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INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014
Modeling of Multiport Systems via a ModeRevealing Transformer
Ehsan Elahi, Hamid Keivani and Alireza Alesaadi

Abstract—Employing an admittance matrix in the frequency
domain by rational functions for power transformers is a wellknown method which improves calculation efficiency. This model
must be passive in order to avoid unstable time domain
simulations. All of the methods have made efforts to overcome the
problem of preserving the passivity of the final model. In this
paper a similarity transformation matrix which better reveals the
eigenvalues of the admittance matrix is presented. The chosen
transformation preserves the passivity and symmetry of the
original data.
Keywords— Electromagnetic Transients, Network Equivalent,
Passivity Enforcement, Pole-Residue Model and Vector Fitting
I. INTRODUCTION
F
requency-dependent modelling of linear devices and
systems is widely applied in the technical fields of power
systems, microwave systems, and high-speed electronic
systems [1]. A large number of methods have been proposed
for construction of a network equivalent. These methods can
be classified into time domain and frequency domain methods
[2]. The solution in the frequency domain is essentially aimed
at the identification of rational functions that approximate the
admittance matrix of the external system seen from boundary
bus(es). Among the various methods of fitting, the vector
fitting (VF) has proved its efficiency and accuracy in different
applications [3].
VF is essentially a robust reformulation of the sanathanan –
koerner iteration [4], using rational basis functions instead of
polynomials and pole relocation instead of weighting [5]. The
model identification involves rational fitting of the given
terminal responses, leading to models that can be included in
EMTP-type programs by recursive convolution [6] or by a
lumped electrical network [7].
This work was supported in part by the Islamic Azad University, Kazeroon
Branch, Kazeroon, Iran.
E. Elahi is with the Department of Electrical Engineering, Kazeroon
Branch, Islamic Azad University, Kazeroon, Iran, (Ccorresponding Author;
Email: [email protected]).
H. Keivani is with the Department of Electrical Engineering, Kazeroon
Branch,
Islamic
Azad
University,
Kazeroon,
Iran
(Email: [email protected]).
A. Alesaadi is with the Department of Electrical Engineering, Kazeroon
Branch,
Islamic
Azad
University,
Kazeroon,
Iran
(Email: [email protected]).
[ISSN: 2045-7057]
Past experiences has shown that simulations involving a
fitted admittance matrix Y can sometimes lead to an unstable
simulation, even though the elements of Y have been fitted
using stable poles only [8]. In the other words, although the
model obtained via vector fitting has guaranteed stable poles,
the model may result in unstable simulations because the
passivity of the model is not assured. Thus assess the passivity
characteristics of the model is needed. For this purpose, it is
common practice to calculate the eigenvalues of a Hamiltonian
matrix which is associated with a state-space formulation of
the model [9].
A difficulty with the Hamiltonian matrix is its size. Its
dimension is two time the number of system states, which
makes the computation of eigenvalues time-consuming for
large models [10]. A second disadvantage is that eigenvalues
corresponding to crossover frequencies are not exactly
imaginary as they are associated with a small real part (noise).
In the case of multi terminal problems, the output response
is sometimes strongly depend on the distribution of the applied
input on the terminals. Such behavior is often observed in the
low-frequency range, due to large differences between shortcircuit and open-circuit modal characteristics. Direct
application of standard modeling techniques to such cases
gives a model with inaccurate representation of the small
modes. This problem is addressed by the method known as
modal vector fitting (MVF) [11] and modal perturbation (MP)
[12] but comes at the cost of long computation times and
increased memory requirements.
In this paper, we introduce a new technique for the multiport
modeling of systems with large differences in modal
characteristics which overcome the high computational costs
of MVF. We show how to compute a suitable transformation
matrix using the given data only, and how to preserve the
essential information of passivity and symmetry. Then
transform the extracted model back to the physical domain via
the inverse information. The advantages of the new approach
are demonstrated for frequency-dependent network equivalent
(FDNE) modeling and wideband modeling of transformers,
focusing on accuracy and computational efficiency [14].
