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ferroresonance oscillation, stabilizing, chaos control, potential transformer, MOSA, neutral earth resistance
Ferroresonance over voltage on electrical power systems were recognized and studied as early as 1930s. Kieny first suggested applying chaos to the study of ferroresonance in electric power circuits [1]. He studied the possibility of ferroresonance in power system, particularly in the presence of long capacitive lines as highlighted by occurrences in France in 1982, and produced a bifurcation diagram indicating stable and unstable areas of operation. Because of importance of this phenomenon, in recent years, many papers described it from various aspects. For example in [2] time delay feedback is used to omit chaotic ferroresonance oscillation in power transformers. The susceptibility of a ferroresonance circuit to a quasi-periodic and frequency locked oscillations has been presented in [3]. Mozaffari has been investigated the ferroresonance in power transformer and effect of initial condition on this phenomena, he analysed condition of occurring chaos in
the transformer and suggested the reduced equivalent circuit for the power system including power switch and trans [4],[5]. The controlling effect of voltage transformer connected in parallel to a MOSA has been illustrated in [6]. Effect of circuit breaker shunt resistance on chaotic ferroresonance in voltage transformer was shown in [7].
In this work, C.B shunt resistance successfully can cause ferroresonance drop out and can control it. Then controlling ferroresonance has been investigated in [8], it is shown controlling ferroresonance in voltage transformer including nonlinear core losses by considering C.B shunt resistance effect, and clearly shows the effect of core losses nonlinearity on the system behaviour and margin of occurring ferroresonance.Effect of linear and nonlinear core losses on the onset of chaotic ferroresonance and duration of transient chaos in an autotransformer is investigated in [9] and control of these phenomena is clearly shown by considering nonlinear core losses effect. In current paper, effect of neutral earth resistance for stabilizing of unstable and high amplitude ferroresonance oscillation is used. Using of this method results improving voltage waveform which leads to protection from insulation, fuses and switchgears. This paper organized as follow: At first reasons of occurrence ferroresonance in transformers is described. Then one type of ferroresonance in potential transformer is explained. Then general introducing of controlling ferroresonance by considering neutral earth resistance and using it in current problem is shown.
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Fig (1) shows the circuit diagram of the power system components at the 275 KV substations. PT is isolated from sections of bus bars via disconnector2 (DS2).
Ferroresonance conditions occurred upon closure of disconnector1 (DS1) with CB and DS2 open, leading to a
1
Hamid Radmanesh,
2Seyed Hamid Fathi,
3Mehrdad Rostami,
4Rozbeh Kamali
1,2
Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran
1
[email protected]
2[email protected]
3
Electrical Engineering Department, Shahed University, Tehran-1417953836 , Iran [email protected]
4
Electrical Engineering Department, Aeronautical University of Science& Technology, Tehran, Iran
[email protected]
system fault caused by failure of the potential transformer primary winding.
Figure.1.Power system one line diagram arrangement resulting to PT ferroresonance
Fig.2 shows the basic ferroresonance equivalent circuit used in this analysis while MOSA is connected in parallel to the potential transformer. The resistor R represents transformer core losses. In [10] accurate model for magnetization curve of core considering hysteresis, was introduced but in current paper the nonlinear transformer magnetization curve was modelled by a single valued seventh order polynomial obtained from the transformer magnetization curve[11].
Figure.2. Basic reduced equivalent ferroresonance circuit
In Fig. 2, E is the RMS supply phase voltage, Cseries is the circuit breaker grading capacitance, and Cshunt is the total phase-to-earth capacitance of the arrangement. The resistor R represents a potential transformer core loss that is found to be an important factor in the initiation of ferroresonance, and MOSA is a nonlinear resistance is connected to the transformer. In the peak current range for steady-state operation, the flux-current linkage can be approximated by a linear characteristic such as %
= λ
where the coefficient of the linear term (a) corresponds closely to the reciprocal of the inductance
( ≅ 1 / % )
.Figure.3. Flux- current characteristic of the transformer core
However, for very high currents the iron core might be driven into saturation and the flux-current characteristic
becomes highly nonlinear, here the
λ −
characteristic of the potential transformer is modelled as in [11] by the polynomial7 (1)
λ λ
+#=
Where =
3 . 14 ,
#=0 . 41
. Fig.3 shows simulation of these iron core characteristic (λ −
) for q=7. Basic potential transformer ferroresonance circuit of Fig.2 can be presented by a differential equation. Because of the nonlinear nature of the transformer magnetizing characteristics, the behaviour of the system is extremely sensitive to change in system parameter and initial conditions. A small change in the value of system voltage, capacitance or losses may lead to dramatic change in the behaviour of it. A more suitable mathematical language for studying ferroresonance and other nonlinear systems is provided by nonlinear dynamic methods. Mathematical tools that are used in this analysis are phase plan diagram, time domain simulation and bifurcation diagram.% & '
MOSA is highly nonlinear resistor used to protect power equipment against over voltages. MOSA can be arranged by cascading several metal oxide discs inside 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 MOSA arrester is modelled by combination of the exponential functions of the form
, 3 ,
α / 1
=
(2)Where: V represents resistive voltage drop, I represents arrester current, K is constant and α is nonlinearity constant.
