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Magnetic Cores Modeling for Ferroresonance
Computations using the Harmonic Balance
Method
N. A. Janssens, Senior Member, IEEE

Abstract-- The determination of the risk of ferroresonance in
actual HV or MV networks and the design of damping devices
need the use of accurate models. In this frame, the nonlinear parts
of the circuits, i.e. the magnetic cores, require a special attention.
This paper describes a modeling of the magnetization curve and
the core losses appropriate for ferroresonance computations using
the harmonic balance method. Tests performed in order to get the
parameters are discussed.
Index Terms— Ferroresonance, Harmonic balance method,
High Voltage, Hysteresis, Magnetization, Modeling, Saturation,
Voltage Transformer
I. INTRODUCTION
F
ERRORESONANCE is a nonlinear phenomenon occurring in electrical circuits involving at least one or several
saturable reactors, capacitances and a power supply. Often,
the saturable reactors are inductive potential transformers. This
phenomenon is characterized by the possible existence of
several stable regimes. These regimes may be either periodic,
with a base frequency equal to the power supply frequency or
to a sub multiple of it, pseudo-periodic or chaotic.
The harmonic balance method, which is a particular case of
the Galerkin method, has proven powerful for the study of
periodic, but also pseudo-periodic regimes, in the above
mentioned circuits [e.g. 1-5]. For this method, a limited
Fourier series is used to represent the periodic (or pseudoperiodic) behavior of the state variables. For instance, the
expression of the magnetic flux in the magnetizing branch of
the nonlinear inductance is :
 t    0    k ,c cosk t    k ,s sin k t 
(1)


I k ,c 
2 T
i t  cosk t  dt
T 0
(2)
and with similar expressions for the sine terms I k , s and for
the DC component I 0 . The coefficients I k , c , I k , s , I 0 are
nonlinear functions of all the coefficients  k , c ,  k, s ,  0 .
The harmonic balance method consists in introducing the
limited Fourier series in the differential equation of the circuit
and forcing to zero the contributions to each considered
harmonic component. So, an algebraic set of nonlinear
equations in the Fourier coefficients is obtained and may be
solved by using a general purpose routine.
The linear part of the circuit may be represented by its
Thevenin equivalents (voltage source Ek and complex
impedance Z k ) for the different frequencies k of the set K .
The use of Thevenin equivalents allows reducing the number
of equations to be solved : there is only one (complex)
equation for each harmonic component for each nonlinear
component, instead of one equation for each harmonic
component for each reactive element (linear or nonlinear). It
also facilitates the choice of an initial approximation of the
solution to be introduced in the computation program.
The use of the harmonic balance method may be extended
to find directly the domain limits in some parameter space of
the various kinds of ferroresonant regimes. This extension
consists in adding one equation to the above mentioned
algebraic set equating to zero the determinant of the Jacobian
of the harmonic balance equations relative to a set K  (not
necessarily identical to the set K ) [1-4].
.
kK
The corresponding Fourier coefficients for the current in this
branch are given by integration from the magnetic
characteristic i  of the inductance :
N. Janssens is with ELIA, Belgian Transmission System Operator, 125,
Rodestraat, 1630 Linkebeek, Belgium (e-mail: [email protected]) and
with the University of Louvain at Louvain La Neuve, Belgium.
II. BASIC MODELING PRINCIPLES
To get accurate results when studying a practical situation,
all the relevant components of the system must be modeled
adequately. In particular, the nonlinear reactors need a special
attention because the ferroresonance phenomenon pushes these
elements in high saturation beyond the validity domain of the
usually available data. The present paper is devoted to the
presentation of a saturable reactor modeling able to get
2
quantitative accurate results when used to study practical
situations.
The iron core magnetization characteristic has of course a
great influence, since the ferroresonance phenomena are
strongly related to this nonlinear function. Very often, the
question to be answered for estimating the risk of
ferroresonance or designing a damping circuit is : what is the
lower bound of the voltage source for which a given kind of
ferroresonant regime exists ? This question has an energetic
background : to what extent can the voltage source bring
enough energy to compensate the losses of a ferroresonant
regime ? From computation and tests, it may be concluded that
the system losses have a great importance on the lower limit of
the voltage source interval where a given ferroresonant regime
exists. A precise modeling of these losses is of prime
importance. It is to be noted that, when the voltage is not close
to these interval limits, the waveforms are not very sensitive to
these losses. Consequently, a validation solely based on
waveform comparisons is not very relevant.
For the time simulation of the system, a static magnetic
hysteresis model, like those described in [6,7] may be used,
associated with a conductance to represent the eddy current
losses and series resistances and leakage inductances. Such a
model could also be used for the harmonic balance method and
proved to provide good results for a low voltage laboratory
test [8]. However, the computation time may be greatly
reduced by using a univocal function i  to represent the
magnetic characteristic. In this case, the terms of the Jacobian
may be expressed by the harmonic components of
d i t 
[8]. For instance, let us consider the term
i  t  
d
 I k1, c
with k1  0 , k 2  0 , k1  k 2 . Using (1), we obtain
  k 2, c
successively :
 I k1, c
  k 2, c


