Modeling of nonlinear and hysteretic iron-core inductors in ATP Nicola Chiesa Department of Electric Power Engineering NTNU, Norway O.S. Bragstadspl. 2E N-7491 Trondheim, NORWAY Voice tel. +47 735 94232 [email protected]Hans Kristian Høidalen Department of Electric Power Engineering NTNU, Norway O.S. Bragstadspl. 2E N-7491 Trondheim, NORWAY [email protected]Abstract - A crucial element of transformer models for transient simulation is the representation of the core. The modeling of non-linear hysteretic inductor required to properly represent a transformer core is a challenge in ATP. The simulation of transient such as inrush and ferroresonance requires a correct handling of nonlinear and frequency dependent losses, accurate hysteresis loop representation, possibility of flux initialization, and a proper automatic initialization by disconnection transients. In addition to the investigation of standard non-linear inductor models of ATP, an advanced hysteretic model based on the Jiles-Atherton theory is implemented in MODELS and tested. The comparison of the models shows several weaknesses and the need of further investigations. A practical table summarizes the main quality and weaknesses of each model together with recommendations useful for the choice of the most appropriate model. Keywords: test report, nonlinear inductor, hysteresis, nonlinear losses, frequency dependent losses, residual flux, Jiles-Atherton. 1 Introduction A transformer iron-core and any other ferromagnetic nonlinear inductance modeled in ATP suffer of low accuracy. The purpose of this paper is to investigate the different possibility that ATP offers for modeling nonlinear inductances. The main limitation is the lack of input data; advanced models require detailed measurements for the estimation of parameters, while standard test report is usually the only source of data. An accurate representation of the losses (nonlinear and frequency dependent) is also a required feature of an accurate model. The first part of this paper addresses a method for dealing with the lack of data by curve fitting. The second part of the paper compares different ATP nonlinear inductor model with focus on losses, shape of the hysteresis loop, and residual flux initialization. An advanced nonlinear inductor model based on Jiles-Atherton theory is implemented in MODELS and tested. The response of the different models to a deenergizarion-reenergization operation is compared.
Transcript
Modeling of nonlinear and hysteretic iron-core inductors in ATP Nicola Chiesa Department of Electric Power Engineering NTNU, Norway O.S. Bragstadspl. 2E N-7491 Trondheim, NORWAY Voice tel. +47 735 94232 [email protected] Hans Kristian Høidalen Department of Electric Power Engineering NTNU, Norway O.S. Bragstadspl. 2E N-7491 Trondheim, NORWAY [email protected] Abstract - A crucial element of transformer models for transient simulation is the representation of the core. The modeling of non-linear hysteretic inductor required to properly represent a transformer core is a challenge in ATP. The simulation of transient such as inrush and ferroresonance requires a correct handling of nonlinear and frequency dependent losses, accurate hysteresis loop representation, possibility of flux initialization, and a proper automatic initialization by disconnection transients. In addition to the investigation of standard non-linear inductor models of ATP, an advanced hysteretic model based on the Jiles-Atherton theory is implemented in MODELS and tested. The comparison of the models shows several weaknesses and the need of further investigations. A practical table summarizes the main quality and weaknesses of each model together with recommendations useful for the choice of the most appropriate model. Keywords: test report, nonlinear inductor, hysteresis, nonlinear losses, frequency dependent losses, residual flux, Jiles-Atherton. 1 Introduction A transformer iron-core and any other ferromagnetic nonlinear inductance modeled in ATP suffer of low accuracy. The purpose of this paper is to investigate the different possibility that ATP offers for modeling nonlinear inductances. The main limitation is the lack of input data; advanced models require detailed measurements for the estimation of parameters, while standard test report is usually the only source of data. An accurate representation of the losses (nonlinear and frequency dependent) is also a required feature of an accurate model. The first part of this paper addresses a method for dealing with the lack of data by curve fitting. The second part of the paper compares different ATP nonlinear inductor model with focus on losses, shape of the hysteresis loop, and residual flux initialization. An advanced nonlinear inductor model based on Jiles-Atherton theory is implemented in MODELS and tested. The response of the different models to a deenergizarion-reenergization operation is compared.
