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1696 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 3, JULY 2008 power Dg is a measure of effect of harmonics generated in the load on the apparent power. Therefore, the power equation of three-phase loads with a nonsinusoidal supply voltage can be written in the form S 2 = P 2 + Q2 + Ds2 + Du2 + Dg2 : These powers and the power equation provide clear information as to how circuits features and phenomena, such as phase shift, change of the load conductance with harmonic order, and the load imbalance and generation of current harmonics in the load due to its nonlinearity or periodic time-variance affect the load apparent power S and its power factor . Moreover, the CPC Power Theory provides fundamentals for compensation of each harmful power, meaning power Q, Ds , Du , and Dg . Observe that the aforementioned powers are not defined in terms of (as recommended by the authors) the Poynting Theorem. By the way, (2) in the Discussion written by de León and Cohen p(t) = a(t) + r(t) provides an excellent illustration on how useless this Theorem is in explaining power properties of electrical loads. When this equation is applied to loads in Fig.1(a) and (b) in the Discussion then, due to the lack of energy storage capability of both loads, the power r(t) = 0, meaning the instantaneous power p(t) of both loads has no components. Consequently, this equation does not provide an answer even for such a simple question: why the load in Fig. 1(a) has power factor = 1, while that in Fig. 1(b) has a power factor equal only to = 1= 2? To be accurate, using only the Poynting Theorem, the concept of the power factor = P=S cannot be introduced because this theorem does not make it possible to calculate the apparent power S . This is not a physical quantity related to the Poynting Vector. In his discussion on the CPC, S-J, Jeon wrote:“The decomposition seems to be meaningful only under strictly restricted conditions…”, which sounds like a rebuke. After that, the author stated that “A theory based on a restricted condition should not be used as a standard to criticize other theory.” These conclusions are wrong due to points 7, 8, and 9. Each scientific statement, in particular theory, can be valid only under strictly defined conditions. The CPC Power Theory, as presented in the discussed paper [1], was developed under the condition that the supply voltage is sinusoidal and symmetrical. The CPC Power Theory of linear, time-invariant loads with asymmetrical, but sinusoidal supply voltage is presented in my paper [6]. The IRP p-q Theory has been formulated for a very wide set of electrical systems, meaning a set of conditions, including nonsinusoidal and asymmetrical voltages and currents. Any theory valid for a set of conditions {C} has to be valid for any subset of the set {C}. To show that such a theory is not valid is enough to prove that it is not valid for some subset of the set {C}. If, as was proven in paper [1], the IRP p-q Theory is not capable of identifying power properties of a load at a symmetrical supply voltage instantaneously, it is not capable, of course, to do it when the voltage is asymmetrical. If it is not able to identify these properties, as it was proven, when the voltage is sinusoidal, then it is not able to do it when this voltage is nonsinusoidal. The CPC Power Theory in my paper was not used as any “standard to criticize other theory…”, but only as a tool that enables identification of power properties of three-phase systems. The objective of my papers was not to “criticize other theory,” but to show that a pair of p and q powers calculated at some instant of time does not p 6) 7) 8) 9) provide information on power properties of three-phase loads. No conclusions can be drawn with respect to the load power properties. The knowledge of these two powers does not allow us to answer the question: is this load active, reactive, balanced, or unbalanced? REFERENCES [1] L. S. Czarnecki, “Instantaneous reactive power p-q theory and power properties of three-phase systems,” IEEE Trans. Power Del., vol. 21, no. 1, pp. 362–367, Jan. 2006. [2] L. S. Czarnecki, “Currents’ physical components (CPC) in circuits with nonsinusoidal voltages and currents. Part 1: Single-phase linear circuits,” Elect. Power Quality Utilization J., vol. XI, no. 2, pp. 3–14, 2005. [3] L. S. Czarnecki, “Energy flow and power phenomena in electrical circuits: Illusions and reality,” Archiv für Elektrotechnik, vol. 82, no. 4, pp. 10–15, 1999. [4] L. S. Czarnecki, “Physical interpretation of the reactive power in terms of the CPC Power Theory,” presented at the 7th Int. Workshop Power Definitions Meas. Under Nonsinusoidal Conditions, Cagliari, Italy, 2006. [5] L. S. Czarnecki, “Currents’ physical components (CPC) in circuits with nonsinusoidal voltages and currents. Part 2: Three-phase linear circuits,” Electr. Power Quality Utilization J., vol. XII, no. 1, pp. 3–14, 2006. [6] L. S. Czarnecki, “Powers of asymmetrically supplied loads in terms of the CPC power theory,” presented at the 7th Int. Workshop Power Definitions Meas. Under Non-Sinusoidal Conditions, Cagliari, Italy, 2006. [7] L. S. Czarnecki, “Currents’ physical components (CPC) concept: A fundamental of power theory,” Przeglad Elektrotechniczny, vol. 84, no. 6, pp. 83–93, Jun. 2008. Discussion of “Transformer Modeling for Low- and Mid-Frequency Transients—A Review” Francisco de León The authors of [1] should be commended for successfully summarizing the state of the art of transformer modeling for the study of electromagnetic transients. They have clearly described the scope and limitations of the available transformer models for the low- and mid-frequency ranges. I would like to complement their excellent work by discussing from the duality-based viewpoint, the physical meaning of the dual Cauer model used to represent the nonlinear iron core in [2]. Additionally, I would like to point out a couple of very recent references presenting advances on the models described in the paper. A note on data availability is also included in this