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European Journal of Scientific Research

ISSN 1450-216X / 1450-202X Vol. 146 No 4 August, 2017, pp.346 - 362

http://www. europeanjournalofscientificresearch.com

Initiation of Ferroresonant Oscillations in Series

Capacitors of Power System

Ataollah Abbasi

Amirkabir University of Technology

Tehran, Iran, 15875-4413

S. Hamid Fathi

Corresponding Author, Amirkabir University of Technology

Tehran, Iran, 15875-4413

Abstract

Ferroresonance can cause severe overvoltages and heavy currents resulting in

damage to power system equipment and customer installations.Ferroresonance is a

complex phenomenon that occurs in power system. This can create thermal and insulation

problems for power system equipments.It can also cause subharmonic oscillations in power

system. This paper, discusses about ferroresonance in series compensation of power lines.

Series capacitors can increase the power carrying capacity of subtransmission and

distribution lines by reducing voltage regulation. When considering Series capacitor

compensation of distribution lines and subtransmission lines careful consideration needs to

be given to capacitor location. Ferroresonance, ohmic reactive value, transient behavior,

short circuit withstand and capacitor protection conventional design approaches include

shunt connected resistors, spark gaps, metal oxide visitors, thyristor controlled reactors and

bypass switches.A small scale nonlinear single phase Ferroresonant circuit was modeled

with realistic per unit component value. Both 3rd

and 2nd

subharmonicFerroresonance modes

were predicted by modeling.

Index Terms: Series Capacitors, Nonlinear Components, subharmonic Ferroresonance,

power transformer

I. Introduction Ferroresonance is an electrical complex nonlinear phenomenon which can causes thermal and

insulation failure in transmission and distribution systems. Ferroresonance may be initiated by

contingency switching operation, lightning, routine switching, no load conditions, series compensation,

load shedding or capacitor banks connection to the secondary transformer involving a high voltage

transmission line[1], [2].

This phenomenon consist of multiple modes with different frequencies such as: fundamental

frequency, sub harmonic, quasi periodic and chaotic[3]–[5]. The abrupt transition or jump from one

steady state to another is triggered by a disturbance, switching action or a gradual change in values of a

parameter.Ferroresonance causes distortional over voltages and over currents in power network which

not only causes damages to transformer but also causes damages to network instruments like surge

arresters and series capacitances[6]–[8].

Initiation of Ferroresonant Oscillations in Series Capacitors of Power System 347

Unlike resonance, which occurs in RLC circuits by linear capacitances and linear inductances

for a particular frequency, ferroresonance is created by nonlinear inductance due to the core of

transformer. Magnetic core of voltage transformer can be considered as a nonlinear inductance and

Composition of line to line and line to earth capacitances and grading capacitors of circuit breaker can

be considered as a linear capacitance[9], [10].

The ferroresonant oscillations is dependent not only on frequency but also on other factors such

as voltage magnitude of the system, initial magnetic flux condition of transformer iron core, total core

loss in the ferroresonant circuit and moment of switching.

Because of nonlinear nature of ferroresonance, ferroresonant systems are considered to be

nonlinear dynamic systems and linear methods can’t be used to analyze them. Thus, investigation and

analysis of this behavior is performed by more complex numerical methods. Analytical approaches

based on graphical solutions were proposed to show bifurcations in single-phase ferroresonant

circuits[11], [12].In ferroresonance,because of nonlinear characteristics of circuit elements the number

of fixed points is more than one. Thus by variation of system parameters fixed points lose their stability

and regain accordingly.

Bifurcation theory is a useful method for identifying system parameters conducive to

ferroresonance[13], [14]. It enables us to describe and analyze qualitative properties of the solutions

i.e., fixed points, when system parameters change. Studying ferroresonance using bifurcation theory

has been performed. Although study of ferroresonance exists in the recent literature, they demand

relatively high computational resources and are only valid for limited cases. Some of these methods are

valid only in limited cases while creating a bifurcation diagram by a continuation method can be more

systematic and save computational effort[13].

Capacitors have been used for the series compensation of transmission and distribution lines for

many years. Series capacitor installations have been described in the literature as far back as 1954[15].