2 Extension of the test report data A core model is usually based on the transformer open-circuit test measurement. Conventionally data at rated excitation is reported. Additional points are required in order to characterize the nonlinearity of the core and built a piecewise linear characteristic. Transformer manufacturers most commonly provide additional data for 90% and 110% excitation levels. Table II in Appendix shows the test report data for a 290 MVA step-up transformer. In this case five results of the open-circuit test are provided, spanning from 75% to 106.25% of the rated voltage. The data provided by the open-circuit test report is sufficient when the purpose of the transient simulation does not involve heavy saturation of the transformer core. Note that the rms V-I characteristic specified in a test report has to be converted to a peak λ-i characteristic by use of the ATP CONVERT/SATURA routine [1,2] or a more advanced method that that can take into account the three-phase coupling of transformers [3,4]. Open-circuit test report seldom reports data above the 110% of excitation level; testing a power transformer in heavy saturation requires a large and stiff source, with rated power comparable to the power of the transformer. Indeed, no testing facility has the capability of performing such test on large units. Thus, the extension process is not straight forward. Fig. 1 shows how a piecewise nonlinear characteristic can be extended by linear extrapolation and curve fitting (saturation curve points are reported in Table III in Appendix). The linear extrapolation method assumes a constant slope of the saturation curve after the last specified segment of the piecewise nonlinear curve. This approach became doubtful when the last points of the piecewise nonlinear curve lie in the 100% to 110% excitation level range: the transformer has not reach the complete saturation during the open-circuit test and a linear extension of the curve will result in a severe underestimation of current for any excitation level above the last specified point. Curve fitting allows the definition of additional artificial points of the saturation characteristic, such that new segments can be added to the piecewise nonlinear curve. While linear extrapolation is the most commonly used, the curve fitting approach should be preferred.
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Fig. 1. Extension of the saturation curve from test report data.
Without specific measurements it is difficult to verify how precisely the curve fitting method can represent the saturated region and mainly relay on the fitting function used. Five different fitting functions are proposed:
ia b i c i
λ =+ ⋅ + ⋅
(1)
( )tanha b i c iλ = ⋅ ⋅ + ⋅ (2)
( )sinhi a b cλ λ= ⋅ ⋅ + ⋅ (3)
ba i c iλ = ⋅ + ⋅ (4)
( )
2
2
E M
SAT E EM
E E
L i ia i ii
L c b i i
λλ
∞
∞
= ⋅ +⎧⎪
⋅ +⎨ = ⋅⎪ + ⋅ +⎩
(5)
where λ represent the flux-linkage. • (1) is a modified version of the Frolich equation, [5]. • (2-4) are equation hard-coded in ATP as user-supplied FORTRAN, [2]. • (5) is a modified version of the equaton proposed by Annakkage, [7].
Fig. 2 shows the curves obtained from the fitting of the open-circuit test report data, while the corresponding parameters are reported in Table IV in Appendix. Beside (2), the rest of the curves give similar expansion of the saturated region. While (2) and (3) give poor agreement in the knee area, (5) shows the best fitting thanks to the higher number of parameters. (1) and (5) should be preferred since they allow a better controll of both the shape of curve in the knee area and the saturation level. Despite the fact that more advanced techniques exist [8], curve fitting is a simple method that require no more than the standard test report data and allow the extension of the saturation characteristic. The authors believe that curve fitting can increase the accuracy of transient simulations like inrush and ferroresonance, and any other study that demand an accurate model of the heavy saturated area of transformers and iron-core devices (the flux in the core can be as high as three time the rated flux during an inrush transient).
