Ferroresonance suppression in power transformers using chaos theory

Electrical Power and Energy Systems 45 (2013) 1–9

Contents lists available at SciVerse ScienceDirect

Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Ferroresonance suppression in power transformers using chaos theory Hamid Radmanesh a,b,⇑, G.B. Gharehpetian b a b

Department of Electrical Engineering, Islamic Azad University, Takestan Branch, Takestan, Iran Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 8 October 2010 Received in revised form 16 August 2012 Accepted 19 August 2012 Available online 5 October 2012 Keywords: Metal oxide varistor Control of chaos Bifurcation Ferroresonance Power transformers

a b s t r a c t The main goal of this paper is the determination of the effect of the metal oxide varistor (MOV) on various ferroresonance modes including fundamental resonance, subharmonic and chaos mode which are generated in electrical power systems. Chaos theory is used for analyzing this effect. Also, the bifurcation, phase plan diagram and time domain simulation are used for this purpose. The proposed power system contains a no-load or lightly loaded power transformer. The magnetization curve of the transformer core is modeled by a single-value two-term polynomial. The core loss is modeled based on the flux of the transformer. The MOV modeled as a nonlinear voltage dependent resistance. The suppression effect of MOV on chaotic ferroresonance in power transformer is studied in this paper. The simulation results confirm that connecting the MOV to the transformer has a considerable suppression effect on ferroresonance phenomena. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The ferroresonance is a nonlinear resonance, which has result in multiple periodic and non-periodic modes in the system behavior. Considering system parameters and the initial condition of the ferroresonant circuit, it may settle to one of the following behaviors such as fundamental, subharmonic, quasi-periodic or chaotic resonances. Usually, the ferroresonance contains a nonlinear inductance and capacitances. The nonlinear inductance typically is a saturate magnetizing inductance of a transformer and the capacitance is a capacitive distribution cable or transmission line connected to the transformer. Ferroresonance phenomenon has been recognized and investigated in many papers as early as 1907 [1–4]. Isolated ferroresonant solutions in transmission lines have been investigated in [5], which presents the detailed analysis of the subharmonic mode of the ferroresonance and its sensitivity with respect to the length of the deenergised line. The study of the periodic ferroresonance in electrical power networks by bifurcation diagrams has been carried out in [6]. The analysis of the lightning-caused ferroresonance in capacitor voltage transformer (CVT) has been given in [7]. In the paper, a dynamic response to lightening and switching has been studied. So, the study investigates the effect of the lightning strike on a tower with a 132 kV CVT. The s-domain model of three winding transformer for modal analysis has been given in ⇑ Corresponding author at: Department of Electrical Engineering, Islamic Azad University, Takestan Branch, Takestan, Iran. Tel.: +98 2164543335; fax: +98 2188212072. E-mail addresses: [email protected] (H. Radmanesh), [email protected] (G.B. Gharehpetian). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.08.028

[8]. The influence of non-differential components to the power system small signal stability region has been studied in [9]. Considering the importance of the initial condition in the nonlinear systems, in [10], the impact of initial conditions on the initiation of the ferroresonance in the model of a 275 kV magnetic voltage transformer has been investigated. The transient response of a practical ferroresonant circuit has been studied in detail in [11]. The iterative approximation technique has been used for the determination of the transient response due to sudden application of a sinusoidal voltage. The analysis of subharmonic oscillations in a ferroresonant circuit with the focused on subharmonic (period-3) ferroresonant oscillations has been given in [12]. A novel analytical solution to the fundamental ferroresonance including power frequency excitation characteristic has been investigated in detail in [13]. A method of protecting the voltage transformer against ferroresonance overvoltages with a compact active load has been developed by [14]. The static VAR compensator (SVC) and the thyristor-controlled series capacitor (TCSC) analytical model, a systematical method for suppressing ferroresonance at neutral-grounded substations and the frequency response of the unfiled power flow controller (UPFC) has been simultaneously studied in [15]. A sensitivity study on power transformer ferroresonance in a 400 kV power system is presented in [16]. In that paper, the model of 1000 MV-400/275/13 kV power transformer has been described and the simulations have been compaired with field test results. The influence of supply, circuit and magnetic material parameters on the occurrence of the fundamental ferroresonance mode in a series inductance– capacitance–resistance (LCR) circuit with a nonlinear inductor has been discussed in [17,18]. The effect of the circuit breaker shunt resistance on the chaotic ferroresonance in voltage transformers

