Portal de Periódicos da CAPES

Technique for inrush current modelling of power transformers based on core saturation analysis

2018; Institution of Engineering and Technology; Volume: 12; Issue: 10 Linguagem: Inglês

10.1049/iet-gtd.2017.1272

1751-8695

Arash Moradi, Seyed M. Madani,

Power Transformer Diagnostics and Insulation

IET Generation, Transmission & DistributionVolume 12, Issue 10 p. 2317-2324 Research ArticleFree Access Technique for inrush current modelling of power transformers based on core saturation analysis Arash Moradi, Arash Moradi orcid.org/0000-0002-8277-4274 Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, IranSearch for more papers by this authorSeyed M Madani, Corresponding Author Seyed M Madani madani104@yahoo.com Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, IranSearch for more papers by this author Arash Moradi, Arash Moradi orcid.org/0000-0002-8277-4274 Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, IranSearch for more papers by this authorSeyed M Madani, Corresponding Author Seyed M Madani madani104@yahoo.com Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, IranSearch for more papers by this author First published: 21 March 2018 https://doi.org/10.1049/iet-gtd.2017.1272Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Energising a power transformer may cause inrush current, which misleads the protection systems. Therefore, the inrush current analysis is important in designing and protecting power transformers. The non-linear behaviour of transformer core saturation makes this analysis difficult. Thus, several researches try to model the core saturation and inrush current. This study presents a new technique based on core flux analysis to develop an equivalent circuit for power transformer during inrush current. For this purpose, a new λ –i equivalent circuit is proposed for saturated core transformer by transformation of conventional v –i circuit. This λ –i equivalent circuit clearly shows the effect of parameters on transformer saturation and inrush current; and provides a powerful insight into these phenomena. Moreover, new equations are developed which can predict inrush current and core-flux envelopes. The predicted waveforms can be compared with measured transformer current to detect transformer internal-fault during inrush current, which is a challenge in transformer protection. This model and the equations are compared with the recorded inrush current waveform of a real transformer, and simulation results. These comparisons verify the efficiency of the model and accuracy of the equations. Nomenclature v sinusoidal grid voltage, pu V1 amplitude of the grid voltage, pu vr voltage drop across resistance r, pu vl voltage drop across inductance l, pu i transformer primary-side current, pu imax (t) inrush current envelope, pu IMax maximum peak of inrush current (MPIC), pu ω angular frequency, rad/s θ0 switching angle, ° r total resistance of the transformer and its system, pu λm transformer core linkage flux, pu λ1 steady-state linkage flux magnitude λl series inductance linkage flux, pu λr core residual DC linkage flux, pu λv sinusoidal steady-state linkage flux, pu λv0 DC linkage flux due to switching, pu λ0 total DC linkage flux (λv0 + λr), pu λri lost linkage flux due to voltage drop on the resistance ri (t), pu λmk λ1 + λ0 − λk, pu Ceq equivalent capacitor in λ –i circuit, pu l transformer and system leakage inductance, pu Req equivalent resistor of λ –i circuit, pu Lm core inductance in linear region, pu Ls incremental inductance of the core in saturation region λk core linkage flux at knee point, pu λmax (t) envelope of core flux peaks, pu λmin (t) envelope of core flux minimums, pu T period of voltage or flux waveform, s α saturation angle, ° Zg grid impedance, pu 1 Introduction Transformers are the vital and widely used instruments in electric power systems. Damage to a transformer results in high maintenance costs. In addition, the unplanned outage of a transformer incurs high expenses to electrical utilities and customers. Recent field experiences show that the majority of transformer failures or outages occur due to large overcurrent on their primary sides [1]. This overcurrent is due to faults or