II. INPUT-OUTPUT TRANSFER FUNCTION
The modelling starts from a given port-admittance matrix
Y(s), which defines the relation between port voltage v and
current i. This matrix can be obtained via calculations or
measurements
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INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014
(1)
The admittance Y is a symmetric matrix, satisfying the
symmetry property (2)
(2)
It is assumed that a rational model for Y has been identified
which satisfies (2). This can be easily achieved by fitting the
elements of Y using the pole relocating algorithm known as
vector fitting (VF)[14]. This leads to a pole-residue model (3)
which can be expanded into a model in standard state-space
form (4).
(3)
(4)
Physicality of the model leads to the following
requirements:
Y is a symmetric matrix. Hence, { }, D, and E are
symmetric.
D and E are real matrices.
The pole and residues are real or come in complex conjugate
pairs.
The poles are in the left half plane [12].
The passivity constraint in terms of the transfer matrix (4) is
equivalent to being Positive-Real (PR) which means that:
(5)
Where denotes complex conjugate. This constraint can be
verified by ensuring that all the eigenvalues of the real section
of the admittance matrix are nonnegative in the whole region
of frequency.
(6)
Where: G(j ) =
eal (Y(j ))
III. MODE – REVEALING TRANSFORMATION
A. Rationale
Any nonsingular matrix T will preserve the eigenvalues of a
matrix Y by the similarity transformation
subjecting
to rational modeling by VF and subsequent
passivity enforcement with relative error control, one can
therefore achieve that the small eigenvalues become more
accurately represented. The model of
must next be
transformed back to the original (physical) basis to give a
model representation for Y.
B. Preservation of Symmetry and Passivity
Diagonalizing Y gives
(8)
In the general case with a frequency-dependent T, it is not
possible to fit the eigenvalues using a stable rational model.
Instead, we will have to use a constant, real transformation
matrix and apply it at all frequencies. This gives us an
eigenvalue decomposition matrix
at the identified
frequency
.
To obtain a real matrix, we follow Brandwajns approach
[16] of rotating each eigenvector by an angle to minimize
its imaginary part in the least – squares sense. For each rotated
eigenvector
(9)
The imaginary part Im
(10)
Differentiating (10) with respect to
= 0 gives the final solution [16]
[ISSN: 2045-7057]
and solving for df ⁄ dθ
(11)
Since can present a minimum and maximum value, we
evaluate (10) at
and at
to determine the correct
value. Finally, we discard the imaginary part
.
A difficulty with the proposed approach so far is that the
transformation
will neither preserve the symmetry nor the
passivity of Y. to overcome this problem, we will approximate
with a real, orthogonal matrix Q. Since orthogonality
implies
, we obtain for (7)
(12)
Using the rule for the transpose of a matrix product, we
obtain
(7)
By making a proper choice of T, it is possible to achieve
that the small eigenvalues become more observable in the
elements of
than in Y. we will therefore refer to the
transformation as a mode revealing transformation (MRT). By
has norm
(13)
Which proves that the transformed matrix is symmetrical.
Q can conveniently be chosen as the matrix Q of the QR
factorization,
where Q is orthogonal and R is an
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INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014
upper triangular matrix. However, in our case, we simply try to
calculate a Q which is as close to T as possible. This problem
is addressed by the orthogonal procrustes problem
(14)
Which seeks to calculate the orthogonal matrix Q which
rotates B as close as possible into A.[15] With the obtained
transformation matrix Q, we calculate the matrix using
(7)
which is fitted by the rational model (3) using vector fitting
(VF).
C. Passivity Enforcement: Fast Modal Perturbation
In [17], it was proposed to perturb the rational model such
that the eigenvalues of Y are perturbed in relation to their size.
Diagonalizing Y gives
(15)
(16)
(22)
IV. WIDEBAND TRANSFORMER MODELING
A. Frequency Sweep Measurements
We use frequency sweep measurements for wideband
modelling of power transformers. In this paper the 30 – KVA
Define abbreviations and acronyms the first time they are
distribution transformer described in [18] is used. See Fig. 1.
