preisach models. For the core under consideration, inferred B /H first-order reversal curves are shown in Fig. 3.
Fig. 3. Inferred B /H first-order reversal curves for the core under consideration and based on experimental acquisition of the limiting curve.
236
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
Fig. 4. Computed results for (a /0.08) and (lar /0.3, lbr / /0.2 and lcr / /0.1).
Fig. 6. Computed results for (a /210.08) and (lar /0.0, lbr /0.0 and lcr /0.0).
Fig. 5. (a) Waveform of supply voltage of phase A, (b) Measured results for (a /0.08) and (lar /0.3, lbr / /0.2 and lcr / /0.1) [2.5 A/ div and 10 ms/div.].
Fig. 7. (a) Waveform of supply voltage of phase A, (b) Measured results for (a /210.08) and (lar /0.0, lbr /0.0 and lcr /0.0) [2.0 A/ div and 10 ms/div.].
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
237
Different residual flux in the transformer iron core is achieved by using proper values of DC excitation after initially demagnetizing the core. Demagnetization has been accomplished by following the steps below: A) Apply and increase an AC voltage supply to the primary windings until the core reaches saturation. B) Slowly decrease the supply voltage down to zero in order to demagnetize the core. C) To ensure that the experiment is exactly under control with predetermined conditions, this demagnetizing process must be carried out before each test. Using the B /H curve as a guide, temporarily apply the appropriate DC current that would lead to the required residual magnetization.
Fig. 8. Computed results for (a /180.08) and (lar / /0.1, lbr /0.2 and lcr / /0.1).
Table 1 Simulated and measured current peaks for (a /0.08) and per unit residual fluxes of (lar /0.3, lbr / /0.2 and lcr / /0.1) Cycle number
1
2
3
4
5
6
7
8
9
10
Ia
Simu. Meas.
9.15 8.8
6.8 6.2
5.4 4.8
4.4 3.9
3.8 3.2
3.45 3.0
3.0 2.8
2.72 2.5
2.5 2.4
2.3 2.1
Ib
Simu. Meas.
/4.0 /4.0
/3.0 /2.8
/2.4 /2.3
/2.0 /2.0
/1.77 /1.8
/1.56 /1.5
/1.4 /1.3
/1.26 /1.15
/1.1 /1.1
/1.0 /1.0
Ic
Simu. Meas.
/5.0 /5.0
/3.6 /3.5
/2.8 /2.8
/2.32 /2.3
/2.0 /1.8
/1.7 /1.6
/1.5 /1.5
/1.33 /1.3
/1.2 /1.1
/1.0 /1.0
Table 2 Simulated and measured current peaks for (a /210.08) and per unit residual fluxes of (lar /0.0, lbr /0.0 and lcr /0.0) Cycle number
1
2
3
4
5
6
7
8
9
10
Ia
Simu. Meas.
/8.3 /8.0
/4.73 /4.2
/3.28 /3.0
/2.5 /2.4
/2.0 /1.6
/1.68 /1.5
/1.44 /1.3
/1.27 /1.2
/1.14 /1.1
/1.0 /0.95
Ib
Simu. Meas.
/4.74 /4.6
/2.72 /2.8
/1.92 /2.0
/1.5 /1.6
/1.22 /1.3
/1.0 /1.1
/0.88 /0.85
/0.78 /0.8
/0.7 /0.7
/0.64 /0.6
Ic
Simu. Meas.
11.4 11.2
6.34 6.6
4.3 4.8
3.2 3.2
2.53 2.8
2.0 2.1
1.76 1.8
1.52 1.55
1.34 1.3
1.2 1.2
Table 3 Simulated and measured current peaks for (a /180.08) and per unit residual fluxes of (lar / /0.1, lbr /0.2 and lcr / /0.1) Cycle number
1
2
3
4
5
6
7
8
9
10
Ia
Simu. Meas.
/11.0 /8.8
/6.28 /5.84
/4.39 /4.24
/3.32 /3.2
/2.72 /2.5
/2.32 /2.24
/2.0 /2.0
/1.8 /1.6
/1.63 /1.6
/1.5 /1.45
Ib
Simu. Meas.
4.8 4.0
2.77 2.56
1.93 1.84
1.48 1.36
1.2 1.2
1.0 0.98
0.88 0.86
0.78 0.75
0.7 0.7
0.65 0.64
Ic
Simu. Meas.
6.6 5.4
3.84 3.6
2.66 2.64
2.0 2.2
1.6 1.7
1.32 1.4
1.11 1.1
096 1.0
0.84 0.9
0.75 0.8
238
A.A. Adly et al. / Electric Power Systems Research 65 (2003) 233 /238
residual fluxes (lar /0.3, lbr / /0.2 and lcr / /0.1). Table 1 contains the peaks of the measured and simulated waves under these conditions. Table 2 gives a comparison between the peaks of calculated and measured results of the inrush currents those are shown in Figs. 6 and 7 for (a /210.08) and (lar /0.0, lbr /0.0 and lcr /0.0). Finally Figs. 8 and 9 show the simulated and measured waveforms for the inrush currents at (a / 180.08) and (lar / /0.1, lbr /0.2 and lcr / /0.1). A comparison between the peaks of the measured and simulated waves under these conditions is given in Table 3.
5. Conclusion This paper presents an accurate method for predicting the magnetizing inrush current taking the hysteresis of the core into consideration. The simulation results by the new method are compared with the experimental results. A good agreement between simulation and experimental results was found under different operating conditions.
References
Fig. 9. (a) Waveform of supply voltage of phase A, (b) Measured results for (a /180.08) and (lar / /0.1, lbr /0.2 and lcr / /0.1) [2.0 A/div and 10 ms/div.].
The inrush current waveforms are recorded by using digital oscilloscope through current transducers. The switch-on angles of the transformer primary voltage are taken into a random process and could be determined by noticing the voltage waveforms on the digital oscilloscope. Under different values of residual flux and switchingon angle the inrush currents are measured and calculated by the proposed method. Figs. 4 and 5 show the simulated and a typical oscillogram for the inrush currents at switching-on angle (a /0.08) and per unit
[1] L.A. Finzi, W.H. Mutschler, The inrush of magnetizing current in single-phase transformer, AIEE Trans. 70 (1951) 1436 /1438. [2] R. Yacamini, A. Abu Nasser, Numerical calculation of inrush current in single-phase transformers, IEE Proc. 128 (6) (1981) 327 / 334. [3] S. Akpinar, M. Coulson, Calculation of transient current in transformers, IEE Proc. 129 (1) (1982) 30 /34. [4] C.E. Lin, C.L. Cheng, J.C. Yen, Investigation of inrush current in transformers, IEEE Trans. On power delivery 8 (1) (1993) 246 / 254. [5] I.D. Mayergoyz, Iterative methods for calculations of static fields in inhomogeneous, anisotropic and nonlinear media, Naukova Dumka, 1979. [6] G. Friedman, I.D. Mayergoyz, Computation of magnetic field in media with hysteresis, IEEE Trans. Magn. 25 (1989) 3934 /3936. [7] A.A. Adly, I.D. Mayergoyz, R.D. Gomez, E.R. Burke, Computation of magnetic fields in hysteretic media, IEEE Trans. Magn. 29 (1993) 2380 /2382. [8] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer Verlag, New York, 1991.