inrush current which is a transient current caused by a sudden change in the transformer excitation voltage [1]. The phenomenon of inrush current in transformers is known for many years and is discussed in many studies [2–10]. Lin et al. [2] investigate inrush current and the effect of residual flux, load power factor, and switching angle. Harmonic analysis of inrush current is investigated in [3]. Yacamini and Abu-Nasser [11] discuss significant impact of inrush current on the grid voltage. They also mention some of the major consequences of inrush current that are voltage drop on the grid resistance, misleading protective relays due to large unbalance current, and motors vibration. Formulas to approximate the shape of inrush current are proposed in [2, 11, 12]. Several transformer models during inrush current are presented in [13–19]. Some of these models are based on electromagnetic-circuit analysis [15, 19]. These models need to have transformer detail data which may be difficult to obtain. Transformer core B –H curve plays an important role in the inrush current analysis. In [20], the B –H curve is approximated by a two-slope representation. Several models use analytical equations for transformer inrush current [11, 12, 14, 16, 21]. In [12], a semi-analytical method is used to determine inrush current waveform, which models the core magnetisation curve λ(i) by a hyperbolic sinusoidal function, that has acceptable accuracy only in the first cycle. Chen et al. [14] model magnetising curve H (B) by a tangent function to find core inductance. This model is not accurate in saturation region, because in this region, the tangent curve approaches infinity, in contrast to the H (B) slope. In [16], hysteresis magnetising curve is modelled based on experimental data. Then, to estimate the inrush current, it uses differential equations. However, modelling the hysteresis curve by experimental data inserts measurement error. The magnitude of the first cycle inrush current is important for transformer design and protection. Therefore, analytical formulas to estimate the peak of inrush current during the first cycle as a function of switching angle are presented in [20–23]. Moreover, due to importance of inrush current envelope for transformer and system protection, analytical formulas to estimate inrush current envelope are presented in [24–26]. These formulas require some detailed data like detailed core dimensions in addition to the electrical parameters and magnetic core characteristics. Although modelling inrush current has been widely investigated by the mentioned research works, further investigations are required to develop an analytical model with acceptable accuracy. The proposed model requires only the transformer and system electrical parameters and the core magnetisation characteristic. This model and the derived formulas can be used to investigate the effects of transformer/system parameters and turn on switching angle on the transformers saturation and inrush current. Moreover, the model and the derived formulas predict the inrush current waveform from the first cycle of the transformer energisation. This predicted waveform can be compared with the real measured current to detect internal fault caused by inrush current, which is a challenge in transformer protection. This can reduce the mal-trips or fail-to-trips in transformer protection. This paper is organised as follows: in Section 3, λ–i circuit is introduced versus conventional v–i circuit. Next, the core magnetising characteristic is modelled by two-piece linear function. Then, a new λ–i equivalent circuit for saturating inductor is presented. Finally, a power system including a transformer is modelled by an equivalent λ–i circuit. In Section 4, new mathematical formulas are developed to estimate upper and lower envelopes of transformer flux, and the envelope of inrush current. In Section 5, the proposed model and envelope formulas are verified by experimental recorded inrush current and simulation results. 