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Impacts of Neutral Earth Resistance on the Stability Domain of
Ferroresonance Modes in Power Transformers including MOV Surge
Arrester
HAMID RADMANESH
Electrical Engineering Department, Aeronautical University of Science&
Technology, Shahid Shamshiri Street, Karaj Old Road, Tehran, IRAN
Tel: (+98-21)64032128, Fax: (+98-21)33311672
[email protected]
Mehrdad Rostami
Electrical Engineering Department,Shahed University, End of Khalij-e-Fars High
way, Infront of Imam Khumaini holly shrine, Tehran-1417953836 , IRAN
Tel: (+98-21)51212020, Fax: (+98-21)51212021
[email protected]
Abstract – this work studies the effect of neutral earth resistance on the controlling
ferroresonance oscillation in the power transformer including MOV surge arrester. A simple case
of ferroresonance circuit in a three phase transformer is used to show this phenomenon and the
three-phase transformer core structures including nonlinear core losses are discussed. The effect
of MOV surge arrester and neutral earth resistance on the onset of chaotic ferroresonance and
controlling chaotic transient in a power transformer including nonlinear core losses has been
studied. It is expected that these resistances generally cause into ferroresonance control.
Simulation has been done on a power transformer rated 50 MVA, 635.1 kV with one open phase.
The magnetization characteristic of the transformer is modelled by a single-value two-term
polynomial with q=7, 11. The core losses are modelled by third order in terms of voltage. The
simulation results reveal that connecting the MOV arrester and neutral resistance to the
transformer, exhibits a great impact on ferroresonance over voltages.
Keywords: MOV Arrester, Neutral Earth Resistance, Control of Chaos, Bifurcation,
Ferroresonance, Power Transformers, Nonlinear core losses effect
I. Introduction
In simple terms, ferroresonance is a series of
“resonance” involving nonlinear inductance and
capacitances. It typically involves the saturable
magnetizing inductance of a transformer and a capacitive
distribution cable or transmission line connected to the
transformer. The word ferroresonance first appeared in
the literature in 1920 [1], although papers on resonance
in transformers appeared as early as 1907 [2]. Practical
interest was generated in the 1930’s when it was shown
that use of series capacitors for voltage regulation caused
ferroresonance in distribution systems [3], resulting in
damaging over voltages. The first analytical work was
done by Rudenberg in the 1940’s [4]. More precise and
detailed work was done later by Hayashi in the 1950’s
[5]. Subsequent research has been divided into two main
areas: improving the transformer models and studying
ferroresonance at the system level. Sensitivity studies on
power transformer ferroresonance of a 400 kV double
circuit is given in [6], Novel analytical solution to
fundamental ferroresonance in [7] investigated a major
problem with the traditional excitation characteristic
(TEC) of nonlinear inductors, in that the TEC contains
harmonic voltages and/or currents, and has been used the
way as if it were made up of pure fundamental voltage
and current. Stability domain calculations of period-1
ferroresonance in a nonlinear resonant circuit have been
investigated in [8]. Application of wavelet transform and
MLP neural network for ferroresonance identification
was done in [9]; in this paper an efficient method for
detection of Ferroresonance in distribution transformer
based on wavelet transform is presented. Using this
method ferroresonance can be discriminate from other
transients such as capacitor switching, load switching,
transformer switching. Impacts of transformer core
hysteresis formation on stability domain of
ferroresonance modes were done in [10]. In this paper,
impacts of various formations of hysteresis on the
stability domain of ferroresonance modes of a voltage
transformer (VT) have been studied. Based on four
different hysteretic and two single-valued polynomial
models, ferroresonance behaviors of the VT are studied.
2-D finite-element electromagnetic analysis of an
autotransformer experiencing ferroresonance was given
in [11]. Experimental and simulation analysis of
ferroresonance in single-phase transformers considering
magnetic hysteresis effects is investigated in [12]. An
accurate hysteresis model for ferroresonance analysis of a
transformer showed an accurate transformer core model,
using the Preisach theory, to represent the core
magnetization characteristic [13]. A new modeling of
MATLAB transformer for accurate simulation of
ferroresonance shows a new modeling of transformers in
Simulink/MATLAB enabling to simulate slow transients
more accurate than the existing models used in the
software [14]. Effect of circuit breaker shunt resistance
on chaotic ferroresonance in voltage transformer was
shown in [15], in this work it has also been shown C.B
shunt resistance successfully can cause ferroresonance
drop out and can control it. Then controlling
ferroresonance has been investigated in [16], it is shown
controlling ferroresonance in voltage transformer
including nonlinear core losses by considering circuit
breaker shunt resistance effect, and clearly shows the
effect of core losses nonlinearity on the system behavior
and margin of occurring ferroresonance. In [17],
electromagnetic voltage transformer has been studied in
the case of nonlinear core losses by applying metal oxide
surge arrester in parallel with it and Simulations have
shown that MOV successfully can cause ferroresonance
drop out. Effect of neutral earth resistance on the
controlling ferroresonance oscillations in power
transformer has been studied and it has been shown that
system has been greatly affected by neutral resistance
[18]. This paper studies the effect of MOV arrester and
neutral earth resistance on the global behavior of a
ferroresonance circuit with nonlinear core losses and
controlling these oscillations.
II. SYSTEM
MODELING
INCLUDING
MOV
ARRESTER WITHOUT NEUTRAL EARTH
RESISTANCE
The system studied here consists of a source feeding an
unloaded power transformer with one supply conductors
being interrupted (Fig.1) [19]. The transformer is then
energized through the capacitive coupling with the other
phases. Fig.2 shows the circuits that feed the
disconnected coil trough the capacitive coupling [19].
1
2
Cm
v1
v2
v3
Cg
3
Figure.1. model of ferroresonance circuit including line capacitance
2
Cm
v1
v2
1
Cm
Cm
Cg
Cg
Cg
Figure.2. Circuit that feeds disconnected coil
In order to derive the mathematical description of the
above circuit, Thevenin’s theorem will be used to obtain
an equivalent circuit. The equivalent capacitance can be
found by decreasing the two remaining voltage source of
phase 1 and 2. In doing so, both corresponding ground
capacitance and transformer windings are shorted and can
be omitted as well. Therefore, the remaining equivalent
circuit will consist of two mutual and one ground
potential. Thus, the equivalent capacitances can be
grouped as [19]
C  C g  2Cm
(1)
And the equivalent source voltage can be derived as
E
Cm
Cg  2Cm
(2)
Under loaded operating conditions, the flux induced in
the primary transformer windings are immediately
compensated by the current flow in the secondary
windings so that the core losses in the transformer can be
neglected. However, in the case of an unloaded (or very
lightly loaded) power transformer, current can develop in
the secondary side and this causes the flux to emanate
from the transformer leg and flow through the iron core.
This in turn increases the transformer core losses and
therefore it can no longer be neglected. Fig.3 shows the
final reduced equivalent circuit with the inclusion of R,
which represents the transformer core losses [19].
C
L
R
E
MOV
the same porcelain housing due to required protecting
voltage. Size of each disc is related to its power
dissipation capacity. The nonlinear V-I characteristic of
each column of the surge arrester is modeled by
combination of the exponential functions of the form
 I
V
 Ki 
I
Vref
 ref
(4)
Where:
V represents resistive voltage drop, I represents arrester
current and K is constant and  is nonlinearity constant.
This V-I characteristic is graphically represented as
follows (on a linear scale and on a logarithmic scale).
5
2
Figure3. Equivalent circuit of system including MOV surge arrester
hysteretic and eddy current characteristics [vm
 iRm ] ,
respectively. The iron core saturation characteristic due
to the explanation above is given by:
x 10
MOV V-I characteristic
1.5
1
MOV voltage
Fig.3. shows the equivalent circuit of system which was
described above. The magnetization branch is modeled
by a nonlinear inductance in parallel with a nonlinear
arrester accordingly and represents the nonlinear
saturation characteristic
[  iLm ] and nonlinear
1/  i