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Fig. 2. Fitting of the saturation curve
3 Nonlinear and hysteretic inductor models in ATP The default nonlinear inductor models available in ATP are:
While the first three mentioned types are fairly common among users, the user supplied FORTRAN type is not widely known. The same saturation curve can be used for both type-98 and type-93. However, if the saturation curve is intended for a type-98, a maximum of five breakpoints is recommended to avoid numerical tracking problems, [2]. The solution method used for type-93 ensures the operation on the correct slope of the characteristic so that many breakpoints can be specified. Both these nonlinear inductor model are single valued and the core losses have to be taken into account with a parallel resistor (R-L representation). Type-96 is a hysteretic reactor, thus with direct handling of the core losses. The hysteresis loop data required by type-96 can be generated in ATP with two routines: HYSDAT and HEVIA, [2,9]. The first routine rescale a hard-coded ARMCO Mh oriented silicon steel characteristic based on saturation current and saturation flux values specified by the user. The second routine uses a piecewise nonlinear single-valued characteristic and active core losses to create a hysteretic characteristic suitable for the type-96. The input data for the HEVIA routine match the information available in a test report and has been preferred in this paper. User-supplied FORTRAN can be easily used in ATPDraw with user supplied library as shown in Fig. 3. They are accessible with special call to type-93 card and special flags -333777, -444777 and -555777 respectivelly for (2-4). Example of their use and how to define the equation parameter can be found in subcase of BENCHMARK DC-7. As reported in [2] users may be able to implement their own user supplied FORTRAN model, though this reqiures ATP executable to be recompiled. For this reason the MODELS language has been chosen as a development platform for a hysteretic model based on the Jiles-Atherton theory. Among the numerous versions of the Jiles-Atherton model present in literature, the variant proposed by Chandrasena [10] has been preferred due to the compatibility with the ATP solution method (trapezoidal rule of integration) and the handling of eddy current losses. The model proposed by Chandrasena has been slightly modified and the set of equations has been described as function of electrical parameter λ-i instead of magnetic parameter M-H; the new formulation fits better with ATP implementation and with the estimation of parameters from test report data.
LIB
X555
C User-supplied fortran follows. This is a regular Type-93 NL inductor until C the time-step loop. Note 3-card characteristic, followed by "9999" bound. C The characteristic parameter usage is: psi = a * i**B + c * i 93X555 .08 35.0 1 -555777. 55.FORTRAN { -555777 = flag; a = 1st param .07 0.0FORTRAN { b = 2nd of 3; c = 3rd of 3 params 1.0 1.0FORTRAN { Dummy third card to protect card 2 9999 { End of user-supplied fortran (see request in cols. 33-39) /INITIAL { initialization card, residual flux can be sprcified 4X555 0.0 0.
Fig. 3. Use of User supplied FORTRAN in ATPDraw.
4 Capabilities and limitations The behavior of the nonlinear and hysteretic inductor models presented in the previous session is compared with focus on:
• no-load current; • Active losses; • shape of the hysteresis loop; • residual flux initialization; • response to a deenergizarion-reenergization transient.
Fig. 4 shows the basic core models investigated in this paper. The models can be divided in R-L and hysteretic. In R-L models the losses can be represented by either a linear or a nonlinear (type-99 or -92) resistor, while the saturation curve is modeled with a nonlinear inductor (type98, -93 or user-defined FORTRAN). Hysteretic models include losses and saturation in the same model; the only available hysteretic model in ATP is the type-96, but additional models can be defined as type-94 elements.
R(i)
R(i)
R(i)
LIB
H
TYPE94 ssor or or
type-96
Hevia
type-94 elem.
Jiles-Atherton
Lin.Res. or type-99
and type-98
Lin.Res. or type-92
and type-93
Lin.Res. or type-92
and user-defined FORTRAN
Fig. 4. Nonlinear and hysteretic inductor models in ATPDraw.