Pioneer power system engineers were seriously examining the merits and demerits of series capacitors

and analyzing the subharmonicferroresonance phenomena in the 1930's.Series capacitors are now in

common use at the transmission level with hundreds of units in service throughout the world. Although

series capacitors have been in use for a long time they have not found widespread acceptance as a

viable economic power system component at the distribution level[16].

Figure 1 shows the typical voltage profile of a very weak distribution line with a lumped load at

the end. The line resistance and reactance is distributed along the line. As the load varies from light

load to full load there is a substantial voltage difference seen by customers. It is the difference between

light load and full load voltage that in many cases limits the capacity of the line.

Figure 1: Voltage profile of a weak distribution line

Distributed line R & L

L lo

ad

R lo

ad

1.0 1.0

1.11.1

0.9 0.9

Light load

I

Full loadLin

e vo

ltag

e P

.U.

0.80.8

348 Ataollah Abbasi and S. Hamid Fathi

Figure 2 shows the effect of series capacitor compensation on the same line. The power line

supplying the transformer is represented by distributed linear inductance and resistance. The circuit

contains a series capacitor to tune out the effects of the line inductance. When used for series

compensation, the capacitance will normally be chosen so as to tune out all or most of the line

inductance at the power frequency.

Figure 2: Voltage profile of a weak distribution line with series capacitor compensation

C

Distributed line R & L

L l

oad

R l

oad

1.0 1.0

1.051.05

0.95 0.95

Light load

I

Full load

Lin

e v

olt

age

P.U

.

The fundamental issues concerning series capacitors are the same today as they were in the

pioneering days of 1930' s. Series capacitors offer the potential to tune out all or part of the series

inductance of lines at the power frequency. This can result in reduced voltage regulation, enhanced

power transfer capability and improved system stability. Series capacitors are particularly attractive in

controlling voltage fluctuations associated with rapidly varying loads. With such significant potential

for enhanced power transfer capability the question arises as to why series capacitors have not found

widespread use and acceptance at the distribution level.

Series capacitors are not in widespread use at the distribution level because of the generation of

ferroresonantover voltages, fault level problems, problems associated with capacitor withstand of

heavy through fault currents and high cost.

Series capacitors can produce subhannonicferroresonantovervoltage and currents. This

phenomenon is generally not well understood by power system engineers with the result that a series

capacitor installation is considered a high risk option or simply not considered at all. The possibility of

serious damage to capacitors, transformers and customer installations by ferroresonance is of real

concern and requires careful management. The modelling work performed in the course of this paper

has highlighted how destructive ferroresonance can be[17].

Effective solutions to the problems offerroresonance and capacitor protection are of great

potential benefit. Analysis of most large power system distribution networks will identify locations of

high voltage regulation where voltage conditions could be improved by series compensation.

A. Improved Voltage Control with Series Capacitors

Voltage control in electric power systems is of fundamental importance in achieving desired power

flows and maintaining voltage levels within specified limits.

Great engineering effort and capital expenditure is invested in power systems to provide

sufficient "system strength" to maintain voltage levels within the required margins. Distribution lines

are normally limited in their load carrying capacity by either thermal current rating considerations or

excessive voltage drop. In general terms distribution systems supplying high load density areas such as

Commercial Business Districts and areas of high density housing development tend to be current rating

limited. Regions of low load density such as rural areas and regional towns tend to be supplied by

distributionsystems that are voltage drop limited.


High voltage regulation in distribution feeders is not the main limitation in itself. It is the

voltage variation between light load and full load that limits the maximum feeder loads that can be

accommodated. With off circuit transformer taps on distribution transformers, the voltage variation in

the high voltage distribution feeder is directly reflected on to low voltage customers. Australian

Standard AS2926 sets maximum voltage variation at +5% to -5% or 0.95 p.u. to 1.05 p.u. in a system.

There are many engineering approaches to overcoming problems of excessive voltage

regulation including augmentation of lines, construction of additional lines, shunt capacitors, voltage

regulators (on load tap changing auto transformers), on load tap changing transformers and

construction of newsubstations. These approaches often involve large capital costs in areas where there

are low load densities. Development of a low cost series compensation arrangement for distribution

systems could provide significant advantages in selected situations[18].