2 New model for transformer saturation and inrush current using core flux analysis Although there exists many model for the transformer saturation and inrush current [8, 14–16], these phenomena still need further investigations. This section proposes an accurate and straightforward model for magnetic saturation in transformer, to analyse and model the inrush current. During normal condition, the transformer core linkage flux λm is less than the saturation level λk. However, after switching-on a transformer, λm usually exceeds the saturation level λk due to residual flux λr and a DC flux caused by switching. 2.1 Transformer saturation analysis Here, a typical power system, Fig. 1, is considered for analysis of transformer core saturation and inrush current phenomena. The single-phase v –i equivalent-circuit of this system after switching-on the transformer is shown in Fig. 2. Fig. 1Open in figure viewerPowerPoint Switching on transformer in power system Fig. 2Open in figure viewerPowerPoint Electrical equivalent circuit for transformer inrush current In this circuit, r and l are the total series resistances and leakage inductances of transformer and its system, respectively. Moreover, the transformer secondary winding is considered to be open circuit which is common for transformer inrush current analysis. Here, the phase grid voltage is assumed as: (1) According to Fig. 2, the flux λ is considered the sum of λm and λl, which can be obtained as: (2) For further analysis, (2) is presented as: (3) where: (4) (5) (6) (7) As seen in (5), DC residual flux linkage λr is an important factor on transformer saturation. When a transformer is de-energised, the magnetising voltage and current approaches zero, while the flux follows the magnetic hysteresis loop of the transformer core. This results in certain remanent DC flux left in the core. Then, at the next energisation of the transformer, this DC flux is added to the flux developed by magnetisation voltage. Depending on the direction of the residual flux; it can have positive or negative effect on transformer saturation [2]. Table 1. Transformation from v –i circuit to λ–i circuit v –i circuit λ–i circuit v λ i i r Ceq (r) = 1/r l Req (l) = l In (6), λv0 is a DC flux due to switching-on the transformer, which can vary from –λ1 to + λ1, as a function of switching angle θ0. In general, after any sudden change in excitation voltage magnitude/phase, such as switching-on, λv0 can be developed [2]. The DC flux λ0, as shown in (5), contains λr and λv0. According to (3), λ consists of λv, and λ0, so the DC flux linkage λ0 may cause λ to enter the saturation region, and causes inrush current, which is a unidirectional current with DC value. According to (7), the integral of voltage drop ri (t) is the resistor lost-flux λri (t). Inrush current is unidirectional, so the resistor's voltage drops with a unidirectional direction, which in turn adds a DC flux to λri (t) in each cycle. According to (3) and (7), λri gradually decreases the total DC flux ǀλ0 − λri ǀ and reliefs the transformer core from saturation. Therefore, the lost-flux λri is important for transformer saturation and inrush current analysis. According to saturation and normal conditions analyses, λm (λm = λ − λl = λ −li) differs from λ only during inrush current interval (when i ≠0). Therefore, the effect of λl (or series inductance) must be considered in calculating the peaks of the inrush current and core flux. 2.2 Introducing λ –i circuits versus v –i circuits In conventional electric circuit named v–i circuit, the variable across each element is voltage v, and the variable flowing-through each element is current i. In contrast, in λ –i circuits, the across-variable is flux linkage λ [which is integral of the element voltage v (t) in time domain], while the flowing-through-variable is current i. Transformation of parameters and variables from v –i to λ –i circuits are presented in Table 1. The proofs of Ceq (r) and Req (l) are presented in (9)–(13). For core saturation and inrush current analysis, fluxes λ are more direct and useful than voltages v, thus, in this paper, λ –i circuits are employed instead of v –i one. 2.3 New equivalent circuit for magnetic saturation An approximate two-piece linear function for transformer core λ –i characteristic is shown in Fig. 3b. This type of core characteristic representation is employed for transformer inrush current modelling in most