This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON POWER DELIVERY 1 Interfacing -Factor Based White-Box Transformer Models With Electromagnetic Transients Programs Bjørn Gustavsen, Fellow, IEEE, and Álvaro Portillo, Senior Member, IEEE Abstract—White-box transformer models are used by trans- former manufacturers during the dielectric design of windings. The models are often based on constant parameters (RLCG ma- trices) with the high-frequency losses accounted for by a scaling of the dc resistance ( -Factor). We show an efficient procedure for interfacing such models with Electromagnetic Transients Program (EMTP)-type circuit simulators via state equations and a Norton equivalent. The approach makes no approximations except for the discretization in the time domain. Diagonalization is utilized for achieving high computational efficiency. Proprietary information about internal voltages is optionally hidden from the user. Internal surge arresters are handled by the EMTP circuit solver by declaring their connection points as external nodes. The model interface includes automated initialization from 50/60-Hz initial conditions. The proposed interfacing capability permits manufacturers to apply their models in complete network simula- tions, and to share their models with users. Index Terms—Electromagnetic Transients Program (EMTP), simulation, transformer, white-box model. I. INTRODUCTION P OWER transformers have, in several instances, suf- fered dielectric failure with overvoltages as a suspected cause, even when normal practices for insulation coordination and protection have been followed. CIGRE formed in 2008 Working Group (WG) A2/C4.39 “Electrical transient inter- action between transformers and the power system” with the objective to clarify the underlying reason for such failures and if possible recommend procedures for avoiding their occurrence. One of the major conclusions from that work is the observation that oscillating overvoltages of relatively low amplitudes that impinge the transformer terminals can result in excessive overvoltages inside the transformer’s windings by resonance as has also been documented in a number of past studies [1]. The Working Group further recommends that the manufacturer should, by request from the customer, provide a terminal equivalent (black-box) model to permit the user to calculate the impinging overvoltages at the transformer terminals using EMTP-type circuit solvers. The impinging overvoltages can Manuscript received September 18, 2013; revised February 03, 2014; ac- cepted June 19, 2014. Paper no. TPWRD-01070-2013. B. Gustavsen is with SINTEF Energy Research, Trondheim N-7465, Norway (e-mail: bjorn.gustavsen@sintef.no). Á. Portillo is with WEG Transformers, Blumenau SC 89068-001, Brazil (e-mail: acport@adinet.com.uy). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRD.2014.2332515 then be used by the manufacturer as input for calculating in- ternal overvoltages using its proprietary (white-box) software. That way, modifications can be made to the adjacent power system and/or transformer design to reduce the probability of future failures. The white-box models can be categorized as lumped circuit models based on a spatial discretization of the windings, or trav- eling wave-type models. The extraction of the model’s parame- ters is, in all cases, based on a detailed description of the trans- former’s geometry and material properties which, in practice, are only known to the manufacturer. From this information, the model’s parameters are extracted via formulae and/or finite-el- ement method computations. Several levels of refinement are possible, including frequency dependency of the model’s pa- rameters as well as the resolution of the spatial discretization. In this paper, we demonstrate how to employ the recom- mended procedure by A2/C4.39 in practice. We consider the case of lumped-parameter white-box models based on constant RLCG parameters, commonly applied among manufacturers. Damping effects are included in the model by applying a con- stant multiplier factor ( -Factor) to all dc resistances. (One may alternatively calculate the resistances at a given, representative frequency.) Based on these matrices, we formulate a model in terms of state equations with current injection from ground to all nodes as input and internal node voltages and branch voltages as output. Next, we develop a computational approach for interfacing this model with EMTP-type circuit solvers, based on convolution and a Norton equivalent. The terminals of the Norton equivalent may include transformer internal nodes for the connection of internal surge arresters. Internal node voltages and branch currents are made available for monitoring as well. Diagonalization is introduced for faster simulations and hiding of proprietary information. Automated initialization from 50/60-Hz steady initial conditions is introduced, assuming the model has sufficiently accurate behavior at that frequency. We show one procedure for enforcing correct 50/60-Hz model behavior for single-phase transformers by introducing the magnetic core flux path in the inductance matrix. Finally, we show a number of pertinent examples demonstrating the new approach to the so-called “Fictitious Transformer” used in the studies by CIGRE Working Group A2/C4.39 [2]. II. WHITE-BOX MODELING A. Lumped Parameter Transformer White-Box Models White-box models are used for analyzing the internal re- sponse of the power transformer to a system voltage transient applied to its terminals. In the lumped parameter modeling 0885-8977 © 2014 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.