Fig. 2 shows the measured elements of the
Y using 348
logarithmically spaced samples between 100HZ and 1MHZ.
B. Modeling
Fig. 3 shows the elements of
as obtained via a
transformation matrix Q calculated at 100HZ. The sample plot
shows a rational approximation of
using N = 60 poles
obtained by VF with inverse magnitude weighting. All
elements are seen to be represented very accurately.
Fig. 4 shows the eigenvalues of
(which are identical to
those of Y). The admittance matrix of the model has
eigenvalues that are very close to the eigenvalues of the data.
Postmultiplying (15) with T and taking first order
derivatives gives for each eigenpair (
)
(17)
Ignoring terms involving
gives
and replacing
with (16)
(18)
Fig. 1. Two-winding transformer(30 KVA).
The perturbation size is made inversely proportional to the
eigenvalue size by using a weighting that is equal to the
inverse of the eigenvalue magnitude [12]
(19)
Equation (19) is built for all modes i and is used as a
replacement for (16). The matrix diagonalization is introduced
in order to reduce the number of free variables, leading to
FMP.
D. Transformation Back to the Physical Domain
The extracted model is finally transformed back into the
physical domain by the inverse transformation of (7), i.e.,
(20)
Application of (20) gives the final mode as
Fig. 2. Elements of Y.
(21)
With
[ISSN: 2045-7057]
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INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014
Fig. 5. Measured versus Simulated voltages.
Fig. 3. Elements of
.
Fig. 4. Eigenvalues of Y.
C. Time Domain Simulation
Fig. 5 shows an example of time – domain results for the
given transformer. In a laboratory measurement, a step-like
voltage was applied to terminal 1(Fig. 1) with terminal #2 and
#3 grounded, and the voltage responses on terminals #4,#5,
and #6 were recorded [18]. The measured voltage on terminal
#1 was then applied as an ideal voltage source in an EMTPlike time-domain simulation using the rational model. It is
observed that the model obtained via the new approach closely
agrees with the measurement whereas the model extracted by
the conventional approach has large errors.
admittance matrix Y. The method is based on applying a
similarity transformation matrix Q to Y to make the
eigenvalues of Y more observable. The transformation matrix
is calculated via eigenvalue decomposition, eigenvector
rotations, and orthogonalization.
1) The transformed matrix
remains symmetrical and it
preserves the eigenvalues and the passivity characteristics of
Y.
2) Applying conventional VF and passivity enforcement
schemes to
with inverse least – squares weighting gives a
much better representation of the small eigenvalues of Y than
what is achieved when applying these techniques directly to Y.
3) The extracted pole – residue model of
is transformed
back to the physical domain by applying the inverse
transformation to each residue matrix individually, giving a
model of .
The new method was demonstrated for the frequency –
domain modeling of a six – terminal distribution transformer
using N = 60 poles. The extracted models were shown to
provide a more accurate representation of the small
eigenvalues of Y compared to a conventional approach based
on direct application of VF Y to, thereby avoiding error
magnification in applications with high impedance
terminations. The new approach was also found to be about 10
times faster compared to the use of MVF.
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V. CONCLUSION
[3]
This paper has introduced a new, straightforward approach
for terminal modeling of multiport systems. The approach is
very useful when the system is characterized by a large ratio
between the large and small eigenvalues of the terminal
[4]
[ISSN: 2045-7057]
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INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014
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[ISSN: 2045-7057]
Ehsan Elahi was born in Kazeroon, Iran, on
September 20, 1989. He received his B.S. degree in
Electrical Engineering from Islamic Azad University,
Kazeroon Branch, Kazeroon, Iran, in 2011, and., M.S.
degree in Electrical Engineering (Power Electronics)
from Kazeroon University, Iran, in 2014. He is
currently as a Lecturer in Department of Electrical Eng,
Kazeroon Branch, Islamic Azad University, Kazeroon,
Iran. His research areas of research interest are power
systems, power systems analysis, transient, and control systems.
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