of the analytical studies [12, 13, 20, 21]. As shown in Fig. 3a, when the core flux linkage exceeds λk, it enters saturation region and creates inrush current. One cycle of the inrush current is shown in Fig. 3c. The inrush current and core flux of the first cycle after switching-on the transformer are shown in Fig. 3d. According to Fig. 3b, the λ–i relation of transformer core can be approximated as [20]: (8) Fig. 3Open in figure viewerPowerPoint Process of core saturation (a) Magnetising flux, (b) Core characteristic, (c) Inrush current, (d) Inrush current and magnetising flux of first cycle In this paper, the approximate λ –i characteristic is modelled by a λ –i electric equivalent circuit shown in Fig. 4a. Fig. 4Open in figure viewerPowerPoint The λ –i equivalent circuits of the transformer core (a ) λ –i equivalent circuit of a saturated core, (b) Model for positive inrush current, ignoring Lm, (c) Model for negative inrush current, ignoring Lm In this equivalent circuit, the linear region [middle row of (8)] is modelled by a resistor Lm. The positive saturation region [lower row of (8)] is modelled by an equivalent series circuit of DC source λk, and resistor Ls. Moreover, to satisfy the condition λm >λk, a diode must be put in series with the source and the resistor. In the same way, the equivalent circuit can be obtained for the negative saturation region λm <−λk. The resistor Lm in Fig. 4a, which represents the core magnetising inductance Lm in v −i circuit, is quite large; consequently, the equivalent circuit can be approximated by ignoring resistance Lm. ‏ Moreover, at the beginning of inrush current, if λ0 is positive, the core flux exceeds only + λk (positive saturation knee point) and the circuit can be reduced to Fig. 4b. In the same way, if λ0 is negative, the circuit shown in Fig. 4c is applicable. These circuits are employed in the rest of this paper. 2.4 New model for analysing transformer saturation and inrush current The conventional v –i equivalent circuit of a transformer and its network are shown in Fig. 2. To achieve a comprehensive λ –i equivalent circuit, all circuit elements must be replaced by their λ –i equivalents. In a v –i circuit, the dropped flux across a resistance λri is (9) According to (9), a resistor in v –i circuit is transformed to a capacitor in λ –i circuit with the following capacity: (10) The flux across this capacitor represents λri which is the lost-flux across the resistor accumulating in each cycle. Moreover, in this circuit, an inductance's voltage and flux are (11) and (12) Thus (13) According to (11)–(13), it is deduced that an inductance l behaves like a resistance in a λ –i circuit, and it causes flux drop only when current i is not zero. The circuit shown in Fig. 2 is represented by its λ –i equivalent circuit shown in Fig. 5. In this equivalent circuit, the AC flux developed by ac grid voltage is represented by an AC flux source λv. The DC flux λ0 is represented by a DC flux source in series with λv. According to (5), λ0 consists of λv0 due to the switching-on and the residual flux λr. Fig. 5Open in figure viewerPowerPoint Proposed model for saturated transformer in a power system As seen in Fig. 5, λ0 is positive, so, the λ –i core equivalent circuit for positive saturation region, Fig. 4b, is employed as a core model. The effect of circuit parameters on the transformer saturation is shown in Fig. 5 clearly. Although this equivalent circuit is applicable for transformer saturation and inrush current analysis, it has some limitations. For elements like power cables which include capacitors, λ –i model may not result in reduction or simplification. Moreover, the model is applicable if a transformers' core characteristic can be accurately model by two-slope line like most of nowadays transformers with large Lm. 2.5 Applying the model on three-phase transformers In the previous section, the proposed model is presented for single-phase transformers. Inrush current of three-phase transformers are investigated by some papers [13, 15, 19, 27]. Based on [27] and EMTP simulations, the proposed single-phase model can be applied on a three-phase transformer, if it is energised from a Y-connected neutral-grounded winding (such as YNynd, Ynd). This is because the decaying DC flux of saturated core induces negligible voltage in the secondary and/or tertiary windings. However, further research and modelling are required to adopt the model for un-grounded three-phase transformers. 