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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