0.5
0
-0.5
-1
-1.5
-2
-5000
0
5000
10000
MOV current
iLm  a  bq
Figure 5. V-I characteristic of MOV surge arrester
(3)
Exponent q depends on the degree of saturation. It was
found that for adequate representation of the saturation
characteristics of a power transformer the exponent q
may acquire values 7 and 11. Fig.4 shows simulation of
these iron core characteristic for q=7, 11.
The Surge Arrester block is modeled as a current source
driven by the voltage appearing across its terminals.
Therefore, it cannot be connected in series with an
inductor or another current source. As the Surge Arrester
block is highly nonlinear, a robust integrator algorithm
must be used to simulate the circuit. MATLAB ode23t
with default parameters usually gives the best simulation
speed. For continuous simulation, in order to avoid an
algebraic loop, the voltage applied to the nonlinear
resistance is filtered by a first-order filter with a time
constant of 0.01 microseconds. This fast time constant
does not significantly affect the result accuracy [17]. In
this paper, the core loss model adopted is described by a
third order power series which coefficients are fitted to
match the hysteresis and eddy current nonlinear
characteristics given in [17]:
(i)
(ii)
1.7
(iii)
(iV)
(V)
 , pu
1.6
1.5
1.4
1.3
1.2
1.1
i  3.14  0.417
9
2
(ii) i    10
(i)
(iii) i