3 Inrush current and core flux modelling For further analysis, based on equivalent circuit, Fig. 5, the capacitor flux λri (the flux lost on the system resistance) must be employed as the state variable of the circuit. 3.1 Formulating λri as a state variable According to (3), the applied flux λv is periodic and DC flux λ0 is constant; therefore, to get rid of sinusoidal variation, the flux variation Δλ(t) is defined as: (14) (15) Δλv (t) = 0 and Δλ0 (t) = 0, therefore: (16) According to Fig. 3d, i is negligible when the core is unsaturated; so, (16) can be changed as follows: (17) As seen in Fig. 3c, transformer inrush current happens when its core flux exceeds the saturation level (18) where L l + Ls. This approximation (18) is accurate when transformer Lm is large, like nowadays transformers. As the changes in λri are very small during each system period T, its differential value can be approximated as dλri/dt ≃Δλri/T. Substituting (18) into (17) yields: (19) Substituting ω = 2π/T yields: (20) In the right side of (20), all quantities are known except λri and α which can be determined versus λri. According to Fig. 3a, at ωt1 = α inrush current is zero and λm. Therefore (21) which is rearranged as: (22) (23) Using an accurate curve fitting of (24) results in: (25) where (26) Applying Tailor approximation of (27) and inserting (27) into (20) yields: (28) Tailor approximation of (27) is more accurate if , which is true in most inrush current cases. Applying definition (26) results in: (29) Substituting α with (25) yields: (30) To solve this equation, (30) is rearranged as: (31) Integrating both sides of (31) results in: (32) where a is: (33) From (32), analytical formula for λri is obtained: (34) The constant c can be determined as: (35) (36) Finally, the state variable λri is obtained from (34) as: (37) 3.2 Inrush current envelope imax (t) To estimate this envelope imax (t), the maximum value of inrush current during each period is determined. Using (18) and choosing λ1 as the maximum value of λ1 (t), results in: (38) Substituting (37) to (38), results in: (39) The formula to estimate inrush current envelope is proposed, as follows: (40) 3.3 Maximum peak of inrush current magnitude IMax The maximum peak of inrush current (MPIC) or Imax (occurs at first cycle after energisation) has the potential to produce mechanical stress and cause malfunction in differential protection relays; thus, it is necessary to estimate the MPIC [20]. Here, the highest peak of transformer core flux (which create MPIC) is assumed as time reference. By substituting t = tpk = 0 into (40), MPIC is estimated as follows: (41) or, in terms of voltage, (41) is rewritten as: (42) 3.4 Upper and lower envelopes of core flux λmax, λmin According to Fig. 3d, the maximum core flux envelope at each period can be determined based on imax (t) (40): (43) For minimum core linkage flux envelope, current i is very small, thus: (44) Equation (43) indicates that λri (t) decreases λmax (t). Therefore, larger loop resistance causes faster saturation removal. The proposed formulas (40), (43), and (44) provide an insight into inrush current analyses and calculations, which can be used in transformer design. 4 Simulation, experimental results, and comparing with other methods In this section, a recorded inrush current waveform of an installed transformer in a 230 kV substation of Isfahan is used as an experimental result, which is shown in Fig. 6. The transformer is energised from its HV side 230 kV. The transformer technical data and grid impedance are reported in Table 2. Other required information like switching angle (θ0) and residual flux (λr) are obtained in related subsections. The transformer and its grid are simulated in EMTP-RV programme through a numerical method described in [28]. This proposed model and these analytical formulas are compared with: the recorded experimental results, EMTP simulation results, and other methods in [12, 20, 21, 23–25]. Fig. 6Open