Energy and Power En gi neering, 2010, 2, 254-261
doi:10.4236/epe.2010.24037 Published Online November 2010 (
Copyright © 2010 SciRes. EPE
Effect of Metal Oxide Arrester on Chaotic Behavior of
Power Tr ansformers
Ataollah Abbasi1, Mehrdad Rostami1, Seyyed Hamid Fathi2, Hamid R. Abbasi3, Hamid Abdollahi1
1Shahed University, Tehran, Iran
2Amir Kabir University, Tehran, Iran
3Iran University of Science and Technology, Tehran, Iran
Received April 28, 2010; revised July 8, 2010; accepted August 16, 2010
This paper investigates the Effect of Metal Oxide Arrester (MOV) on the Chaotic Behaviour of power trans-
formers considering nonlinear model for core loss of transformer. The paper contains two parts. In part (1):
effect of nonlinear core on the onset of chaotic ferroresonance in a power transformer is evaluated. The core
loss is modeled by a third order power series in voltage. in part (2): Effect of Metal Oxide Arrester(MOV) on
results in part (1) will be studied. The results reveal that the presence of the arrester has a mitigating effect on
ferroresonant chaotic overvoltages. resulted bifurcation diagrams and phase plane diagrams have also been
Keywords: Chaotic Ferroresonance, MOV, Nonlinear Core Loss, Bifurcation Diagram, Phase Plane Diagram
1. Introduction
Ferroresonance is a complex nonlinear electrical phe-
nomenon that can cause dielectric and thermal problems
to components power system. Electrical systems exhib-
iting ferroresonant behaviour are categorized as nonlin-
ear dynamical systems. Therefore conventional linear
solutions can not be applied to study ferroresonance. The
prediction of ferroresonance is achieved by detailed
modelling using a digital computer transient analysis
program [1]. Ferroresonance should not be confused with
linear resonance that occurs when inductive and capaci-
tive reactances of a circuit are equal. In linear resonance
the current and voltage are linearly related and are fre-
quency dependent. In the case of ferroresonance it is
characterised by a sudden jump of voltage or current
from one stable operating state to another one. The rela-
tionship between voltage and current is dependent not
only on frequency but also on other factors such as sys-
tem voltage magnitud e, initial magnetic flux condition of
transformer iron core, total loss in the ferroresonant cir-
cuit and moment of switching [2]. Ferroresonance mey
be initiated by contingency switching operation, routine
switching, or load shedding involving a high voltage
transmission line. It can result in Unpredictable over
voltages and high currents. The prerequisite for ferrore-
sonance is a circuit containing iron core inductance and a
capacitance. Such a circuit is characterized by simulta-
neous existence of several steady-state solutions for a
given set of circuit parameters. The abrupt transition or
jump from one steady state to another is triggered by a
disturbance, switching action or a gradual change in val-
ues of a parameter. Typical cases of ferroresonance are
reported in Refs. [4-6]. Theory of nonlinear dynamics
has been found to provide deeper insight into the phe-
nomenon. Refs. [7-9] are among the early investigations
in applying theory of bifurcation and chaos to ferroreso-
nance. The susceptibility of a ferroresonant circuit to a
quasiperiodic and frequency locked oscillations are pre-
sented. in Ref [10]. The effect of initial conditions is also
investigated. Ref. [11] is a milestone contribution high-
lighting the effect of transformer modeling on the pre-
dicted ferroresonance oscillations. Using a linear model.
authors of Ref. [12] have indicated the effect of core
loss in damping ferroresonance oscillations. The impor-
tance of treating core loss as a nonlinear function of
voltage is highlighted in Ref. [13]. An algorithm for cal-
culating core lo ss from no-load characteristics is given in
Ref. [14]. The mitigating effect of transformer with lin-
ear core loss model connected in parallel to a MOV ar-
rester is illustrated in Ref. [15]. Ferro resonance in ca-
pacitive voltage transformer using nonlinear models
Copyright © 2010 SciRes. EPE
along with (Current/Voltage) limiting filters has been
discussed in Refs [16-19]. Evalua tion of chaos in voltage
transformer, effect of resistance key on the chaotic be-
havior voltage transformer and and subharmonics that
produced with ferroresonance in this type transformer are
studied in Refs. [20-23]. In all of done researches until
now, the effect of Metal Oxide Arrester on the chaotic
ferroresonance behaviour of power transformer with
nonlinear core loss model have not be done. Therefore,
present paper addresses the Effect of Metal Oxide Ar-
rester on chaotic behav iour of power transformer consid-
ering nonlinear core loss.
2. Circuit Descriptions and Modeling
without MOV
The three-phase diagram for the circuit is shown in Fig-
ure 1.
The 1100 kV transmission line was energized through
a bank of three single-phase as reported in autotrans-
formers Ref. [3]. Ferroresonance occurred in phase A
when this phase was switched off on the low-voltage side
of the autotransformer; phase C was not yet connected to
the transformer at that time. The autotransformer is mod-
eled by a T-equivalent circuit with all impedances re-
ferred to the high voltage side. The magnetization branch
is modeled by a nonlinear inductance in parallel with a
nonlinear resistance which represented by nonlinear
saturation characteristic ()
and nonlinear hys-
teresis and eddy current characteristics ()
vi, re-
spectively. The hysteresis and eddy current characteris-
tics are calculated from the no-load characteristics by
applying the algorithm given in Ref. [14]. The iron core
saturation characteristic is given by:
 (1)
For three different values of q, Equation (1) is plotted
in Figure 2 Coefficients s1 and s2 are selected as fol-
11.0067, .0001
7.0067, .001
5.0071 .0034
for qss
for qss
for qss
 