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Fig 4. V-I characteristic of MOSA
Figure.4 shows V-I characteristic of MOSA is simulated by the given parameters in this paper.
( )
Mathematical analyses of the equivalent circuit by applying KVL and KCL laws are done and equations of the system can be presented as bellowed:
Arrester can be expressed by the so- called “alpha”
equation:
α /
3
1, =
(3)
+
= +
−
−
−
2 2
7 ( )
) 1 (
1 cos 1
4 4
# 4 4
4 6 5
λ λ λ
λ ω λ
ω
α
(4)
Where ω is supply frequency, α, k are MOSA parameters also in equation (1) a=3.4 and b=0.41 are the seven order polynomial sufficient [11]. The time behaviour of the basic ferroresonance circuit is described by (4).Results for one parameter sets showing a chaotic response of ferroresonance oscillation. Table (1) shows base values used in the analysis and parameters values are given in table (2).
Table (1): base values of the power system used for simulation Base value of input voltage
3 /
275
KVBase value of volt-amperes 100 VA
Base angular Frequency
2 π 50
rad/secTable (2): Parameters used for various states simulation
System behavior Parameters
Cseries
(nf) Cshunt
(nf) Rcore
(MΩ) Rn
(kΩ) ω (rad/sec)
E (KV) Subharmonic
Ferroresonance
3 0.1 1900 50 314 275
Table (3): initial conditions of the propose system
λ λ λ
0
1.4144p.u 0.5p.u
* +
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In this case of the power system study, the system which was considered for simulation is shown in fig.5.
Figure.5. Basic reduced equivalent ferroresonance circuit including MOSA considering neutral earth resistance
In Fig. 5, Rn is the neutral earth resistance. Typical values for various system parameters is considered for simulation were kept the same by the case 1, while
neutral resistance is added to the system grounding, and its value is given bellowed:
Ω
=
5 50
The differential equation for the circuit in fig.5 can be presented as follows:
( )
( )
(
1)
%
%
%
%
5 #
#1 5 4 5 4
5 4 4 5 4
6 4 5
4 4
λ λ
λ λ
α λ ω ω
α
α α
+
−
−
−
+
−
+ + +
−
=
−
1
6
2 2 1 1
2 2
1 cos 2
(6)
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In this section of simulation, power system is considered without neutral resistance and time-domain simulations were performed using the MATLAB programs. One state of ferroresonance is studied in two cases, 1) without considering neutral earth resistance, and 2) with considering neutral resistance.
- "
"
Phase space and waveform of voltage for Subharmonic response were shown in figure (6.a) and (6.b). The phase plane diagram clearly shows the effect of MOSA on the system behaviour, MOSA clamps the ferroresonance over voltages, and keeps it on 2.3p.u and doesn’t allow to across from this point.
)
* * + , ( - %% . "#
Figure (6.a). Phase plan diagram for subharmonic ferroresonance motion without neutral earth resistance
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/
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! /
$% &
+ " , ( %% .
Figure (6.b). Time domain simulation for subharmonic ferroresonance motion without neutral earth resistance
According to the fig.3, V-I characteristic of MOSA is shown when voltage of transformer is crossed from 2.2p.u, MOSA causes across high current from its terminal, and over voltages is damped by this nonlinear varistor. By parameters value of table (2) system is simulated, and it is shown that over voltages clams by considering MOSA in parallel to the transformer.
- "
In this case of simulation, effect of neutral earth resistance on the system behaviours is investigated. Phase space and waveform of voltage for quasiperiodic response were shown in figure (7.a) and (7.b). The phase plane diagram clearly shows the torus trajectory characteristic of the quasiperiodic waveform. Chaotic ferroresonance changes to the quasiperiodic resonance by considering neutral earth resistance as shown in figs.
(7.a), (7.b).
2 0 / / 0 2
)
* * + , ( %% .
Figure (7.a). Phase plan diagram for quasiperiodic motion considering neutral earth resistance effect
! / 0 1 2 3
$% &
+ " , ( %% .
Figure (7.b). Time domain simulation for quasiperiodic motion considering neutral earth resistance effect
By comparing this case of simulation with the simulation results of fig (6.a), it clearly shows that amplitude of subharmonic resonance decreases to 0.8p.u, and subharmonic resonance is changed to quasiperiodic resonance. Because of the neutral earth resistance, order of the nonlinear equation (5) is increased, this increasing in the nonlinear differential equation is changed the type of the previous equation to the duffing equations, so behaviour of this case is shown as torus behaviour. Effect of neutral earth resistance is clearly obvious, because amplitude of ferroresonance over voltages has been
decreased from 2.2p.u to 0.8p.u. Table (4) shows parameters value of the second case of simulation when neutral earth resistance is considered on the system structure.
.