in figure viewerPowerPoint Recorded inrush current of a real transformer measured by CT Table 2. Transformer data Parameter Value voltage 230/66/20 kV rating 160 MVA type YNyn0d11 frequency 50 Hz transformer resistance (H –M) 0.28875 Ω transformer resistance (M –L) 0.045832 Ω transformer resistance (H –L) 0.046399 Ω transformer reactance (H –M) 9.77804 Ω transformer reactance (M –L) 4.183069 Ω transformer reactance (H –L) 2.306136 Ω equivalent grid impedance (Zg) 0.47Ω + j5.61Ω Slope of B –H curve in saturation region 0.404 4.1 Verifying inrush current waveform of the proposed model by experimental data For simulation of the real transformer inrush current, the initial condition, λ0, λv0, and λr must be estimated. From Fig. 6, MPIC is estimated as Imax = 194.5 A, which is equal to the peak-to-peak value of the first cycle. Next, based on (41), the DC flux is estimated as λ0 = 0.46 pu. From the magnetic point of view, the total value of λr + λv (which is equal to λ0) determines the inrush current waveforms. Here, the switching-on angle is assumed as θ0 = 70°, which results in λv0 = 0.34 pu. Then, based on (6), residual flux becomes λr = 0.12 pu. This information is employed for all the simulations. Moreover, based on the recorded waveform, Fig. 6, the system frequency during inrush current is estimated as 50.022 Hz. The recorded inrush current shown in Fig. 6 is the output of a CT; consequently, part of its DC value is removed due to CT response and saturation. To compensate the effect of CT in Fig. 6, for each period, the waveform is shifted up such that the experimental inrush current during unsaturated intervals (which is named dwell-time) become zero, as shown in Fig. 7. Fig. 7Open in figure viewerPowerPoint Inrush current, and its envelope (θ0 = 70° and λr = 0.12 pu) (a) First few cycles, (b) One second after switching-on, (c) Two seconds after switching-on Fig. 7 shows the inrush current waveforms obtained through: experimental data (after CT effect compensation), EMTP, this proposed model, and the model presented in [12]. The envelope of inrush current obtained by the proposed formula (40) is shown in the same figures. Figs. 7a–c show the inrush current waveforms of: the first few cycles, after 1 and 2 s, respectively. As shown in Fig. 8, comparing inrush current of this proposed model with the experimental recorded data, the peak error of the proposed model is ∼2%, while that of method in [12] is ∼8%, during the first cycle. Moreover, in Fig. 8, the coefficient of determination, named R2 or R -squared [29], depicts how well the proposed model estimates the real data, compared to EMTP and [12]. The value of R2 for EMTP and the proposed model is close to 0.98 that proves the accuracy of the proposed model. Fig. 8Open in figure viewerPowerPoint Comparing accuracy of: EMTP, the proposed model, and [12] in estimating the real transformer inrush current (Fig. 6), using (a) Peak error, (b) Coefficient of determination (R2) 4.2 Verifying inrush current waveform of the proposed model by EMTP simulation results Fig. 9 shows inrush current waveforms obtained through: EMTP-RV (in blue) and the proposed model simulations, for the 230 kV transformer. This comparison once more proves the accuracy of this proposed model for inrush current. Fig. 9Open in figure viewerPowerPoint Inrush current and its envelope obtained by the proposed model comparing to the transformer simulation results (θ0 = 0 and λr = 0) 4.3 Verifying the proposed inrush current envelope formula by experimental and simulation results As shown in Fig. 7, the maximum error of the proposed envelope formula (40) compared with the experimental results is 6.5%. While compared with EMTP simulation result, it is <2%. This error difference (6.5–2)% is due to transformer parameter error. 4.4 Comparing the proposed inrush current envelope formula by other methods Holcomb [24] proposes the following formula to estimate the inrush current envelope: (45) Another estimation formula is developed in [24]: (46) Equations (45) and (46) require saturation angle α which must be calculated in every cycle. Moreover, the saturated core inductance is required, which is difficult to access. Therefore, comparing to methods of [24, 25], this paper's proposed formula (40) directly estimates the envelope of inrush current, with no need to calculate α and L. The maximum error of this proposed formula (40) is 6.5%. This maximum error is lower than that of the methods in [24, 25], which are 8.5 and 9.2%, respectively. 