2

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9

has been studied in [19]. The suppression technique of the ferroresonance phenomenon in the coupling capacitor of the voltage transformer has been given in [20]. The impact of hysteresis and magnetic couplings on the stability domain of ferroresonance in asymmetric three-phase three-leg transformers has been discussed in [21]. Mitigating the ferroresonance of 161 kV electromagnetic potential transformers by damping reactors in gas-insulated switchgear has been presented in [22]. The frequency domain analysis of a power transformer ferroresonance has been studied in [23]. The aim of the current paper is to show the controlling effect of MOV on clamping the ferroresonance overvoltages and the application of MOV as a practical solution for protecting power transformers against ferroresonance overvoltages. 2. Power system modeling The ferroresonance phenomenon, in most situations, consists of a capacitance and an inductance and there is no definite resonant frequency, so more than one response is possible for the same set of parameters including fundamental, subharmonic and chaotic resonances [24]. In this section, the ferroresonance equivalent circuit is a power transformer connected to the power system. But, one of the three poles of the circuit breaker is open and only transformer two phases are energized. This switching action produces induced voltage in the opened phase. This voltage causes ferroresonance overvoltages, if the distribution line is capacitive. This case here consists of a source with one conductor being interrupted as shown in Fig. 1 [25]. Fig. 2 shows the circuit that feeds the disconnected coil trough the capacitive coupling of the power system [25]. The circuit shown in Fig. 2 has been analyzed using the venin’s theorem. By shorting the two remaining voltage sources of phases 1 and 2 the capacitance of the remaining part of the power system can be derived. So, Cg and transformer windings are short circuited and phases 1 and 2 having the same voltage and can be omitted. So, the remaining circuit will consist of two Cm and one Cg. Thus, the equivalent capacitances can be given as Eq. (1) where Cg is ground capacitance and Cm is mutual capacitance of the power system.

C ¼ C g þ 2C m

ð1Þ

And the equivalent source voltage is given by equation:

E¼

Cm Vl C g þ 2C m

ð2Þ

In unloaded or very lightly loaded transformer, the current causes the flux flow through the iron core. This phenomenon increases the transformer core loss and therefore it should not be

1

2

Cm

v1

1

Cm Cm

v2

Cg

3 Cg

L Cg

Fig. 2. Circuit that feeds disconnected coil.

neglected. Fig. 3 shows the reduced equivalent circuit including R2, where it modeles the core loss including eddy current and hysteresis loss. C models the total remaining capacitance and Lcore represents the power transformer disconnected coil. The magnetization curve is modeled by a nonlinear inductance in parallel with a nonlinear resistor representing the saturation and hysteresis and eddy current characteristics, respectively. Any damping can be added to the circuit, which may cause the ferroresonance voltage and current elimination. This damping effect can be a resistive source impedance, transformer loss and corona loss. But, the load applied to the secondary of the transformer is much important. A lightly-loaded or unloaded transformer fed through capacitive source impedance is a major candidate for ferroresonance as shown in Fig. 1. Due to high currents and core fluxes, the ferroresonance can overheat the transformer. The high temperature may weaken the winding insulation and cause a transformer failure. In extra high voltage systems, ferroresonance voltages occur during the first cycles and cause insulation coordination problems. Because of non-linearity nature of these circuits, the analytical solution must be determined using time domain methods. So, a computer-based numerical integration method is applied using time domain simulation programs such as the MATLAB. In this paper, the iron core saturation characteristic is modeled by equation:

iLm ¼ au þ buq

ð3Þ

where q depends on degree of saturation. The adequate value of saturation parameters of a power transformer have been derived by Watson and listed in Table 1 [26]. For deriving the differential equation of the ferroresonance circuit we have:

i1  i2  i3 ¼ 0 Vl ¼

ð4Þ

du _ ¼ R2 ði1  i2 Þ ¼u dt

ð5Þ

where Vl is voltage of transformer, u is flux linkage, R2 is transformer core loss, i1 and i2 has been shown in Fig. 1. The nonlinear C

2

Cm v1

v2

v3

R1 Cg

+Vc -

i1

3

e(t)

Fig. 1. Model of ferroresonance circuit including line capacitances.