 
 
The exponent q depends on the degree of satur ation. It
has been found that for adequate representation of the
saturation characteristics of a power transformer the ex-
ponent q may take the values 5, 7, and 11 in Ref. [8].
According to Figure 2, for q = 11, flux linkage curve
via magnetization current in saturation section has less
slope than what can be seen for q = 5&7. Thus, ferrore-
sonance effects appear earlier. Besides, increasing q will
increase both number of stable and unstable points. Then
Figure 1. .System modeling.
Figure 2. Nonlinear saturation characteristic for three
values q.
changing control parameters may expose unstable points
like saddle points and chaotic attractors as well.
The core loss is modeled by a switched resistor; which
effectively reduced the core loss resistance by a factor of
four at the time of onset of ferroresonance. In this paper,
the core loss model adopted is describ ed by a third order
power series whose coefficients are fitted to match the
hysteresis and eddy current nonlinear characteristics
given in [3]:
01 23
Rmm mm
ihhvhvhv  (2)
Per unit value of ()
i for this case given in (3)
.000001 .0047 .0073 .0039
ivvv (3)
Using presented nonlin ear functions of core losses and
current-flux linkage, plot of ic (total current) vs. λ in
nominal operation for different values of q (5,7,11) are
shown in Figures 4, 5, 6 respectively.
Core current-flux linkage curve includes both eddy
current and hysteresis losses. As it can be seen, increas-
ing q will decrease loop width, which consequences re-
ducing losses. Existing even order power loss function
causes unsymmetrical loop shape which can be seen in
last presented figures.
The circuit in Figure 1 can be reduced to a simple
form by replacing the dotted part with the Thevenin
equivalent circuit as shown in Figure 7.
Copyright © 2010 SciRes. EPE
Figure 3.V-I characteristic of nonlinear core loss.
Figure 4. Core current – Flux linkage curve for q=1 1 wi-
tout MOVand nonminal operation.
Figure 5. Core current – Flux linkage curve for q = 7 witout
MOVand nonminal operation.
Transformer data in [3] Eth and Zth are:
130.1 ;1.012430.5
th th
The resulting circuit to be investigated is shown in
Figure 8 where Zth represents the Thevenin impedance.
The behavior of this circuit can be described by the fol-
Figure 6. Core current – Flux linkage curve for q = 5 witout
MOVand nonminal operation.
Figure 7. Thevenin circuit of Figure 1.
lowing system of nonlinear differential equations:
The behavior of this circuit can be resented by the fol-
lowing system of nonlinear differential equations:
 
12 0123
ss hhPhPhP
pV C
 
With these parameters:
635.1,78.72 ,8067,
.0188, .07955,
.0014556.68 .
basebase base
Spu pu
In (4), (5)
, P
and c
V has been taken as state
variables as follows:
12 3
; ;
 (6)
Px x
 
11210122 23 2
xsx hhxhxhx
Px C
 
 
23 1
21 23
q q
th c
et V PRssh hPhPhPLsPqsP
PLh hPhP
 
 
 
 
 
 
23 1
32 211210122232212212
21 2232
q q
Px Lh hxhx
 
 
 