4.1 No-load current, losses and shape of the hysteresis loop The no-load current is the sum of the magnetization current and the current due to the core losses. Core losses are an important aspect to take into accout; their nonlinear and frequency dependent nature makes the implementation in a transient simulation program not straight forward. Core losses can be devided in two parts: hysteresis losses and losses due to eddy current in laminations. Hysteresis losses per cycle are frequency indpendent, but they are affected by the excitation level. Eddy current losses vary non linearly with frequency. The most used and straight forward way to take losses into account it to use a linear resistor that match the losses at rated condition. A nonlinear resistor may also be used to represent the nonlinear behaviour. Finally, hysteretic models directly include hysteresis losses, but eventually require additional handling of eddy current losses. Fig. 5 contains the comparison between simulated and test report active losses. When a linear resistor is used, it is common to fix the resistance value to match the rated losses. This gives overestimation of losses for lower level of excitation and underestimation for higher level of exitation. Nonlinear resistors (type-99 and type-92) can be used to have an accurate matching of the losses at the terminal. Type-96 hysteretic inductor gives a result similar to the linear resistor up to the second last point; for value higher than this exitation level, the characteristic became single valued and the losses remain constant (Fig. 7c). The accuracy of the Jiles-Atherton model in fitting the losses is quite poor; this is probably due to a lack of data for the estimation of the correct model parameters. However, also [10] shows some difficulties in a proper matching of core losses with this model; the Jiles-Atherton theory is quite mature and widely axepted, but has room for improvement in losses fitting. Fig. 6 provides the comparison between simulated and test report rms no-load current. The overall agreement is quite satisfatory. Both nonlinear inducance and losses representation are taken into account here; better agreement can be achieved with a better curve fitting and losses modeling.
Rms Voltage [kV]
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e Lo
ss [k
W]
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Test ReportLinear ResistorType-96Type-99/92Jiles-Atherton
Fig. 5. Active losses comparison.
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s C
urre
nt [A
]
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Fig. 6. No-load current comparison.
Active losses and rms values only give an average indication of the behavior over one period. The analysis of instantaneous quantities is required in order to better understand the differences of the various models. The shape of the hysteresis loops as shown in Fig. 7 is suitable for this purpose. The hysteresis loops are found by stepping up the excitation voltage according to the test report and calculation of the corresponding flux linkage in TACS. When an equation is used as a replacement for a piecewise characteristic, the hysteresis loops become smoother (compare Fig. 7a with 7b, and Fig. 7c with 7d). True hysteretic models (Fig. 7c and 7d) have a better relocation of the losses in the full area of the hysteresis loop, while models that represent losses with a resistor in parallel to a nonlinear inductor (Fig. 7a and 7b) have a tendency to underestimate the losses in the vicinity of the knee area (this can be seen from the narrow width of the hysteresis loops in this area of the curve).
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Fig. 7. Hysteresis loop function of the excitation level.
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Fig. 8. Effect of the nonlinear resistor.
The area of the hysteresis loop represents the losses per cycle and provides information on the losses distribution along a cycle. Fig. 8 shows how the losses distribution changes when a nonlinear resistor is used instead of a linear one. While a nonlinear resistor models the losses at the terminal more accurately (Fig. 5), the losses accumulate at the base of the hysteresis loop. A nonlinear resistor characteristic is defined as a v-i curve: the differential resistance became lower (higher losses) as the voltage level increases. The losses reach their highest value at voltage peak, which correspond to the zero of the flux-linkage and to the base of the hysteresis loop. Fig. 9 presents how the hysteresis loops behave as function of frequency. Each model has been tested at values of frequency 25, 50, 200 and 400 Hz. As the frequency increases, also the applied voltage has to increase to obtain a constant flux density in the core (V/f constant). Therefore, when a parallel linear or nonlinear resistor is used, it results in an over estimation of the losses. A frequency dependent resistor may be used to compensate this undesired behavior. The type-96 hysteretic inductor does not show modification of the hysteresis loop as the frequency changes. This is the expected behavior as it only takes into account hysteresis losses (hysteresis losses per cycle are frequency independent). The original Jiles-Atherton model is also frequency independent. However, the implemented model allows to take into account classical and excessive eddy current losses in addition to the hysteresis losses. The curves illustrated in Fig. 9d have only a qualitative meaning since no information was available to accurately calculate the parameters that characterize the frequency dependency behavior of the model. 4.2 Residual flux initialization The residual flux is the flux trapped in a magnetic core after a transformer is disconnected from the network. The value of the residual flux greatly affects the peak of inrush current that develop at the subsequent re-energization. The ability to initialize an inductor model is of great value to test different re-energization scenario. The type-93 element that has a special initialization card; user-defined FORTRAN elements can advantage from the same initialization card as shown in Fig. 3. The type-96 hysteretic inductor and the Jiles-Atherton based model provide the option to set a residual flux value. The type-98 nonlinear inductor model lacks of this feature. A possible workaround to create an initial step in the flux value (in order to resemble a residual flux) is to use a DC source to create a voltage impulse. The DC source has to be active for one time step and have amplitude:
RESPEAKV
timestepλ
= (6)
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a. Linear resistor
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Fig. 9. Hysteresis loops function of frequency (V/f constant).