R2

Fig. 3. Equivalent ferroresonance circuit.

Vl

3

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9 Table 1 Typical value of q and its coefficient.

x20 ¼

q

Coefficient (a)

Coefficient (b)

11

0.0028

0.0072

With the above definition of p, Eq. (13) can be written as:

d / d/ d/ þ / ¼ a  b/10 þ p/  /11 þ d cos s ds2 ds ds

u_ di1 au þ bu11  R1  R2 C dt C

If R1  R2 and

di1 dt

  1 þ R1 ða þ 11bu10 Þ R2 C



u_ þ

au þ bu11 _ ¼ eðtÞ C

ð7Þ

Using the method of Krylov and Bogoliubov from [30] and Migulin [31,32], and taking the first and second derivatives of the equation (21) and substituting it into equation (20), it can be shown that:

ð8Þ

The above equation is a variant of the Duffing equation, with the magnetization curve having an index of n = 11. For e = Esin xt, Eq. (8) can be written as:

€ þ f1 ðuÞu _ þ f2 ðuÞ ¼ xE cosðxtÞ u

ð9Þ

where

f 1ðuÞ ¼

f2 ðuÞ ¼



   1 þ R1 ða þ 11bu10 Þ R2 C

ð10Þ

au þ bu11 C

ð11Þ

Eq. (11) can be time scaled by setting s = xt. 2

d u f1 ðuÞ du f2 ðuÞ E þ þ ¼ cos s ds2 x ds x2 x

ð12Þ

Eq. (12) can be amplitude scaled by letting / ¼

u s

2

d / f1 ð/Þ d/ f2 ð/Þ E þ þ ¼ cos s ds2 x ds sx2 sx

ð13Þ

ð14Þ

The nonlinear damping component of the Eq. (13) is:

f1 ðs/Þ

x

¼

  10 1 11R1 bs /10 aR1 þ þ R2 C x x 1

ð15Þ

With the following definitions:

  1 aR1 þ ; R2 C x rffiffiffiffiffiffiffiffiffiffi 2 10 x C ¼ b

a¼

1

b ¼ 11bR1 s10 ;

d¼

E

xs

;

s ð16Þ

Eq. (15) can be written as:

f1 ðs/Þ

x

¼ a þ b/10

where

x20 x2

ð22Þ

The average rate of changing in frequency is calculated as:

Ah_ ¼

pA A11 d þ 11  cos h 2 2 2

ð23Þ

To determine the component of the solution, the singular points should be found and the motion in their neighborhood must be checked. The nature of trajectories in the neighborhoods of singular points shows when a small noise in the steady-state motion grows; that is, they show the stability of the steady-state motion. The steady-state motions occur when, which corresponds to the singular points of Eqs. (22) and (23). So, the steady-state amplitude and the frequency of oscillation of the system are obtained by equating the right-hand sides of Eqs. (22) and (23) to zero.

pA A11 d ¼ 11  cos h 2 2 2

ð24Þ

  d a 1 1 sin h ¼ A þ b 9  n A11 2 2 2 2

ð25Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi d2 1 1 p ¼ 10   a þ 2b 9  11 A10 2 2 2 A2 A10

ð26Þ

where p is the measure of detuning of the system from resonance. The plot of A as a function of p for a given E and system parameters R and C is called a frequency response curve. Each point of this curve corresponds to a singular point in a different state plane; there is one state plane for each combination of parameters. For the case of no series resistance (R1 = 0) Eq. (26) can be written as:

p¼

A10 210



sffiffiffiffiffiffi d2 A2

 ða2 Þ

ð27Þ

and

ð17Þ

For a small difference between the natural frequency and excitation frequency, which is defined by a detuning factor p, where p is the deviation of the oscillation from the frequency of the system, we can write:

p¼1

  d dA 1 1  b 9  11 A11 A_ ¼ sin h  2 2 2 2

ð21Þ

Eliminating h from both Eqs. (24) and (25) we calculate the value if p as:

The nonlinear restoring component of Eq. (13) is:

f2 ðs/Þ a b n1 11 ¼ /þ s / s x2 C x2 C x2

The flux linkage is now considered to vary slowly both in amplitude and in phase, and can be considered to be of the form:

/ ¼ AðsÞ cosðhðsÞÞ

and



ð20Þ

ð6Þ

P didt3 then:

di1 di2 du ¼ dt du dt

€þ u

ð19Þ

2

differential equation of the ferroresonance circuit is given in following equation:

_  € ¼ eðtÞ u

a C x2

ð18Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 E 1 p ¼ 10   Asx R2 C x 2 A10

ð28Þ

We use this analysis for deriving the FFT plots and frequency analysis in the following sections. The stable attracting point exhibits fundamental as well as harmonic and subharmonic regions of operation. The stability of the different portions of the response curves can be determined by investigating the nature of the singular points of Eqs. (4.24) and (4.25). This is done by eigenvalue analysis of the Jacobian matrix of Eqs. (24) and (25) [33].

4

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9

2 6 JðA; hÞ ¼ 4

3  PA þ 2111 An 2 7 h i 5 a þ 11b 1  1 A11 2 29 211

h i  a2 A1  b 219  2111 A11  2p þ 11 A10 29

ð29Þ

Table 2 Parameters of simulated power system.

For R1 = 0.0 and n = 11, matrix of (29) can be written as:

" JðA; hÞ ¼

 a2 A1

 PA þ 4:882  104 A11 2

 2p þ 5:3711  103 A10

a

#

ð30Þ

A k2 þ kð0:5a þ 0:5aAÞ  ða2  p2 Þ þ 2:929  103 pA11 4  2:622  106 A21 ¼0

ð31Þ

Solving the above quadratic equations will result in two solutions. The real part of the eigenvalue is then used for phase plan analysis. The simulated curves for different q coefficients are shown in Fig. 4. In this paper nonlinear core loss model is defined as a nonlinear resistance [27]:

R2 ¼

Rmin  Rmax

a þ Rmax 1 þ ujuj

Actual value

Per unit value

Sbase Vbase Ibase Rbase C

25 MVA 635 kV 131 A 484 X 7.77 nF 377 (rad/s) – – 0.159Vl

– – – – 1 pu 26 pu 1.4 pu – –

x a

2

The eigenvalues of the above matrix are calculated by setting jkI  JðA; hÞj ¼ 0

Parameters

K E

3. Metal oxide varistor model A surge arrester is a device designed to protect power system equipments from overvoltages or sparks. A surge protector cause to limit the overvoltage by either blocking or shorting to ground any unwanted voltages above a safe threshold. Power system equipment especially power transformer protection is playing an important role in normal operation of the power system. The MOV is one of the important components of overvoltage protection system. This paper describes modeling of MOV in parallel to

ð32Þ L

rated

where Rmin is minimum resistance of the transformer core, Rmax is maximum resistance, urated is nominal flux linkage and its value is more than 1. Parameters value of this core model for transformer under study are Rmin = 0.2, Rmax = 1.6, urated = 1.8 and a = 9. Other core models effect has been studied previously in [28,29]. The typical values of power system parameters used in simulations are listed in Table 2. For the initial conditions, we have:

uð0Þ ¼ 0:0; v l ¼

pffiffiffi du ð0Þ ¼ 2 dt

Rl

C

ð33Þ RB

The Bode diagram of the system is the plot of the coefficient matrix of the system voltage versus frequency to show the frequency response of the magnitude and Bode phase of the studied system. The Bode plot of state space variables is used in this paper, study the effect of inputs on the outputs in magnitude and phase.

Fig. 5. Equivalent circuit of MOV.

1.6 1.4 1.2

flux (pu)

RL

1 0.8 0.6 q=11

0.4

q=9

0.2

q=7 q=5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

i (pu) Fig. 4. Nonlinear characteristics of transformer core with different values of q.