2
T

2
T
T
0
d i 
cosk1  t  dt
d    k 2, c
 0 i  cosk 2  t  cosk1  t  dt
T
1
1
I k 1k 2,c  I k1k 2 ,c
2
2
Section IV will pay attention to the measurement of the
magnetic characteristic.
In order to have a clean convergence of the iteration
process, the function i   must be continuous. On the other
hand, the modeled magnetic characteristic must be very close
to the measured curve. For this purpose, an analytical
expression like, for instance, a polynomial will, in general,
show discrepancies with respect to the real curve or exhibit
oscillations in the ( i   ) plane. Therefore, we found more
adequate to use a parabolic spline, consisting of successive
parabola segments with a continuous slope at the nodes.
Section IV will show an example of such a modeling.
Besides the magnetization curve, another important aspect
is the modeling of the core losses. During the computation
process to solve the algebraic system of equations, at the
beginning of each iteration, a set of Fourier coefficients is
given from the previous iterations (for the first iteration, the
user or an auxiliary routine provides an initial point). This
gives a flux (and voltage) behavior of the core magnetizing
branch. From there, it is possible to determine the value of two
linear conductances G F and GH such that the losses in these
conductances are equivalent respectively to the eddy current
losses and the hysteresis losses according to an appropriate
model. Doing so, the waveforms will differ slightly from those
obtained using a model to be used for a time simulation.
However, these differences will be very small considering that
the cores are made of soft magnetic materials with limited
losses and that the saturation has a much larger effect on the
waveform. The important point is that, globally for the whole
oscillation period, the core losses are accurately modeled. The
computation of the conductances G F and GH from basic
data is developed in section III and an example of core losses
as a function of the saturation is shown in section IV.
III. MAGNETIC CORE LOSSES
Experimental studies [9] have shown that the power W
dissipated in silicon steel plates, for a distorted wave and in the
frequency range under interest here, can be written as the sum
of two terms relative to the hysteresis losses WH and the eddy
current losses W F :
W  WH  WF
of the form :
WH  f   wH  min , max 
(3)
(4)
loops
(5)
WF  WF U rms 
The symbols W (upper case letters) designate the powers
(energy per second) while the w (lower case) designate the
energy dissipated per cycle. In (4), the sum deals with all the
hysteresis loops (major and minor), f is the fundamental
frequency of the oscillation, min and  max are the extreme
values of the flux. In (5), U rm s is the r.m.s. voltage across the
magnetizing branch.
A. Eddy current losses
We suppose that the function W F0 giving the eddy currents
losses for a sine wave of the flux (directly related to the
voltage) at the grid frequency f 0 as a function of its peak
value
 max
conductance
is known. The associated eddy current
GF0
WF0  max  
max  is then related to WF0
2 2
 2 f 0  max
G 0 
by :
(6)
2
Let us now consider a distorted wave characterized by its
frequency components   k  (peak values) where the
numbers k may be fractions. The peak value  max of a sine
wave corresponding to the same r.m.s. value of the voltage is
given by :
F
max
3
 max 
 k 2  2k
(7)
k
Hence, by taking :