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Fig. 10. Hysteresis loops alteration due to residual flux initialization.
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The residual flux acts as a DC offset of the flux and the expected behavior is the saturation of the core for only either the positive or negative half period, depending on the sign of the residual flux. Fig. 10 shows the effect on the hysteresis loops when the models are initialized with a residual flux 10% of the rated flux. All the single-valued nonlinear inductors model of ATP can be initialized either with an initialization card or with a DC source and perform as expected and illustrated in Fig. 10a and 10b. The Jiles-Atherton model also reproduces the expected behavior both if initialized with its internal parameter (Fig. 10c) or by a DC source. Finally, the type-96 does not operate correctly if internally initialized; better result is obtained employing the DC source initialization technique (Fig. 10d). 4.3 Deenergizarion-reenergization transient Each model is tested with the ATPDraw circuit presented in Fig. 11 to verify its capability to automatically initialize the residual flux after a disconnection transient. The simulation first generates a ringdown transient followed by an inrush transient. A ringdown transient is a natural LC response that appears as the stored energy dissipates whenever a transformer is deenergized, [11]. Residual fluxes are due to the remnant magnetization of the core, after a transformer has been deenergized. The residual flux pattern is mostly unknown or not known precisely due to the complexity of the ringdown transient itself: an advance and accurate model is required to correctly simulate a ringdown transient.
100 nF
0.15 Ohm
16 kV50 Hz
top = 13 ms
tcl = 35 msCore Model
Fig. 11. Deenergizarion-reenergization circuit.
Fig. 12 presents the results of the performed simulations. Fig. 12a shows the common behavior of most of the nonlinear ATP inductor models: the whole energy dissipates in the parallel R-L elements so that no energy remains trapped in the core and no residual flux is generated. The result of any simulation performed with such models is independent of the disconnection instant since the remanent magnetization of the core will rapidly go to zero. The user-defined FORTRAN model -555777 gives the impression to reach some value of residual flux even if it is part of the parallel RL category. A longer duration of the disconnection transient has confirmed that this residual flux is not a characteristic of the model, but it is artificially generated by the simulation algorithm: it will eventually decay to zero if the reenergization is postponed for a sufficient time. It is difficult to explain the reason of this behavior without information about the implementation of this model. True hysteretic models should be used to obtain a better representation of the ringdown transient, as demonstrated in Fig. 12c and 12d. In this case the residual flux stabilizes at a constant value after few milliseconds from the disconnection operation. When the switch closes, the response of the four models to a reenergization transient is compared in Fig. 12. In order to have a more realistic comparison of the inrush transient a residual flux is artificially created by applying a voltage impulse to the Type-93 model. In this way a similar initial condition of the flux is ensured for all the models. The response of the model to the reenegization is quite dissimilar due to the different characterization of the saturation curves in complete saturation. Type-93 and type-96 model uses linear extrapolation of the last defined segment of the characteristic, while user-defined FORTRAN relay on curve fitting. Thanks to the test report extension procedure presented in the first part of this paper, the type-93 model performs similarly to the -555777 model; without a preliminary curve fitting the type-93 model would have given much lower current values. In addition to
curve fitting, the Jiles-Atherton model defines an air-core inductance (proportional to μ0) for excitation above the saturation level. This results in a much higher inrush current. The decay of the flux DC offset and the inrush current are influenced by the voltage drop on the series resistor, thus a faster decrease occurs for higher inrush currents.