0.8

5

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9

three regions: low, medium and the high current regions. At low currents, the MOV can be treated as a high value resistor (RL) and at very large currents the low value resistance (RB) of the MOV dominates the MOV response. In between, (RI) has the ideal MOV property, which is strongly nonlinear. The ideal varistor characteristic (V–I curve) in the range from few lA to tens of kA is approximated by the following interpolation:

Table 3 Parameter used for simulation of MOV. Coefficient

S20K250 MOV

B1 B2 B3 B4 C (pF) L (nH)

2.6830619 0.0261918 0.0006173 0.0045183 700 13

logðuÞ ¼ B1 þ B2 logðiÞ þ B3 expð logðiÞÞ þ B4 expðlogðiÞÞ

where i is the current of the MOV and u is the voltage across it. The parameters B1, B2, B3 and B4 depend on the type of MOV. The parameters B1, B2, B3, B4, C and L are given by the manufacturer. The parameters used in simulations for the disk varistor S20K250 are listed in Table 3. MOV consists of cascaded several metal oxide disks inside the same porcelain housing considering the required protecting voltage level. Size of each disk is highly related to its thermal power dissipation capacity. The V–I characteristic of the MOV is modeled by the following equation:

C

R1

i1

+Vc -

R2

e(t)

Vl

V I ¼ K i ð Þ1=a V ref Iref

Fig. 6. Equivalent circuit of system including MOV.

transformer for fast suppression of ferroresonance. MOV is frequently used for the surge protection in low-voltage power installations or can be used as single protective device or may be installed as a component in a complex protection circuit containing different stages. In this case, when it is connected in parallel to the transformer, the voltage and current characteristics of the transformer have to share with the MOV according to the energy handling capacity of the transformer. Fig. 5 shows an equivalent circuit for a MOV, where L is an inductance of conducting leads and C is the capacitance of the device package and zinc oxide material. Considering the conduction mechanism of the MOV, the resistive region of the voltage–current characteristic can be divided into

Voltage (pu)

(a)

ð35Þ

where V is the resistive voltage drop, I is the MOV current, K and a are constant parameters. Now, the MOV model can be added to the system model as shown in Fig. 6. The differential equation for the new ferroresonance circuit shown in Fig. 6 can be written, as follows:

_  € ¼ eðtÞ u

u_ R2 C

 R1

 a di1 au þ bu11 1 1 _ Þa  ðu   C k dt C

The general requirements for the ferroresonance are induced voltage, magnetizing inductance, capacitance and low damping. Considering the diversity of transformer winding and core configurations, system connections, system and stray capacitances and the non-linearity involved, the conditions under which

X: 106.2 Y: 3.702

2 0 -2

0

20

40

60

80

100

120

140

160

180

200

Time (msec)

Voltage (pu)

(b)

4 2 0 -2 -4 -2

-1.5

-1

-0.5

ð36Þ

4. Simulation results

4

-4

ð34Þ

0

0.5

1

1.5

2

Flux (pu) Fig. 7. (a) Time domain simulation and (b) Phase plan presenting ferroresonance overvoltage of system for q = 11 (without using MOV).

6

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9

ferroresonance can occur are completely different. But, in the case of occurrence of ferroresonance, MOV connected to the primary or secondary side of the transformer, can protect it against the ferroresonance overvoltages. In this section, two cases are simulated. A sudden change in parameters of the system causes a nonlinear behavior known as ‘‘chaos’’. It describes the time behavior of a system when the behavior is non-periodic. Three important factors should be, to determine the type of the power system behavior. The first one is the time equations here given by Eqs. (6) and (36). The second one is the parameters of the system listed in Table 2. And the last one is a dynamical tool such as phase plan projection and bifurcation diagrams. Fig. 7a and b shows the time domain simulation and phase plan diagram of the simulated system. In Fig. 7b, the ferroresonance overvoltage of the transformer (vertical axis) is shown against the flux of the transformer (horizontal axis). The voltage of the ferroresonance equivalent circuit (E) is set to 2.5 pu in these simulation. Fig. 8a and b shows the FFT and Bode diagram of the same voltage for q = 11, respectively. In these