 
Copyright © 2010 SciRes. EPE
2.1. Simulation Results and Discussion for
Circuit without MOV
Time domain simulations were performed using fourth
order Runge_/Kutta method and validated against Matlab
Simulink results. The initial conditions derived from
steady-state solution of Matlab are:
12 3
0,0 ;1.67;1.55)
xpuxpu 
The major analytical tools that in this paper are used to
study chaotic ferroresonance are Phase Plane and bifur-
cation di a gram.
The phase plane analysis is a graphical method, in
which the time behavior of a system is represented by the
movement of state variables of the system in a state
space coordination against time. As time evolves, the
initial poin t follows a trajectory. If a trajectory closes on
itself, then the system produces a periodic solution. In
the chaotic system, the trajectory will never close to it-
self to shape cycles. A bifurcation diagram is a plot that
displays single or multiple solutions (bifurcatio ns) as the
value of the control parameter is in creased.
Figures 9-11 shows phase plane diagrams of chaotic
behaviour for various values of q.
The chaotic behaviour has been intensified with in-
creasing values of q and this behaviour has been shown
in these figures.
In Figures 12-14 core current – flux linkage curves in
chaotic ferroresonance mode has been shown. Appearing
high frequency oscillations, causes loop width wider than
normal operation of power transformer for all q values.
To study voltage effect on system behavior, system
has been enegrgized with different supply voltages. Fig-
ures 12-14 show resulted core current-flux linkage
curves. All figures are obtained in normal operation of
transformer and not in ferroresonance or other abnormal
situations. Also Figures 15-17 show the bifurcation dia-
gram of chaotic behaviours for these three of values of q.
The nonlinear core loss model creats a mitigation on
the chaotic ferroresonance behavior in transformer. This
Figure 8. Circuit of ferroresonance investigations without
Figure 9. Phase plane diagram for q = 5 without MOV.
Figure 10. Phase plane diagram for q = 7 without MOV.
Figure 11. Phase plane diagram for q = 11 without MOV.
mitigation is due to delaying terms in nonlinear model.
Also presence of nonlinear model for core loss in dy-
namic equation shows that resulted chaotic ferroreso-
nance behavior of system in comparison with the one
with linear core loss model, shows that routing to chaos
from period doubling will be smother, more regular and
also more distinguishable.
The hatched area under 1.p.u in bifurcation diagrams
are due to transmission lines parameters effects. In Ref
[8] results has been simplified by elimination of trans-
mission line parameters, which hatched area cannot be
3. Circuit Descriptions and Modeling with
The system considered for analysis consists the MOV
Copyright © 2010 SciRes. EPE
Figure 12. Core current – Flux linkage curve for q = 5 wi-
tout MOV in chaotic condition.
Figure 13. Core current – Flux linkage curve for q = 7 wi-
tout MOV in chaotic condition.
Figure 14. Core current – Flux linkage curve for q = 1 1
witout MOV in chaotic condition.
arrester connected across the transformer winding. The
related equivalent circuit is as shown in Figure 18.
The nonlinear characteristic of the arrester is modelled
as [15]:
ov m
 (10)
i, is the current through the arrester in p.u.,
V is the voltage across the transformer winding, k and
are arrester parameters. The differential equation for
the circuit in Figure 18 can be derived as:
  
23 1
2120123 212
q q
th cd
et V PRssh hPhPhPLsPqsP
 
 
 
 
 
 
 
 
01 23
, p
and c
v are assumed as state variables as follows:
12 3
; ;
 (13)
Px x
221 012232112
cs s
E VLaxqbxxRhhhxhxaxbxx
dx x
 
 
 
2012 223211
Copyright © 2010 SciRes. EPE
Figure 15. Bifurcation diagram for q = 5 without MOV.
Figure 16. Bifurcation diagram for q = 7 without MOV.
Figure 17. Bifurcation diagram for q = 11 without MOV.
Figure 18. Circuit of ferroresonance investigations with
The numerical param ete rs for M OV is:
25 2.5101k
Figures 19, 20, 21 show the time domain simulation
phase plane diagram of system states and curve of core
current-flux linkage, including arrester for E = 5;
The MOV cuts off over voltage so a result of ferrore-
Figure 19. Time domain signal for E = 5 and q = 11 With
Figure 20. Phase plane diagram for E = 5 and q = 11 With
Figure 21. Core current – Flux linkage curve for q = 11
while MOV exists in circuit.
sonance chaotic unstable trajectory and eliminates chaos
Figures 22, 23, and 24 are the bifurcation diagrams by
applying MOV surge arrester that show chaotic region
mitigates. The tendency for chaos exhibited by the sys-
tem increases while q increases too
Refer to Figures 21-24 the MOV in some cases may
cause ferroresonace dropout. For q = 5, 7 MOV chaotic
condition chang es to periodic behavior in system, but for
q = 11 independent chaotic regions which can be created
under MOV nominal voltage have survived. Also the
pollutions under 1.p.u in bifurcation diagrams due to
effect of transmission lines parameters, despite existence
of MOV would not be su rvived.
Copyright © 2010 SciRes. EPE
Figure 22. Bifurcation diagram for q = 5 with MOV.
Figure 23. Bifurcation diagram for q = 7 with MOV.
Figure 24. Bifurcation diagram for q = 11 with MOV.
4. Conclusions
The dynamic behavior of a transformer is characterized
by multiple solutions. The system shows a greater ten-
dency for chaos for saturation characteristics with lower
knee points, which corresponds to higher values of ex-
ponent q. The presence of the arrester results clamping
the Ferroresonant over voltages in studied system. The
arrester successfully suppresses or eliminates the chaotic
behaviour of proposed nonlinear core loss model. Inclu-
sion of nonlinearity in the core loss reveals that so lutions
are more realistic and been improved when compared
with linear models.
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Appendix: Nomenclature
a, b, c index of phase sequence
h0, h1, h2, h3, coefficient for core loss nonlinear func-
n index for the neutral connection
s1 coefficient for linear part of magnetizing curve
s2 coefficient for nonlinear part of magnetizing curve
q Inde x of n onl inearity of the magnet i zi ng curve
Zth Thevenin’s equivalent impedance
C linear capacitor
Rm core loss resistance
L nonlinear magnetizing inductance of the trans-
i instantaneous value of branch current
v instantaneous value of the voltage across a branch
eth(t) instantaneous value of Thevenin voltage
e instantaneous value of driving source
p time derivative operator
Eth R.M.S. value of the Thevenin voltage source
x state variable
λ flux linkage in the nonlinear inductance