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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 ferroresonancecomputations 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

ERRORESONANCE is a nonlinear phenomenon occur-

ring 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 pseudo-

periodic) behavior of the state variables. For instance, the

expression of the magnetic flux in the magnetizing branch of

the nonlinear inductance is :

( ) ( ) ( )( )

++=Kk

skck tktkt sincos ,,0 (1)

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.

( )( ) ( ) dttktiT

IT

ck cos2

0, = (2)

and with similar expressions for the sine terms skI , and for

the DC component 0I . The coefficients ckI , , skI , , 0I are

nonlinear functions of all the coefficients ck, , sk, , 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 kE and complex

impedance kZ ) for the different frequencies kof 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 nonlinearcomponent, 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 (notnecessarily identical to the set K) [1-4].

.

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

Magnetic Cores Modeling for Ferroresonance

Computations using the Harmonic Balance

Method

N. A. Janssens, Senior Member, IEEE

F


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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 energeticbackground : 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 representthe magnetic characteristic. In this case, the terms of the

Jacobian may be expressed by the harmonic components of

( )( ) ( )( )

d

tidti = [8]. For instance, let us consider the term

ck

ckI

,2

,1

with 01 k , 02k , 21 kk . Using (1), we obtain

successively :

( ) dttkd

id

T

I

ck

T

ck

ck

1cos

2

,20

,2

,1

=

( ) ( ) ( ) dttktkiT

T 1cos2cos

2

0 =

ckkckk II ,21,212

1

2

1+ +=

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 closeto 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 FG and HG 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 differslightly 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 FG and HG

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 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 HW and the

eddy current losses FW :

FH WWW += (3)

of the form :

( )=loops

HH wfW maxmin , (4)

( )rmsFF UWW = (5)The symbols W (upper case letters) designate the powers

(energy per second) while the w (lower case) designate theenergy 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), rmsU is the r.m.s. voltage across the

magnetizing branch.

A. Eddy current losses

We suppose that the function0

FW giving the eddy

currents losses for a sine wave of the flux (directly related to

the voltage) at the grid frequency 0f as a function of its peak

value max is known. The associated eddy current

conductance ( )max0FG is then related to

0FW by :

( ) ( ) ( )

2

22max

20

max0

max0

f

GW FF = (6)

Let us now consider a distorted wave characterized by its

frequency components { }k (peak values) where thenumbers kmay be fractions. The peak value max of a sine

wave corresponding to the same r.m.s. value of the voltage is

given by :


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=k

kk22

max (7)

Hence, by taking :

=

k

kFF kGG220 (8)

an expression of 2rmsFF UGW = 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 function0

FG with a sufficient accuracy.

B. Hysteresis losses

The purpose is to determine a conductance HG 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 expressed using a conductance ( )max0

HG functionof the maximum flux of the evolution for the grid frequency

200 =f . The loss for one cycle is related to the

conductance by the relation :

( ) ( )max02

max0max0

HH Gw = (9)

For practical cases, the representation of the function0

HG

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 ofthe 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 themechanism 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 1w :

( )maxmin1 ,wwH = (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 :

( )dmH ww ,2= (11)

Since, in general, the loss for unsymmetrical loops is not

known, we must restrict ourselves to the use of the

function ( )max0Hw given by (9) for symmetrical loops.

We will assume that :

a) for 0 >d (large loops), the extreme values are located

on the limit cycle. Hence,

( ) ( ) ( )( )min0max0maxmin12

1, HH www =

or ( ) ( ) ( )( )dmHdmHdm www += 0022

1, (12)

b) for 0


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be done between the highest flux and current obtained by the

low 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 wastaken 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 1I 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 cI1 exhibits a more usual hysteresis loop shape.

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

0 500 1000 15000.3

0.4

0.5

0.6

0.7

current

flux

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 1R and leak inductance

1L with respect to the secondary and tertiary windings may be

computed by regression : from the recordings relative to the

primary winding feeding, the parameters ,1R and ,1L ,

where = 2 or 3 according to the reference to the secondary

of the tertiary winding, are chosen in order to minimize the

integral :

dtudt

idLiRuJ

T

=

0

21

,11,11

Figure 4 shows the results for the inductance 1L (circle

markers for 2,1L and square markers for 3,1L ). It may be seen


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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

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.

120 140 160 1800

25

50

75

100

secondary or tertiary voltage

seriesinductance

Fig. 4. Primary leak inductance

0 50 100 150 2000

0.03

0.06

0.09

0.12

secondary or tertiary voltage

shuntconductance

Fig. 5. Shunt conductance for the modeling of the core losses

V. CONCLUSIONS

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 acontinuous 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.


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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, asshown in [2, 3, 5].

VI. REFERENCES

[1] N. Janssens, Calcul des zones d'existence des rgimes ferrorsonantspour un circuit monophas , IEEE Canadian Communications andPower Conference, Montral 18-20 Oct. 1978, Cat No 78 CH 1373-0REG 7, pp 328-331

[2] N. Janssens, A. Even, H. Denol, 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

[3] N. Janssens, V. Vanderstockt, H. Denol, 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

[4] N. Janssens, Th. Van Craenenbroek, D. Van Dommelen, F. Van De

Meulebroeke, Direct calculation of the stabili ty domains of three-

phase ferroresonance in isolated neutral networks with grounded-

neutral 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)

[5] 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)

[6] 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.

[7] N. Janssens, Static models of magnetic hysteresis , IEEE

Transactions on Magnetics, Vol MAG-13 n5, Sept 1977 pp

1379-1381.

[8] N. Janssens, Hystrsis magntique et ferrorsonance. Modles

mathmatiques et application aux rseaux de puissance , PhD thesis,

UCL (Universit Catholique de Louvain, Belgium), June 1981

[9] T. Nakata, Y. Ishihara, M. Nakano, Iron losses of silicon steel core

produced by distorted flux , Electrical Eng. in Japan, Vol 90, n1,

1970, pp 10-20

VII. BIOGRAPHY

Nol Janssens (S2001) 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.

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