G F  G F0   k 2  2k 


 k

(8)
2
an expression of WF  GF U rms
of the form (5) is obtained
corresponding to the data (6) for a sine wave.
Measurements on magnetic cores showed that a polynomial
of the third degree is able to represent the function G F0 with a
sufficient accuracy.
B. Hysteresis losses
The purpose is to determine a conductance G H valid for
any periodic evolution using easily obtainable data. These data
are :
a) the hysteresis losses for a symmetric evolution with respect
to the origin for a pure sine applied voltage. These losses are
0
expressed using a conductance GH
max  function of the
maximum flux of the evolution for the grid frequency
f 0   0 2 . The loss for one cycle is related to the
conductance by the relation :
0
2
0
max     0  max
 max 
wH
GH
(9)
0
For practical cases, the representation of the function G H
by a third order polynomial has been found to be sufficiently
accurate.
b) the smallest value  0 of the half amplitude (peak to peak)
of a closed loop such that both extremes are located on the
limit cycle. This value illustrates the “speed” to approach to
the other side of the limit cycle after a change in the sense of
the flux and current variation. This value may be estimated by
the examination of the hysteresis loops. For lack of this, one
may choose half of the flux of a point located in the saturation
knee.
To represent a magnetic evolution, we will use the Preisach
model, taking the flux as independent variable and assuming
that the Girke coefficient is infinite [8 pp 82-88]. In this frame,
a periodic evolution will be composed of a set of closed
hysteresis loops. For a given periodic evolution of the flux
(available at the beginning of each iteration when using the
harmonic balance method), a first computation step is to
associate all the extreme values by pairs, according to the
mechanism of the Preisach model.
The loss for one cycle is the sum of the losses for each
individual loop. The loss of one loop is a function of its
minimum and maximum values, expressed by introducing a
function w1 :
wH  w1  min ,  max 
(10)
In the following, we will use an alternative formulation making
use of the mean flux  m and the half width  d of the loops :
wH  w2  m ,  d 
(11)
Since, in general, the loss for unsymmetrical loops is not
known, we must restrict ourselves to the use of the
0
function wH
 max  given by (9) for symmetrical loops.
We will assume that :
a) for  d   0 (large loops), the extreme values are located
on the limit cycle. Hence,
1 0
max   wH0 min 
w1  min ,  max   wH
2
1 0
0
or w2  m ,  d   wH  m   d   wH
(12)
m  d 
2
b) for  d  0 (small loops), we will assume that w2 is the




product of the loss of a symmetrical loop and a function f to
take the unsymmetrical character into account :
0
(13)
 d   f  m 
w2  m ,  d   wH
The continuity of the loss for  d   0 defines the function f :
1
0
 m   0   wH0  m   0 
f  m  
wH
(14)
0
2 wH  0 