(a)

simulations, MOV has not been modeled. In Figs. 9 and 10, the model of the MOV is considered in simulations. Fig. 9a and b shows the same variables of Fig. 7a and b after installation of MOV. According to these figures, the amplitude of the overvoltages is limited (clamped) to 1.48 pu. The 1.48 pu is a voltage level, which is determined by MOV. This voltage level can be changed by changing the type of MOV. The effect of MOV on FFT and Bode diagram of the voltage of the transformer for q = 11 is shown in Fig. 10a and b, respectively. The same effect of MOV on overvoltages can be seen in these cases, too. When the magnitude of the input voltage is 2.5 pu, the trajectory of the system has chaotic behavior and the amplitude of the flux and ferroresonance overvoltage reaches to 7 pu. Fig. 9 shows the closed trajectory of the system, which indicates a subharmonic ferroresonance oscillation. These plots clearly show the controlling effect of MOV. The MOV successfully reduces the overvoltage amplitude to 1.48 pu. By comparing these plots with Figs. 7–10, it is obvious that the chaotic ferroresonance is changed to the

Voltage (pu)

1 X: 0.34 Y: 1

0.8 0.6 0.4 0.2 0

0

1

2

3

4

5

6

4

5

6

Normalised Frequency

(b) Voltage (pu)

10

0

-2

10

10

-4

0

1

2

3

Normalised Frequency Fig. 8. (a) FFT and (b) Bode diagram of ferroresonance voltage of system for q = 11 (without using MOV).

(a) Voltage (pu)

2 1 0 -1 -2

0

20

40

60

80

100

120

140

160

180

200

Time (msec)

(b) Voltage (pu)

2 1 0 -1 -2 -1.5

-1

-0.5

0

0.5

1

1.5

Flux (pu) Fig. 9. MOV effect of (a) Time domain simulation and (b) phase plan presenting ferroresonance overvoltage of system for q = 11.

7

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9

(a) Voltage (pu)

1

X: 0.155 Y: 0.3163

0.5

0

0

1

2

3

4

5

6

4

5

6

Normalised Frequency

(b) Voltage (pu)

100 10-2 10-4 10-6

0

1

2

3

Normalised Frequency Fig. 10. MOV effect on (a) FFT and (b) Bode diagram of ferroresonance voltage of system for q = 11.

(a)

Voltage of Transformer (pu)

subharmonic resonance. To generate the bifurcation diagram, the value of the peak voltage of the transformer should be studied as a function of the control parameter. In this paper, the control parameter is the input voltage of the ferroresonance equivalent circuit, i.e. E. For value of q, i.e. 11, the effect of MOV on bifurcation diagrams is studied. Three of these bifurcation diagrams at fixed frequency and including transformer nonlinear core loss are shown in Figs. 11a and b. The chaotic behavior is obvious in these figures. 5000 samples are plotted as function of the voltage of the transformer. In these diagrams from left to right, period-1 is obvious, and then period-2 oscillation can be seen and so on. In the chaotic region, several periodic windows appear and suddenly the

transition occurres from period-1 oscillation to the chaotic and ferroresonance behavior. In Fig. 11b the MOV effect on chaotic ferroresonance is plotted for q = 11. In Fig. 11b, before 0.5 pu, there is a nonlinear region with q = 11, then the voltage of the transformer has a period-1 oscillation. By increasing the core non-linearity, the system behavior remains in the periodic oscillation and MOV can cause ferroresonance drop out. By comparing the bifurcation diagrams of Fig. 11b with the bifurcation diagrams of Fig. 11a, it can be said that the effect of MOV is very important. By varying the parameters, many ferroresonance modes are possible but only some selected cases are shown here. A bifurcation clearly shows a jump from one mode of ferroresonance to another in vast variation of the control parameter. Also, a simulation is

2.5 2 1.5 1

chaotic region

period-5 period-3

0.5 0

0

0.5

1

1.5

2

2.5

3

Input voltage (pu) 1.5

Voltage of Transformer (pu)

(b)

X: 2.91 Y: 1.48

1.45 1.4

period-1

1.35

Noisy Region 1.3 1.25

0

0.5

1

1.5

2

2.5

3

Input Voltage (pu) Fig. 11. Bifurcation diagram of system with q = 11 for ferroresonance overvoltage (a) without and (b) with MOV.

