0885-8977 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRD.2019.2930758, IEEE Transactions on Power Delivery 1 Abstract—Switching-on a power transformer creates Energizing Inrush Current (EIC) which in turn leads to Sympathetic Inrush Current (SIC) in adjacent In-service Transformers (ITs). Sympathetic Inrush current changes the waveform and prolongs the duration of its origin EIC. This makes the analysis and estimation of EIC complicated. Occasionally, the stress of EIC can cause internal short-circuit in Energizing Transformer (ET). Detecting short-circuit fault current during EIC is a challenge for transformer differential protection. On the other hand, SIC may mislead the earth-fault protection of in- service-transformer. This paper, first, proposes a new dc equivalent-circuit based on the dc-component of inrush-current, which is efficient in analyzing concurrent EIC and SIC. Next, based on this equivalent circuit, new predictive formulas for EIC and SIC are derived. By employing these formulas, the transformer internal fault during inrush current can be detected based on the comparison between the predicted and measured waveforms, which prevents differential protection from fail-to- trip. Also, the predicted SIC waveform can be compared with the measured waveform to prevent IT earth-fault protection from mal-trip, during inrush current. Finally, the proposed equivalent circuit and formulas are verified using the EMTP simulation results of a real grid, under different system conditions. Index Terms— Transformer protection, inrush current, sympathetic inrush current, power system transient, dc model. I. NOMENCLATURE vg Grid voltage igr Grid current, equal to ien + isy ien Energizing inrush current (EIC) isy Sympathetic inrush current (SIC) ia, ib, ic The current of phases a, b and c ima, ima, ima The magnetizing current of phases a, b and c i0 (ia+ib+ic)/3 lg Grid inductance le ET primary side leakage inductance ls IT primary side leakage inductance Lm Transformer nonlinear core inductance Ls Saturated core inductance rg Grid resistance re ET primary side resistance rs IT primary side resistance This work was supported by Esfahan Electrical Regional Company. The authors are with the Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran (e-mail: moraadi.arash@gmail.com; madani104@yahoo.com, m.madani@eng.ui.ac.ir) ZP Transformer’s Primary side impedance Zs Transformer’s secondary side impedance Zg Grid impedance ZT Equal to Zg+Zp Zg0 Grid zero sequence impedance λm Transformer core flux λg, λe, and λs Dc flux due to the voltage drop on rg, re and rs λd λg+ λe + Λk – Λ0 Λk Core characteristic knee point flux Λ0 Initial flux, equal to Λv0+ Λr Λ1 Core flux magnitude Λr ET residual flux Λv0 Initial flux due to switching ET α[n] Saturation angle (deg), equal to (α+2πn) α0 Saturation angle at the first cycle, n=0 (deg) t0 ET switching on time (s) t1 SIC beginning time (s) ω Angular frequency (rad) T Voltage waveform period (s) θ Voltage waveform angle (rad) θ0 Voltage waveform angle at switching instant (rad) ig Grid current dc component, equal to ie + is ie EIC dc component is SIC dc component i ֘ e EIC alternative component i ֘ s SIC alternative component leg ET dc equivalent inductance lsg IT dc equivalent inductance Λeg ET dc equivalent flux Λsg IT dc equivalent flux αope Optimum saturation angle for EIC (deg) αops Optimum saturation angle for SIC (deg) τi Circuit time constant before beginning of SIC s1 and s2 Circuit eigenvalues after beginning of SIC A and B State and initial values matrices a, b, c and d Constants of EIC and SIC predictive formulas CB Circuit Breaker Predictive Formulas to Improve Transformer Protection During Inrush Current Using the Proposed dc Equivalent Circuit Arash Moradi, Member, IEEE, Seyed M. Madani, Senior Member, IEEE