c) for  d   0 , the function w2  m ,  d  is continuous.
However, its partial derivative with respect to  d is not. In
order to have a clean convergence of the iterative
computations including this model, in a small band of  d
around 0 , an interpolation using a third order polynomial is
used in order to have the continuity of the function and its
derivative.
Finally, having defined the function w2  m ,  d  for all
loops by means of the available data as explained here above,
the conductance G H is given, for an evolution whose
fundamental frequency is f , by :
f
GH 
 w2  m ,  d 
2 2
2  f 0  k 2  2k loops
(15)
k
IV. MEASUREMENTS FOR PARAMETER IDENTIFICATION
The ferroresonance phenomena are associated, in general,
with a very high saturation of the magnetic cores. Hence, the
magnetic flux spreads largely outside the iron cores. As a
consequence, the core and windings design (location, shape),
as well as the transformer tank and the internal shields play a
role in the high saturation range.
Therefore, the best way to measure the model parameters is
to use a complete device fed through the HV primary side. Of
course, this requires measurements in a HV laboratory. If this
is not possible, measurements can be done on a magnetic core
fitted with two low voltage windings (one for the magnetizing
current, the second one for the voltage – and flux –
measurement). For the high saturation region, the asymptotic
behavior of the magnetization curve may be approached by a
straight line obtained as follows : the slope is the transformer
HV primary winding in air minus the primary leak inductance ;
the flux offset for zero current corresponds to the full
magnetization of the core. Some interpolation must be done
between the highest flux and current obtained by the low
4
In order to determine the magnetization curve and the
losses, an adjustable sine voltage was applied to the primary
winding. Its amplitude covered the interval from zero to about
3 times the nominal voltage. Primary voltage and current
waveforms were recorded as well as the secondary and tertiary
winding voltage. Figure 2 shows the relation between the
maximum flux and maximum current of these periodic
waveforms, either by integrating the secondary voltage to
obtain the flux (circle markers), or the tertiary voltage (square
markers). The first one is higher than the second one because
the secondary winding is located closer to the primary than the
tertiary. Similar tests were done by feeding the voltage
transformer through the secondary winding. In this case, the
capacitance of the primary bushing loaded greatly the voltage
transformer. The current in the magnetizing branch is then
obtained by the difference between the secondary and the
primary current. Hence, for evolutions in the unsaturated
region, the precision on the result is rather bad. The magnetic
characteristic obtained with the secondary winding feeding and
using the tertiary flux is also shown on figure 2 (cross
markers).
Figure 3 illustrates the construction of the parabolic spline
modeling the magnetization curve. The step curve represents
the slope (current vs. flux) of the broken line of figure 2
obtained for the primary winding feeding and the secondary
winding flux. Nodes (stars on fig. 3) are chosen, usually
among the data points, in order to build a broken line whose
integral approximates closely the magnetization curve. Starting
from an initial slope, this broken line is constructed by a
Fig. 1. Magnetic flux – current evolution
0.7
0.6
flux
voltage measurements and the asymptotic straight line of the
magnetic characteristic. However, the extrapolation of the
losses in the high saturation region may lead to significant
errors.
The results given below relate to a voltage transformer
designed for the 245 kV voltage level. The measurements were
done in a HV laboratory by feeding the device either by the
primary or by the secondary winding. Special care was taken
for the measurements : shielding of the connection cables
between the measurement points and the recording devices,
digital synchronous recording with a sampling rate of 50 kHz
(1000 points per period), same input impedance of the various
channels, check of the transmission delay of the channels. The
losses were also measured using a wattmeter. For unsaturated
periodic evolutions, the measurement given by the wattmeter
matched very well with the computation based on the voltage
and current waveforms. However, for highly saturated
evolutions, the power factor was about 0.02 and the error on
the measurement of the wattmeter reached about 50 %.
Since the primary winding has tens of thousands of turns,
its capacitance may not be neglected. Figure 1 shows the flux –
current recording for the nominal voltage applied to the
primary. The current I1 is the primary current, the flux being
obtained by integration of the secondary voltage. It may be
seen that the behavior of this inductive voltage transformer is
essentially capacitive. Taking into account the primary
winding capacitance (modeled by a shunt capacitance in
parallel with the magnetizing branch), the flux - corrected
current I1c exhibits a more usual hysteresis loop shape.
0.5
0.4
0.3
0
500
1000
1500
current
Fig. 2. Magnetization curve
progression from a zero flux in the direction of the saturated
region. In the small flux region, the broken line does not
attempt to approximate the step function, since the hysteresis
effect induces a lowering of the slope that is to be neglected in
the frame considered here.
The primary winding resistance R1 and leak inductance L1
with respect to the secondary and tertiary windings may be
computed by regression : from the recordings relative to the
primary winding feeding, the parameters R1, and L1, ,
where  = 2 or 3 according to the reference to the secondary
of the tertiary winding, are chosen in order to minimize the
integral :
2
T 
d i1