8

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9

6

Voltage of Transformer (pu)

(a)

X: 1.492 Y: 5.628

5 4 3 2 1 0

0

0.5

1

1.5

1

1.5

Capacitance (pu) 2

Voltage of Transformer (pu)

(b)

X: 0.021 Y: 1.632

1.5

1

0.5

0

0

0.5

Capacitance (pu) Fig. 12. Bifurcation diagram of system after varying capacitance with q = 11, ferroresonance overvoltage (a) without and (b) with MOV.

developed to very slowly ramp the capacitance of the system. Due to nonlinearities, the capacitance is ramped both upward and downward, to ensure that as many ferroresonance modes are discovered as possible. Using the bifurcation, the ferroresonance for capacitances of is simulated for both cases with and without using MOV. This corresponds to waveforms of periods 1, 2, 3, 5 and chaotic. Period-3 simply means that the waveform takes three periods of the forcing function to repeat it contains 1/3 harmonics. Fig. 12a and b shows the result of one of simulated bifurcation diagrams for the capacitance as control parameter. Fig. 12a shows the ferroresonance case and Fig. 12b shows the effect of MOV on ferroresonance overvoltages for capacitance variations. The diagram correctly predicts the existence of all modes of ferroresonance at the correct values of capacitance. The actual waveforms simulated are very close for the period’s one, two, and three. Period five is generally correct. 5. Conclusion The dynamic behavior of a transformer in the case of occurance of ferroresonance has been studied including non-linearity in the core loss. Some modes of ferroresonance oscillation have been derived, and then it has been shown that the system greatly affected by MOV. The presence of the MOV results in clamping the ferroresonance in studied system. The MOV successfully limits the chaotic overvoltage to its marginal level. Therefore, in case of switching operation or phase opening causes ferroresonance oscillations in power networks, the transformer can be protected by using the MOV. It has been also shown that the core saturation plays a decisive role in determining the type of ferroresonance oscillations. The results show that a change in the value of the control parameter may originate different types of ferroresonance oscillations, MOV can clamp ferroresonance overvoltages, and the system shows a tendency to damp chaotic oscillations while saturation characteristic with lower knee point has been considered.

References [1] Mork BA, Stuehm DL. Application of nonlinear dynamics and chaos to ferroresonance in distribution systems. IEEE Transactions on Power Deliv 1994;9:1009–17. [2] Bethenod J. Sur le transformateur et résonance. L’Eclairae Electriq 1907:289–96. [3] Butler JW, Concordia C. Analysis of series capacitor application problems. AIEE Trans 1937;56:975–88. [4] Hayashi C. Nonlinear oscillations in physical systems. New York, NY: McGrawHill Book Company; 1964. [5] Saravanaselvan R, Ramanujam R. Isolated ferroresonant solutions in transmission lines in the same right-of-way. Electric Power Energy Syst 2012:1–5. [6] Ben Amar F, Dhifaoui R. Study of the periodic ferroresonance in the electrical power networks by bifurcation diagrams. Electric Power Energy Syst 2011;33:61–85. [7] Bakar AHA, Rahim NA, Zambri MKM. Analysis of lightning-caused ferroresonance in Capacitor Voltage Transformer (CVT). Electric Power Energy Syst 2011;33:1536–41. [8] Varricchio Sergio Luis, Gomes Jr Sergio, Rangel Ricardo Diniz. Three winding transformer s-domain model for modal analysis of electrical networks. Electric Power Energy Syst 2011;33:420–9. [9] Jia Hongjie, Yu Xiaodan, Cao Xiaodong. Impact of the exciter voltage limit to small signal stability region of a three-bus power system. Electric Power Energy Syst 2011;33:1598–607. [10] Milicevic Kruno, Emin Zia. Impact of initial conditions on the initiation of ferroresonance. Electric Power Energy Syst 2009;31:146–52. [11] Elkamchouchi H, Zaky AA. Transient response of a ferroresonant circuit, vol. 7(1). Butterworth & Co (Publishers) Ltd.; 1985. [12] Kavasseri Rajesh G. Analysis of subharmonic oscillations in a ferroresonant circuit. Electric Power Energy Syst 2005;28:207–14. [13] Li Yunge, Shi Wei, Li Furong. Novel analytical solution to fundamental ferroresonance power frequency excitation characteristic. IEEE Trans Power Deliv 2006;21(2):788–93. [14] Piasecki W, Florkowski M, Fulczyk M, Mahonen P, Nowak W. Mitigating ferroresonance in voltage transformers in ungrounded MV networks. IEEE Trans Power Deliv 2007;22(4):2362–9. [15] Lima L, Semlyen A, Iravani RM. Harmonic domain periodic steady-state modeling of power electronics apparatus: SVC and TCSC. IEEE Power Eng Rev 2002:56. [16] Charalambous C, Wang ZD, Osborne M, Jarman P. Sensitivity studies on power transformer ferroresonance of a 400 kV double circuit. Gener Trans Distri, IET 2008;2(2):159–66. [17] Barbisio E, Bottauscio O, Chiampi M, Crotti G, Giordano D. Parameters affecting ferroresonance in LCR electric circuits. IEEE Trans Magn 2008;44(6):870–3.