4.5 Comparing the proposed maximum peak of inrush current MPIC formula with other formulas Wang et al. [20] propose the following formula for the MPIC: (47) This formula is similar to the equation presented in [23], when substituting λ = N × B × A and θ0 = 0° to calculate the worst case of inrush current as a function of magnetic flux density: (48) Moreover, a long formula is presented in [21], which is developed in the same way as of (47). Table 3 presents and compares MPICs obtained by this and three other methods [19, 20, 22] for three different transformers. The transformer and simulation data of table's first row are described in Table 2. Moreover, the data of table's second and third rows are described in [13] (three-phase transformer) and [20] (single-phase transformer), respectively. Table 3. Maximum peak of inrush current: calculation versus EMTP simulations and other methods (θ0 = 0° and λr = 0.8 pu) Data EMTP [22, 19] [20] (41) 1 Table 2 Imax (A)/Error (%) 2305/0 1995 13.4 2141/7.1 2339/1.5 2 1 MVA 10/ 0.38 kV Imax (A)/Error (%) 9453/0 8110 14.2 10,275/8.7 9642/2.1 3 30 kVA 120/280 V Imax (A)/Error (%) 1811/0 1539 12.1 1923/6.2 1834/1.3 This comparison concludes that the proposed formula has higher accuracy to estimate MPIC than the other mentioned formulas. 4.6 Verifying the proposed model core flux waveform and its envelope formulas by EMTP Fig. 10 shows core flux waveform obtained through this proposed model and EMTP simulation (in blue). Moreover, the proposed formulas for upper and lower envelope of the core flux (43), and (44) are shown in Fig. 9. This figure confirms the accuracy of the flux waveform and its envelops estimated through the proposed model and the envelope formulas. Fig. 10Open in figure viewerPowerPoint Transformer core flux, its upper and lower envelopes obtained by the proposed model comparing to the transformer simulation results (θ0 = 0 and λr = 0) 5 Conclusion In this paper, λ –i circuit is introduced at first, which simplifies transformer flux and current analysis compared to the conventional v –i circuit. Next, a transformer and its power system are modelled by an equivalent circuit. Then, analytical formulas are derived to estimate: MPIC, envelopes of inrush current, and core flux. The proposed model and the formulas are verified by: an experimental recorded inrush current of a real transformer and EMTP simulation results. Moreover, the proposed formula for inrush current envelope and MPIC are compared with the experimental data and EMTP simulation results. Compared to the existing methods, this proposed λ –i equivalent circuit is accurate, while consists of few elements. It provides a clear insight into the mechanism of the core saturation and inrush current, which is helpful for transformer design and protection. It directly results in the flux of saturated core and the inrush current. The proposed formulas (40)–(44) clearly show the effects of transformer parameters like series resistance, leakage inductance, knee point flux, core characteristic, and system inductances on transformer saturation and inrush current. Moreover, this model and the derived formulas can predict inrush current waveforms, which improve transformer protection to detect internal faults during inrush current. 7 References 1Zou, M., Wenxia, S., Ming, Y. et al.: 'Improved low-frequency transformer model based on Jiles–Atherton hysteresis theory', IET Gener. Transm. Distrib., 2017, 11, (4), pp. 915– 923 2Lin, C.E., Cheng, C.L., Huang, C.L. et al.: 'Investigation of magnetizing inrush current in transformers. I. Numerical simulation', IEEE Trans. Power Deliv., 1993, 8, pp. 246– 254 3Lin, C.E., Cheng, C.L., Huang, C.L. et al.: 'Investigation of magnetizing inrush current in transformers. II. Harmonic analysis', IEEE Trans. Power Deliv., 1993, 8, pp. 255– 263 4Persson, M., Baig, W., Thiringer, T.: 'Measurements and modelling of three- and five-limb transformer behaviour during large