J 
 u  dt
 u  R1,  i1  L1, 
0  1
dt


Figure 4 shows the results for the inductance L1 (circle
markers for L1,2 and square markers for L1,3 ). It may be seen
that the leak inductance increases with the saturation level.
This results from the dispersion of the flux outside the core for
a high saturation level. It may also be seen that the leakage of
5
100
series inductance
75
50
25
0
120
140
160
180
secondary or tertiary voltage
Fig. 4. Primary leak inductance
0.12
shunt conductance
0.09
0.06
0.03
0
0
50
100
150
200
secondary or tertiary voltage
Fig. 5. Shunt conductance for the modeling of the core losses
V. CONCLUSIONS
Fig. 3. Construction of the parabolic spline
the primary winding with respect to the tertiary winding is
greater than with respect to the secondary winding, because
this last one is closer to the primary winding.
The recorded waveforms allow to compute the losses.
From the primary current and the secondary or tertiary voltage,
the core losses may be computed as a function of the voltage
amplitude. The ratio of the losses and the square of the voltage
give an equivalent conductance G . It may be seen on figure 5
(circle markers for the secondary voltage, square markers for
the tertiary voltage) that this conductance is far from being
constant.
The computation of the risk of ferroresonance and the
design of damping devices for practical situations need an
accurate modeling of the system. In particular, due to the high
saturation level reached in the iron cores, the saturable reactors
and transformers require a special attention. In this frame, the
most important aspects of these components are the saturation
curve and a good representation of the losses.
A suitable model for the magnetization curve is a parabolic
spline, made of a succession of parabola segments with a
continuous derivative. This allows a large flexibility to
reproduce very closely the experimental data. On the other
hand, the continuous character of its derivative leads to a clean
convergence process of the ferroresonant regimes
computations.
6
When using the harmonic balance method, the core losses
may be modeled by two linear conductances, one for the eddy
currents and the other for the hysteresis losses. Their value is
adapted at each iteration of the Fourier coefficients
computation to take into account the amplitude and the
waveform of the magnetic flux behavior.
This modeling succeeded to get an accurate determination
of the domains of existence in some parameter space of the
various ferroresonant regimes for HV and MV systems, as
shown in [2, 3, 5].
VI. REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
N. Janssens, « Calcul des zones d'existence des régimes ferrorésonants
pour un circuit monophasé », IEEE Canadian Communications and
Power Conference, Montréal 18-20 Oct. 1978, Cat No 78 CH 1373-0
REG 7, pp 328-331
N. Janssens, A. Even, H. Denoël, P-A. Monfils, « Determination of the
risk of ferroresonance in high voltage networks. Experimental
verification on a 245 kV voltage transformer », Sixth International
Symposium on High Voltage Engineering, New Orleans, Aug 28 - Sep
1, 1989, paper 11.03
N. Janssens, V. Vanderstockt, H. Denoël, P-A. Monfils, « Elimination
of temporary overvoltages due to ferroresonance of voltage
transformers : design and testing of a damping system », CIGRE
Session 1990, Paris, Report 33-204
N. Janssens, Th. Van Craenenbroek, D. Van Dommelen, F. Van De
Meulebroeke, « Direct calculation of the stability domains of threephase ferroresonance in isolated neutral networks with groundedneutral voltage transformers », IEEE Trans. on Power Delivery, Vol 11,
n° 3, 1546-1553, July 1996. (presented at the IEEE PES Summer
Meeting 1995, Portland, paper 95 SM 420-0 PWRD)
T. Van Craenenbroeck, D. Van Dommelen, C. Stuckens, N. Janssens,
P.A. Monfils, « Harmonic balance based bifurcation analysis with full
scale experimental validation », IEEE 1999 Transmission &
Distribution Conference, New Orleans, USA, 1999, CD-ROM
(6 pages)
N. Janssens « Mathematical modelling of magnetic hysteresis »,
Proceedings of the COMPUMAG Conference on the computation of
magnetic fields, Oxford, 31 March - 2 April 1976, pp 191-197.
N. Janssens, « Static models of magnetic hysteresis », IEEE
Transactions on Magnetics, Vol MAG-13 n°5, Sept 1977 pp
1379-1381.
N. Janssens, « Hystérésis magnétique et ferrorésonance. Modèles
mathématiques et application aux réseaux de puissance », PhD thesis,
UCL (Université Catholique de Louvain, Belgium), June 1981
T. Nakata, Y. Ishihara, M. Nakano, « Iron losses of silicon steel core
produced by distorted flux », Electrical Eng. in Japan, Vol 90, n°1,
1970, pp 10-20
VII. BIOGRAPHY
Noël Janssens (S’2001) was born in Louvain, Belgium, in 1948. He is
Electrical Engineer from the University of Louvain (1971) and obtained a
Ph.D degree in 1981 (modeling of magnetic hysteresis and study of
ferroresonance). From 1981 to 1983, he worked at ACEC (Charleroi) as head
for R&D in the On Load Tap Changer department. From 1978 to 1981 and
from 1984 to 1995, he was with Laborelec, where his main fields of interest
were the modeling, simulation and control of power systems. Since 1996, he
is at the Belgian National Dispatching. He is also teaching at the University
of Louvain (Louvain La Neuve) in the Electrical Engineering Department.