Table (4): Parameters used for various states simulation in the case of considering neutral resistance
] Cseries
(nf) Cshunt
(nf) Rcore
(MΩ) Rn
(kΩ) ω (rad/sec)
E (KV) Subharmonic
Ferroresonance
3 0.1 1900 50 314 275
- % . + '
In this paper, it is shown the effect of variation in the voltage of the system on the ferroresonance overvoltage in the PT, and finally the effect of applying neutral resistance on this overvoltage by the bifurcation diagrams.
Table (5): Parameters value is used for plotting bifurcation diagram
System behavior
Cseries
(nf) Cshunt
(nf) Rcore
(MΩ) Rn
(kΩ) ω (rad/sec)
E (KV)
α k
Bifurcation diagram, figs(8.a)
0.1 0.5 1900 - 314 275-
1375
25 2.5101
Bifurcation diagram, figs(8.b)
0.1 0.5 1900 50 314 275-
1375
25 2.5101
Figures (8.a) and (8.b) clearly shows the ferroresonance overvoltage in PT when voltage of system increase to 5 p.u. In the bifurcation diagram of fig (8.a), system behaviours are shown in the case of considering MOSA, as previously described, MOSA clamps ferroresonance over voltages to 2.2p.u, when input voltage has been increased to 5p.u, ferroresonance appears for some value of input voltages, MOSA can controlled these over voltages and keep its amplitude under 2.2p.u.
! ! / /
$% &
$%&
- + 4 , ( 5 %% .
Figure (8.a). Bifurcation diagram for voltage of transformer versus voltage of system, without considering neutral earth resistance effect
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! / 0 1 2 3
' % 4 $% &
$%&
- + 4 , 5 %% .
Figure (8.b). Bifurcation diagram for voltage of transformer versus voltage of system, considering neutral earth resistance effect
In this plot, before 1p.u ferroresonance appears, after that between 1p.u to 3p.u period3 occurred, then until 4.5p.u, period3 oscillation is changed to subharmonic resonance but MOSA doesn’t allow ferroresonance over voltages goes up more than 2.2p.u. By applying neutral earth resistance to the system configuration, bifurcation diagram of fig. (8.a) is changed to fig.(8.b). Important effect of neutral earth resistance is that ferroresonance over voltages is controlled and quasiperiodic route to chaos is take placed, finally, neutral earth resistance successfully can control these over voltages.
/
In this work it has been shown that system is greatly affected by neutral earth resistance. The presence of the neutral resistance results in clamping and controlling the ferroresonance over voltages in studied system. The neutral resistance successfully eliminates the chaotic behaviour of proposed model and system shows less sensitivity to initial conditions in the presence of the neutral resistance.
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[1] C. Kieny, Application of the bifurcation theory in studying and understanding the global behavior of a ferroresonant electric
power circuit, 666 7 ' ! 8' 1991,
pp. 866-872.
[2] B.Stojkovska, A.Stefanovska, R.Golob " 7 " #
4 " 4 ", IEEE Porto.
Power Tech Conference, vol.2, Page(s):6 pp, 2001.
[3] W.L.A. Neves, H. Dommel, on modeling iron core nonlinearities,
666 ' ! 9, 1993, pp. 417-425.
[4] S. Mozaffari, M. Sameti, A.C. Soudack, Effect of initial conditions on chaotic ferroresonance in power transformers, 66
:;< ' 7 # ' !
*=='1997, pp. 456-460.
[5] S. Mozaffari, S. Henschel, A. C. Soudack, Chaotic ferroresonance in power transformers, ! 66 < '
7 #!' ! *=-' 1995, pp. 247-250.
[6] Radmanesh, H.; Khalilpour, J.;” Controlling Chaotic Ferroresonance in Voltage Transformers by Application of MOV Surge Arrester,” 2010 3rd International Conference on Computer and Electrical Engineering, (ICCEE 2010), Chengdu, China, vlo5, page 516-520.
[7] Radmanesh. Hamid, Controlling ferroresonance in voltage transformer by considering circuit breaker shunt resistance including transformer nonlinear core losses effect,
5 56 > ' , ! ?
@! /' ' 6 0 4 -+.
[8] Radmanesh, Hamid.; Rostami, Mehrdad.;” Decreasing Ferroresonance Oscillation in Potential Transformers Including Nonlinear Core Losses by Connecting Metal Oxide Surge Arrester in Parallel to the Transformer”, 5
4 !56! !4 ! ' , ?'@!/ Issue 6A 4 0 @ ,0 -+*+.
[9] Radmanesh. Hamid, Controlling Chaotic Ferroresonance oscillations in Autotransformers Including Linear and Nonlinear
Core Losses Effect, 5
!5!6!6 ' , ! / @! 8' ' @ ,0764 -+*+.
[10] A.Rezaei-Zare, M.Sanaye-Pasand, H.Mohseni, Sh.Farhangi, R.Iravani “ "
B 0 C $
” IEEE Transaction on Power Delivery, vol. 22,no.
2,pp.919-928, 2007.
[11] H. Radmanesh, M. Rostami, "Effect of Circuit Breaker Shunt Resistance on Chaotic Ferroresonance in Voltage Transformer,"
Advances in Electrical and Computer Engineering, vol. 10, no. 3, pp. 71-77, 2010.