voltage and frequency disturbances', IET Gener. Transm. Distrib., 2016, 10, pp. 334– 340 5Rico, J.J., Acha, E., Madrigal, M.: 'The study of inrush current phenomenon using operational matrices', IEEE Trans. Power Deliv., 2001, 16, pp. 231– 237 6Adly, A.A.: 'Computation of inrush current forces on transformer windings', IEEE Trans. Magn., 2001, 37, pp. 2855– 2857 7Wu, Q., Jazebi, S., Leon, D.F.: 'Parameter estimation of three-phase transformer models for low-frequency transient studies from terminal measurements', IEEE Trans. Magn., 2017, 53, pp. 710– 718 8Bronzeado, H., Yacamini, R.: 'Transformer inrush calculations using a coupled electromagnetic model', IEEE Proc. Sci. Meas. Tech., 1994, 141, pp. 491– 498 9Smith, K.S., Ran, L., Leyman, B.: 'Analysis of transformer inrush transients in offshore electrical systems', IEEE Proc. C Gener. Transm. Distrib., 1999, 146, pp. 89– 95 10Chen, X.S., Neudorfer, P.: 'Digital model for transient studies of a three-phase five-legged transformer', IEEE Proc. C Gener. Transm. Distrib., 1992, 139, pp. 21– 27 11Yacamini, R., Abu-Nasser, A.: 'Numerical calculation of inrush current in single-phase transformers', IEEE Proc. B Electr. Power Appl., 1981, 128, pp. 327– 328 12Vanti, M.G., Bertoli, S.L., Cabrai, S.H.L. et al.: 'Semianalytic solution for a simple model of inrush currents in transformers', IEEE Trans. Magn., 2008, 44, pp. 1270– 1273 13Chen, X., Venkata, S.S.: 'A three-phase three-winding core-type transformer model for low-frequency transient studies', IEEE Power Eng. Rev., 1997, 17, pp. 85– 86 14Chen, S.D., Lin, R.L., Cheng, C.K.: 'Magnetizing inrush model of transformers based on structure parameters', IEEE Trans. Power Deliv., 2005, 20, pp. 1947– 1954 15Chiesa, N., Hidalen, H.K., Mork, B.A.: 'Transformer model for inrush current calculations: simulations, measurements and sensitivity analysis', IEEE Trans. Power Deliv., 2010, 25, pp. 2599– 2608 16Faiz, J., Saffari, S.: 'Inrush current modeling in a single-phase transformer', IEEE Trans. Magn., 2010, 46, pp. 578– 581 17Abdulsalam, S.G., Xu, W., Neves, W.L.A. et al.: 'Estimation of transformer saturation characteristics from inrush current waveforms', IEEE Trans. Power Deliv., 2006, 21, pp. 170– 177 18Naghizadeh, R.A., Vahidi, B., Hosseinian, S.H.: 'Modelling of inrush current in transformers using inverse Jiles–Atherton hysteresis model with a neuro-shuffled frog-leaping algorithm approach', IET Electr. Power Appl., 2012, 6, pp. 727– 728 19Liu, J., Dinavahi, V.: 'Detailed magnetic equivalent circuit based real-time nonlinear power transformer model on FPGA for electromagnetic transient studies', IEEE Trans. Ind. Electron., 2016, 63, pp. 1191– 1202 20Wang, Y., Abdulsalam, S.G., Xu, W.: 'Analytical formula to estimate the maximum inrush current', IEEE Trans. Power Deliv., 2008, 23, pp. 1266– 1268 21Jazebi, S., de Leon, F., Wu, N.: 'Enhanced analytical method for the calculation of the maximum inrush currents of single-phase power transformers', IEEE Trans. Power Deliv., 2015, 30, pp. 2590– 2599 22Zirka, S.E., Moroz, Y.I., Arturi, C.M. et al.: 'Topology-correct reversible transformer model', IEEE Trans. Power Deliv., 2012, 27, pp. 2037– 2045 23Girgis, R.S., Tenyenuis, E.G.: 'Characteristics of inrush current of present designs of power transformers'. IEEE Power Eng, Soc. General Meeting, 2007, 12, pp. 1– 6 24Holcomb, J.E.: 'Distribution transformer magnetizing inrush current', Trans. AIEE, Part III: Power Appar. Syst., 1961, 80, pp. 697– 702 25Specht, T.R.: 'Transformer magnetizing inrush current', Trans. Am. Inst. Electr. Eng., 1951, 70, pp. 323– 328 26Bertagnolli, G.: ' Short-circuit duty of power transformers: the ABB approach' ( Golinelli, Formigine, 1996) 27Nakra, H., Barton, T.: 'Three phase transformer transients', IEEE Trans. Power Appar. Syst., 1974, 93, pp. 11– 12 28 EMTP-RV T.T.: ' 3 phase, 3 winding transformers', in ' EMTP-RV user manual. EMTP-EMTPWorks devices documents' ( EMTP-RV, Quebec, Canada, 2005, 2nd edn.), pp. 1– 7 29Cameron, A.C., Windmeijer, F.A.G.: 'An R-squared measure of goodness of fit for some common nonlinear regression models', J. Econ., 1997, 77, pp. 329– 342 Citing Literature Volume12, Issue10May 2018Pages 2317-2324 FiguresReferencesRelatedInformation