B. Improved System Stability with Series Capacitors

Series capacitors can increase the stability of power systems by reducing the effective impedance of

lines. Reduced line impedance has the effect of increasing system fault levels and increasing the

strength of interconnection of a distributed network of generators.Of course, there are cases

thatcapacitor installation disrupts the stability of system, which can be referred to as:

1. Ferroresonance:Ferroresonance is a well-documented hazard of series capacitors in distribution

networks. Ferroresonance can result in severe overvoltages in capacitors, distribution

transformers and customer installations.

2. Subsynchronous Resonance: Subsynchronous resonance is a potential hazard with series

capacitors. Subsynchronous resonance involves a low frequency exchange of energy between a

series capacitor and a generator. Subsynchronous resonance can cause the mechanical failure of

generator shafts.

3. Asynchronous Resonance: Asynchronous resonance is another potential problem whereby

motors can lock onto subharmonic frequencies on starting and consume abnormally high

currents[19].

In first section of this paper, use of capacitor in transmission and distribution lines is

investigated and its advantages and disadvantages completelyare explained, as we have seen, one of the

disadvantages of series capacitors is creating of ferroresonance in the circuit.

In second section of this paper a small scale nonlinear single phase Ferroresonant circuit was

modeled with realistic per unit component value.

II. Circuit Description and Modeling Figure 3 shows the standard series ferroresonant circuit where the linear LRC circuit elements are in

series with a saturable transformer. Inorder to gain a basic understanding of ferroresonancebehavior the

transformer can be considered as an open circuit when it is not saturated and a short circuit when it is

saturated. The transformer no load case is being considered. The transformer flux linkage is governed

by equation (1).

(1)

When the transformer is not saturated the current flow is near zero with the result that the

capacitor cannot charge or discharge and hence the capacitor stays at near constant voltage. The

voltage applied to the transformer is the source voltage plus a contribution from the capacitor voltage.

This voltage after a short period of time causes the transformer to saturate. Transformer saturation

results in an effective short circuit across the primary transformer terminals (Vp = 0) and the supply

voltage is suddenly applied across the series LRC elements. The capacitor will charge up toward the

applied supply voltage.

pv dtλ = ∫


Figure 3: Series ferroresonant circuit

LV

V(t)

L R C

RVCV

PV

The rate at which the capacitor voltage can change to form a repetitive pattern of transformer

saturation is a critical factor in determining the possibility offerroresonance. During any ferroresonance

the transformer can only remain in saturation for part of a full cycle. If the circuit is capable of

changing the capacitor voltage by a significant amount (in the order of 5% of the supply voltage)

during part of a cycle then ferroresonance is possible.

In order to predict the possibility of ferroresonance it is useful to examine the ratio of the

natural circuit frequency to the power frequency as defined by equation 2.

(2)

(3)

(4)

(5)

The other key indicator is the X/R ratio of the circuit.Equation 4 shows that the X/R ratio is a

direct measure of the ratio of the decay time constant of the natural circuit behavior (tc) to the power

frequency period (T).

The nonlinear nature of ferroresonance makes it a difficult and complex task to predict.

Computer modelling is the most common way of predicting the possibility of ferroresonance in a

particular circuit.Modelling has shown that the frequency ratio and X/R ratio can provide a simple

method of determining by inspection the possibility offerroresonant states. Table 1 below has proved to

be a good predictor as to how these two simple ratios influence the risk of ferroresonance. The table

and comments below relate to typical power system conditions where a small but significant line

resistance is in the ferroresonant circuit and the transformeris operating near its design voltage and

hence close to saturation[20], [21].

A. Influence of the Frequency Ratio on Ferroresonance Behavior

For the establishment of sustained ferroresonance the circuit must be capable of significantly changing

the capacitor voltage by charging through the seriesinductance on resistance over a small part of a

cycle. For example if the power frequency is 50 hertz the capacitor voltage must be able to change its

voltage by a significant amount in a period much less than 0.02 seconds. The frequency ratio is a key

indicator as to the capability of the circuit to exhibit ferroresonantbahavior.

Table 1: Predictor of Ferroresonance

fr<<1 fr≈ 1 fr>>1

X/R << 1 No Ferroresonance No Ferroresonance No Ferroresonance

X/R ≈ 1 No Ferroresonance Ferroresonance Possible Ferroresonance Possible

X/R >> 1 No Ferroresonance Ferroresonance Possible Ferroresonance Possible

rfβ

ω=

2c

Lt

R=

2

C CC

t tX Lf t

R R T

ω πωπ= = = =

21

2

R

L c Lβ

= −


Case1 fr<<1: Due to the inherent low natural frequency of the circuit the capacitor voltage of

the ferroresonant circuit can change by only a very small amount during any interval the transformer is

in saturation. Under these conditions ferroresonance is not possible.