 0.28  0.7211  102
(iV) Transformer
(V) i
 11  10 2
0.2 0.4 0.6 0.8 1 1.2
i, pu
Figure. 4. Nonlinear characteristics of transformer core with different
values of q [16]
III. Metal Oxide Surge Arrester Model
Surge Arrester is highly nonlinear resistor used to protect
power equipment against over voltages. MOV can be
arranged by cascading several metal oxide discs inside
iRm  h0  h1 Vm  h2Vm  h3Vm
2
(5)
Per unit value of
3
(iRm ) is given in (6)
iRm  0.000001  0.0047Vm  0.0073Vm  0.0039Vm
2
(6)
The differential equation for the circuit in fig. 3 before
connection of MOV arrester can be presented as follow:
3
vl 
d
dt
(13)
  h0 
u
 1  C 
u    dE 

 2 
 dt 
(7)
ic  C
dvc
, vc  E  vl
dt
(8)
ic  il  iRm
(14)
y (t )  CX (t )
(9)
dE
dv
C
C l 
dt
dt
2
3
h0  h1vl  h2vl  h3vl  a  bq  iMOV

y (t )  v(t )  0

(10)
dvl

dt
dE 1
1
v 
2
3
 h0  h1vl  h2vl  h3vl  a  bq   l 
dt C
C
k

 

det  A  0
 x1 (t ) 
1

 x2 (t )
(15)
(16)
1




det a  bx1 (t ) q 1
(h1  h2 x2 (t )  h3 x2 (t )2   0




C
C
(17)
(11)
Where E is the peak value of the voltage source as shown
in fig.3 and  is the flux linkage of the transformer.
Typical values for system parameters without MOV
arrester due to the equation (3) are as given below:
Table 1: Typical value of q and its coefficient
q
11
7
Coefficient (a)
0.0028
3.14
Coefficient (b)
0.0072
0.41

(h1  h2 (1.41)  h3 (1.41) 2 )
p



C

q  a

C

2  p  q  0
1, 2 
(18)
 p  p 2  4q
2
(19)
Bode Diagram
60
Magnitude (dB)
  377 rad/sec
Cpu  0.7955
Vbase  635.1 kv , I base  78.72 A
Phase (deg)
d
 (0)  0.0; vl 
(0)  2
dt
 and d / dt
0
-4
as state
10
-3
10
-2
10
-1
10
0
10
Frequency (rad/sec)
Fig.5. Bode diagram of the given parameters
IV. Simulation Results


1
dE
h0  h1 x 2 (t )  h2 x 2 (t ) 2  h 3 x 2 (t ) 3  ax1 (t )  bx1 (t ) q 
C
dt
0
 x1 (t )  

 x (t )   a  qb
x1 (t ) q 1
 2  
 C
90
-90
 x1 (t )  
 x (t )  x (t )
 2
1

v  d / dt
 x2 (t )  v
x 2 (t )  
20
0
180
Initial conditions:
The state-space formulation
variables is as given below:
40
(12)

  x1 (t )   1u
 (h1  h2 x2 (t )  h3 x2 (t ) 2 )  
 