Fig. 12. Ringdown and inrush transient simulation. 5 Conclusions The purpose of this paper has been to compare the features and the capability of available inductor models and a possible way to implement advanced hysteretic models in ATP. Methods for overcoming limitations due to limited amount of data and flux initialization have been presented. An optimum model for each circumstance cannot be selected due to the lack of comparison with measurements and several weaknesses of each model. The two major problems that need to be recalled are the faulty flux initialization of the type-96 hysteretic inductor and the non suitable R-L representation for deenergization transient simulations. Table I summarizes the most important quality of each model and may be helpful for choosing the correct model to use in a simulation. The outcome of this investigation suggests that ATP lacks of an advanced hysteretic model. The Jiles-Atherton model tested has good potential, but more work has to be done to refine the losses model and the estimation of parameters from standardly available data. In addition to the Jiles-Atherton model, another widely accepted model that may be worth to investigate and test in ATP is the Preisach model, [12]. Future work will include the extension from a single-phase equivalent model to a topologically correct three phase core model where several hysteretic inductors need to interact together.
TABLE I MODELS COMPARISON.
Parallel R-L description True hysteretic model
Nonlinear inductance: Type-98 or -93 User-defined FORTRAN Type-96 Type-94, Jiles-Atherton
Fitting equation: Any, only required for test report extension procedure.
(2-4) & possibly other.
Not compatible with HEVIA & HYSDAT routines (possible tuning of the slope of the last segment).
(5), but any can be used by the J-A model.
Losses model: Parallel linear or nonlinear resistor. Included in the model. Included in the model.
No-load losses: Linear resistor is commonly used; very good with nonlinear resistor.
Poor, constant losses after the second last point.
Poor, difficult parameters estimation. Requires model improvement.
Shape of hysteresis loop: Piecewise, bad losses distribution.
Smooth, bad losses distribution.
Piecewise, ok losses distribution.
Smooth, good losses distribution.
Core losses frequency dependency: Bad, require frequency dependent resistor. Only hysteresis losses.
Hysteresis. Also eddy current losses but require estimation of extra parameters.
Flux initialization: Internal initialization for type-93 and user-defined FORTRAN (type-98 requires DC source workaround).
Built-in initialization does not work. Requires DC source workaround.
Ok with both built-in initialization and DC source workaround.
Residual flux after deenergization: Flux goes to zero
Flux goes to zero (very slow decay for -555777, “artificial” residual flux).
OK, but not accurate. OK, accuracy to be tested with measurement.
Complete saturation:
Poor, linear extrapolation after the last segment. Can be improved with test report extension procedure.
OK, curve fitting.
Poor, linear extrapolation after the last segment (Can be improved with manual tuning of the slope of the last segment).
OK, curve fitting.
Parameters estimation: Test report & CONVERT routine.
Test report & Curve fitting.
Test report, CONVERT & HEVIA (or HYSDAT) routines.
Possible but nor straightforward from test report. (Better from measured hysteresis loops).
Computational time: + ++ (+++ for -555777) ++ +++++ (uses MODELS, not hard-coded).
Appendix
TABLE II
GENERATOR STEP-UP TRANSFORMER TEST-REPORT.
Main Data [kV] [MVA] [A] Coupling HS 432 290 338 YN LS 16 290 10465 d5
References [1] Dommel, H. W.; et.al.: Electromagnetic Transients Program Reference Manual (EMTP
Theory Book,. Portland, OR: Prepared for BPA, Aug. 1986. [2] ATP Rule Book. Leuven EMTP Center, Jul. 1987. [3] Chiesa, N.; Høidalen, H. K.: On the calculation of flux linkage/current-characteristic or
[4] Neves, W. L. A.; and Dommel, H. W.: Saturation curves of delta connected transformers from measurements, IEEE Trans Power Delivery, vol. 10, no. 3, pp. 1432 – 1437, 1995.
[5] Chiesa, N.: Power Transformer Modelling: Advanced Core Model, M.SC. thesis, Poltecnico di Milano, Italy, 2005.
[7] Annakkage, U.D.; McLaren, P.G.; Dirks, E.; Jayasinghe, R.P.; Parker, A.D.: A current transformer model based on the Jiles-Atherton theory of ferromagnetic hysteresis, Power Delivery, IEEE Transactions on , vol.15, no.1, pp.57-61, Jan 2000.
[8] Abdulsalam, S.G.; Wilsun Xu; Neves, W.L.A.; Xian Liu: Estimation of transformer saturation characteristics from inrush current waveforms, IEEE Transactions on Power Delivery, v 21, n 1, Jan. 2006, p 170-7
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