discussion. Fig. 1 shows a cut of a transformer iron-core lamination. It helps to illustrate the physical significance of the dual Cauer model used for the representation of the eddy current effects in the core. One can appreciate that the transversal inductances correspond in the duality sense to magnetic flux paths (or reluctances) for sections of the lamination. Therefore, nonlinear effects, due to saturation and hysteresis, could be included in each inductance. In the figure, one can also note that the paths of the eddy currents are represented by resistances. In 2002, Holmberg et al. modeled the eddy current phenomena in the windings of a coil using a Cauer model derived from the duality principle between magnetic and electric circuits [3]. While the series Manuscript received September 3, 2003; revised May 24, 2005. Paper no. TPWRD-00455-2003. The author is with Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRD.2008.924191 0885-8977/$25.00 © 2008 IEEE IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 3, JULY 2008 1697 Closure on “Transformer Modeling for Low- and Mid-Frequency Transients—A Review” J. A. Martinez and B. A. Mork Fig. 1. Illustrating the physical significance of the dual Cauer circuit for the representation of eddy currents in the transformer iron core. Foster equivalent circuit proposed in [2] is only a terminal model, the elements of the Cauer circuit proposed in [3] can be related to currents and magnetic fluxes in the duality sense. The circuit proposed in [3] is similar to the one shown in Fig. 1 with similar physical interpretation. In 2004, Chandrasena et al. presented an improved model for hysteresis based on the Jiles–Atherton theory which includes eddy currents in the same model [4]. Finally, to estimate the parameters of a transformer model that is adequate for the calculation of electromagnetic transients, one requires knowledge of internal construction details. Some parameters cannot be obtained from terminal measurements only. Transformer design information is proprietary and, therefore, is seldom available to engineers performing system studies. Degeneff et al. suggested that transformer manufacturers could deliver a set of equivalent circuits with various degrees of sophistication [5]. It is perhaps now the time to assemble a working group to establish guidelines for transformer modeling for electromagnetic transient studies. The authors’ comments on the aforementioned items would be greatly appreciated. We want to thank the discusser for his comments since they give us the opportunity to clarify some parts of our paper. We also thank him for the complementary bibliography, although the paper by Chandrasena et al. was published when our paper was already in print. As Dr. De León knows very well, the list of references included in our paper is just a small sample of the work performed on transformer modeling. In any case we agree that [1] and Fig. 1 of this Discussion [2] provide some physical meaning of the dual Cauer equivalent. As for the incorporation of nonlinear effects, they can certainly be represented in the dual Cauer model, as already pointed out in the paper. The number of nonlinear components will depend on the number of sections included in the model. As mentioned in the paper, this number should not be too high for low- and mid-frequency transients; only one or maybe two of them could be used to represent nonlinear effects. Parameter estimation was also part of our paper, but since a paper on the same subject by the IEEE Task Force on Data for Modeling System Transients was in progress, only a short section was related to it. We agree that knowledge about some construction details may be needed to develop an accurate enough transformer model, and this information is even more critical for developing low-frequency models. The suggestion that transformer manufacturers could deliver equivalent circuits is very interesting but this will not occur unless the standards require it and there are good guidelines on how to proceed. We take the opportunity that this discussion gives us to invite Dr. De León to join the IEEE Task Force on Data for Modeling System Transients, as the development of standards on modeling guidelines for the various power components, as suggested by him, is a future task of this group. REFERENCES [1] J. A. Martinez and B. A. Mork, “Transformer modeling for low- and mid-frequency transients–A review,” IEEE Trans. Power Del., vol. 20, no. 2, pt. 2, pp. 1625–1632, Apr. 2005. [2] F. de León, “Discussion of ‘Transformer modeling for low- and midfrequency transients–A review’,” IEEE Trans. Power Del., vol. 23, no. 3, pp. 1696–1697, Jul. 2008 REFERENCES [1] J. A. Martinez and B. A. Mork, “Transformer modeling for low- and mid-frequency transients—A review,” IEEE Trans. Power Del., vol. 20, no. 2, pt. 2, pp. 1625–1632, Apr. 2005. [2] F. de León and A. Semlyen, “Time domain modeling of eddy current effects for transformer transients,” IEEE Trans. Power Del., vol. 8, no. 1, pp. 271–280, Jan. 1993. [3] P. Holmberg, M. Leijon, and T. Wass, “A wideband lumped circuit model of eddy current losses in a coil with a coaxial insulation system and a stranded conductor,” IEEE Trans. Power Del., vol. 18, no. 1, pp. 50–60, Jan. 2003. [4] W. Chandrasena, P. G. McLaren, U. D. Annakkage, and R. J. Jayasinghe, “An improved low-frequency transformer model for use in GIC studies,” IEEE Trans. Power Del., vol. 19, no. 2, pp. 643–651, Apr. 2004. [5] R. C. Degeneff, P. J. McKenny, and M. R. Gutierrez, “A method for constructing reduced order transformer models from detailed lumped parameter transformer models,” IEEE Trans. Power Del., vol. 7, no. 2, pp. 649–655, Apr. 1992. . Manuscript received September 3, 2003; revised June 30, 2005. Paper no. TPWRD-00455-2003. J. A. Martinez is with the Department d’Enginyeria Elèctrica, Universitat Politècnica de Catalunya, Barcelona 08028, Spain. B. A. Mork is with the Department of Electrical Engineering, Michigan Technological University, Houghton, MI 49931 USA. Digital Object Identifier 10.1109/TPWRD.2008.924192 0885-8977/$25.00 © 2008 IEEE