 x2 (t ) 1
C

1
Ferroresonance in three phase systems can involve large
power transformers, distribution transformers, or
instrument transformers. The general requirements for
ferroresonance are an applied source voltage, a saturable
magnetizing inductance of a transformer, a capacitance,
and little damping. The capacitance can be in the form of
capacitance of underground cables or long transmission
lines, capacitor banks, coupling capacitances between
double circuit lines or in a temporarily-ungrounded
system, and voltage grading capacitors in HV circuit
breakers [20]. In this section of simulation, system has
Bifurcation Diagram with q=7 including Nonlinear core losses and MOV effect
2.5
Voltage of Transformer(perunit)
been considered with MOV arrester and Time-domain
simulations were performed using the MATLAB
programs. Figs.6 and 7 show the phase plan diagram of
system states with MOV arrester for E=3 p.u. Figs.8 and
9 show the corresponding bifurcation diagrams of the
system states with MOV arrester and including nonlinear
core losses for q=7, 11 which depicts chaotic behavior.
Phase Plan Diagram with q=7 including Nonlinear core losses and MOV effect
2
1.5
1
0.5
2.5
2
0
0
1
2
3
Input voltage(perunit)
4
5
6
Figure8. Bifurcation diagram with q=7, without MOV arrester
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
0.5
Flux Linkage of Transformer
1
1.5
2
Figure6. Phase diagram of system with q=7, with MOV arrester
In Fig.6 Input voltage has been plotted against the flux of
the power transformer. When the magnitude of the input
voltage is 3p.u, trajectory of the system has chaotic
behavior and amplitude of the flux and ferroresonance
overvoltage reaches up to 3p.u. by increasing in the
degree of core nonlinearity, q, the behavior of the system
has being more chaotic as shown in fig.7. In this case
q=11 and oscillation in the voltage of the transformer
goes up. Fig.7 shows the Phase plan diagram that has
been simulated by q=11, it shows that chaotic resonance
is highly dependent on the transformer modeling.
Phase Plan Diagram with q=11 including Nonlinear core losses and MOV effect
Fig.8 shows the abnormal phenomena in the power
transformer with the core model nonlinearity q=7, it
clearly shows when the q degree changes from 7 to 11;
ferroresonance begins in 2p.u. by comparing fig.9 with
fig.8 margin of the beginning ferroresonance has been
decreased. Fig.9 shows the bifurcation diagram of the
over voltage with q=11, it shows that with q=11, system
behavior reaches up to 1.6 p.u.
Bifurcation Diagram with q=11 including Nonlinear core losses and MOV effect
1.8
1.6
Voltage of Transformer(perunit)
Voltage of Transformer
1.5
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
Input voltage(perunit)
4
5
6
Figure9. Bifurcation diagram with q=11, without MOV arrester
2.5
2
Table (2) shows different values of E, considered for
analyzing the circuit in the absence of MOV arrester.
Voltage of Transformer
1.5
1
0.5
Table 2: Simulation results without neutral earth resistance
q/E
1
2
3
4
5
6
Chaotic
Chaotic
7
P1
P1
P3
P3
0
-0.5
-1
11
P1
*Pi: Period i
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
0.5
Flux Linkage of Transformer
1
1.5
P1
P1
chaotic
P3
P3
2
Figure7. Phase diagram of system with q=11, with MOV arrester
Another tool that can show manner of the system in vast
variation of parameters is bifurcation diagram. In fig.8,
voltage of the system has been increased up to 6p.u and
ferroresonance has been begun in 3p.u, it is shown that if
input voltage goes up due to the abnormal operation or
switching action, ferroresonance occurs, because of the
nonlinearity in core losses, ferroresonance began in the
big value of the input voltage.
V. System Modeling With MOV Arrester And
Neutral Earth Resistance
In the case of simulation, time domain simulations were
performed using fourth order Runge-Kutta method and
validated against MATLAB SIMULINK. In this case, the
system which was considered for simulation is shown in
fig.10.
Phase Plan Diagram of for q=7, Including NonLinear Core Losses and Neutral Resistance
C
3
Rcore
Lcore
vl
MOV
Voltage of Transformer
2
1
0
-1
-2
E
-3
-2
Rneutral
vRn
-1.5
-1
-0.5
0
0.5
Flux Linkage of Transformer
1
1.5
2
Figure11. Phase diagram of system with q=7, with MOV arrester and
neutral earth resistance
Phase Plan Diagram of for q=11, Including NonLinear Core Losses and Neutral Resistance
3
Figure.10. Equivalent circuit of system with MOV arrester and neutral
earth resistance
Typical values for various system parameters has been
considered for simulation were kept the same by the case
1, while neutral earth resistance has been added to the
system and its value is given below:
The differential equation for the circuit in fig.10 can be
presented as follows:

dvl dE 1 
v  
2
3


h0  h1vl  h2 vl  h3vl  a  bq   l  
dt
dt C 
 k  
dv
dv
d
d
2 dvl
 Rn .(h1 l  2h2 vl l  3h3vl
a
 qbq1
dt
dt
dt
dt
dt