H. Radmanesh, G.B. Gharehpetian / Electrical Power and Energy Systems 45 (2013) 1–9 [18] Model for ferroresonance analysis of a transformer. IEEE Trans Power Deliv 2008;23(3):1448–56. [19] Radmanesh H, Rostami M. Effect of circuit breaker shunt resistance on chaotic ferroresonance in voltage transformer. Adv Electric Comput Eng 2010;10(3):71–7. [20] Ajaei FB, Sanaye-Pasand M, Rezaei-Zare A, Iravani R. Analysis and suppression of the coupling capacitor voltage transformer ferroresonance phenomenon. IEEE Trans Power Deliv 2009;24(4):1968–77. [21] Moses PS, Masoum MAS, Toliyat HA. Impacts of hysteresis and magnetic couplings on the stability domain of ferroresonance in asymmetric threephase three-leg transformers. IEEE Trans Energy Convers 2011;26(2):581–92. [22] Tseng KH, Cheng PY. Mitigating 161 kV electromagnetic potential transformers’ ferroresonance with damping reactors in a gas-insulated switchgear. Gener Trans Distrib – IET 2011;5(4):479–88. [23] Charalambous CA, Wang ZD, Jarman P, Sturgess JP. Frequency domain analysis of a power transformer experiencing sustained ferroresonance. Gener Trans Distrib – IET 2011;5(6):640–9. [24] Hamid Radmanesh, Mehrdad Rostami. Impacts of new suggested ferroresonance limiter on the stability domain of ferroresonance modes in power transformers considering metal oxide surge arrester effect. Iran J Electric Electron Eng 2011;7(4):283–91.

9

[25] Mozaffari S, Sameti M, Soudack AC. Effect of initial conditions on chaotic ferroresonance in power transformers. IEE Proc Gener Trans Distrib 1997;144:456–60. [26] Dick EP, Watson W. Transformer models for transient studies based on field measurement. IEEE Trans Power Apparat Syst 1981:409–17 [PAS-100]. [27] Emin Z, Tong YK. Ferroresonance experience in UK: simulations and measurements. In: International conference on power systems transients (IPST), Brazil, June 2001. [28] Radmanesh Hamid, Gharehpetian GB, Fathi Hamid. Ferroresonance of power transformers considering non-linear core loss and metal oxide surge arrester effect. Electric Power Compon Syst 2012;40(5):463–79. [29] Al-Anbarri K, Ramanujam R, Saravanaselvan R, Kuppusamy K. Effect of iron core loss nonlinearity on chaotic ferroresonance in power transformers. Electric Power Syst Res 2003;65(1):1–12. [30] Kantorovich LV, Krylov VI. Approximate methods of higher analysis. Interscience Publishers, Inc.; 1958. [31] Migulin A. Basic theory of oscillations. Moscow: Mir Publishers; 1983. [32] Chakravarthy SK, Nayar CV. Series ferroresonance in power systems. J Electric Power Energy Syst 1995;17(4):267–74. [33] Nayfeh A, Mook D. Nonlinear oscillation. John Wiley&Sons; 1979.