 1
 1   dv 
   l 
 k   dt 
)
(20)
Where λ is the flux linkage and V is the voltage of
transformer.
Voltage of Transformer
2
1
0
-1
-2
-3
-1.5
-0.5
0
0.5
Flux Linkage of Transformer
1
1.5
Figure.12. Phase diagram of system with q=11, including MOV
arrester and neutral earth resistance
The use of geometric graphical methods like phase plane
projections and bifurcation can be applied to obtain a
better understanding of ferroresonance. Ferroresonance in
the above three phase nonlinear core distribution
transformer can be periodic or non periodic. Some of the
periodic modes of ferroresonance may contain sub
harmonics, but still have strong power frequency
components, but take longer than one fundamental cycle
to repeat. This occurs more typically for very large values
of q. The “higher energy modes” of ferroresonance
involving relatively large capacitances and little damping
can produce a non-periodic voltage on the open phase(s).
These voltage waveforms can be quite similar to those of
Doffing’s equation.
VI. SIMULATION RESULTS
Bifurcation Diagram with q=7 Including NonLinear Core Losses and Neutral Resistance
1.8
1.6
1.4
Voltage of Transformer
The nonlinear behavior of ferroresonance falls into two
main categories. In the first, the response is a distorted
periodic waveform, containing the fundamental and
higher-order odd harmonics of the fundamental
frequency. The second type is characterized by a nonperiodic, or chaotic, response. In both cases the
response’s power spectrum and simulation results contain
fundamental and odd harmonic frequency components. In
the chaotic response, however, there are also distributed
frequency harmonics and sub harmonics. Figs.11and 12
shows the corresponding phase diagram for the system in
fig.10. Also figs.13 and 14 show the bifurcation diagrams
for corresponding system including MOV arrester and
neutral earth resistance. It is shown that chaotic region
mitigates by applying neutral earth resistance.
-1
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
Input Voltage(perunit)
4
5
6
Figure13. Bifurcation diagram with q=7, with MOV arrester and
neutral earth resistance
In figs. 13 and 14, system behavior is shown by the
proper bifurcation diagram. In fig13, the degree of
transformer core is q=7, in this figure, one jump has
occurred in the trajectory of the system and voltage of the
transformer has period 3 oscillation.
In fig.14, q=11 and by considering MOV arrester in the
system, there is no nonlinear phenomena and voltage of
the transformer has a period 3 oscillation. By increasing
the core nonlinearity, behavior remains in periodic
oscillation and MOV arrester can cause ferroresonance
drop out.
Bifurcation Diagram with q=9 Including NonLinear Core Losses and Neutral Resistance
Voltage of Transformer
1.5
makes over voltage on power equipment and is a
dangerous phenomenon. MOV surge arrester is a
nonlinear resistance and can cause ferroresonance
dropout for some range of system parameters, but MOV
only can limit these over voltages and cannot remove it
successfully. By connecting neutral earth resistance to the
neutral point of the power transformer is shown that
system has been greatly affected by this resistance. The
presence of the neutral earth resistance results in
clamping the ferroresonance non-conventional over
voltages in the power system. This resistance can remove
the ferroresonance oscillation completely and system
shows less sensitivity to the initial conditions.
1
IX. REFERENCES
[1]
0.5
0
0
1
2
3
Input Voltage(perunit)
4
5
6
Figure14. Bifurcation diagram with q=11, with MOV arrester
Table (3) includes the set of cases which are considered
for analyzing the circuit including MOV arrester. For
cases including MOV arrester, it can be seen that
ferroresonance drop out will be occurred.
Table 3: Simulation results (with MOV arrester)
q/E
1
2
3
4
5
6
7
P2
P1
P1
P1
P2
P2
9
P1
P1
P2
P3
P3
P3
11
P1
P1
P3
P3
P3
P3
*Pi: Period i
VII. Comparative discussions
In this paper power transformer has been studied with
nonlinear core losses model and finally effect of MOV
surge arrester and neutral resistance on the controlling of
nonlinear phenomena and chaos has been investigated. In
the real system, transformer core has many kind of losses
and is strictly nonlinear, the model has been used in this
work was chosen from [19] , therefore other mathematic
and real models for studying the transformer behavior
when one unwanted phenomena such as switching action
and lightning has been occurred. it has been shown the
nonlinear core losses can cause system works better and
margin of occurring ferroresonance has been decreased.
By considering neutral resistance, ferroresonance over
voltages has been ignored and even if unwanted
phenomena appear, power transformer can works in the
safe operation region and there is no dangerous condition
in the power system.
VIII. CONCLUSIONS
This paper investigated the ferroresonance oscillation in
the power transformer. Ferroresonance in power system
B.A. Mork, D.L. Stuehm, Application of nonlinear dynamics and
chaos to ferroresonance in distribution systems, IEEE
Transactions on Power Delivery 9 (1994) 1009_/1017
[2] J. Bethenod, “Sur le transformateur et résonance,” L’Eclairae
Electrique, pp. 289–296, Nov. 30, 1907.
[3] J.W. Butler and C. Concordia, “Analysis of series capacitor
application problems,” AIEE Trans., vol. 56, pp. 975–988, Aug.
1937.
[4] R. Rudenberg, Transient Performance of Electric Power
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