Case2fr≈1: Under these conditions the capacitor voltage will change by a significant but limited

amount during any interval the transformer is in saturation. Under these conditionsferroresonant states

are possible. Because of the limited change in capacitor voltage at each point of transformer saturation,

the capacitor voltage tends to change in steps that can generate repeating wave forms with subharmonic

fundamentals. Ferroresonance is possible.

Case3 fr>>1: Under these conditions the capacitor voltage can change rapidly and track the

supply voltage during any interval the transformer is in saturation. Under these conditions

ferroresonant states are possible.

(6)

B. Influence of the X/R Ratio on Ferroresonance Behavior

The X/R ratio is a measure of the transient damping characteristic of circuit. A high line resistance

results in a highly damped system with a small X/R ratio.

Case1 X/R<< 1: Where the X/R ratio is much less than 1 the circuit is highly damped by the

line resistance and the generation offerroresonance is not possible.

Case2 X/R ≈1: Under these conditions the circuit is moderately damped and if ferroresonance

establishes it is likely to produce steady state repeating waveforms.

Case3 X/R >> 1: Under these conditions the circuit is veryunderdamped. Where ferroresonance

establishes with a large X/R the circuit may create either repeating or non-repeating

waveforms.Non repeating chaotic circuit behavior is possible.

When analysing series compensated circuits it is very useful to calculate the frequency ratio and

the time constant ratio to gain an understanding of the ferroresonance possibilities.

In the case of series compensation, the series capacitor will normally be chosen so as to tune

out all or most of the line inductance at the power frequency. This means that the frequency ratio fr will

typically be unity or slightly less than unity.

For distribution lines, typical X/R ratios are in the range of 0.1 for small diameter steel

conductor lines to 3 for high capacity lines with bundled conductors.

Modelling has shown that in series single phase compensated distribution lines with typical

ranges of frequency ratio and X/R ratio, the most common type of ferroresonance is the subharmonic

fundamental type with stable repeating waveforms.

Another major source of ferroresonance in power systems is the result of single phase switching

where there is naturally occurring phase to earth capacitance. In general the phase to earth capacitance

is small creating a frequency ratio fr much greater than unity.

Generation of chaotic ferroresonance in power systems is rare but has been identified as

possible in the literature. The generation of chaotic ferroresonance has been reported in an

Electronic circuit by Deane and Hamill. The resonance was generated with a square wave

voltage generator operating at high frequency.

III. Simulation Results A. Circuit Modeling

In order to investigatethe ferroresonant models, a series compensated circuit was modeled with the

component values as shown in Figure 4. The series capacitor was selected to tune out the line

inductance at 50 Hz and demonstrate the generation of both odd and even subharmonics. The

transformer was modelled with no load by equation (6).

50.1 0.892i λ λ= +


Figure 4: Series ferroresonant circuit

V(t)

L= 0.02H

R= 1.55 Ω

C= 507µF

PV

i

The transformer losses were modelled by a 14400Ω resistance across the primary transformer

terminals. The supply frequency is 50 Hz. The transformer used is a single phase 11KV/400V rated at

3MVA. The 400 volt transformer winding is used in the circuit with no connected load. The supply

voltage was sinusoidal. The resulting transformer magnetizing curve is shown in Figure 5.

Circuit represents the series compensation of a distribution line with approximately 11% line

voltage drop at full transformer load. The circuit characterises the compensation of a very high

impedance line. Deviation from standard per unit distribution line values were used to produce the

desired range of ferroresonant conditions.

Fundamental mode of circuit behavior is not unusual and simply reflectsincreasing transformer

magnetizing current with increasing applied voltage.

Figure 5: transformer model magnetizing curve

With running and repeating of simulation in over a range of supply voltages, the 3rd

subharmonic mode is sustainable only over a range of supply voltages from 0.84 to 1.3p.u. Figures 6

and 7 show the Time Domain Modelled voltage, current and flux

Waveformsat a supply voltage of 0.84p.u.

Figure 6: 3rd

subharmonicferroresonance – voltage waveforms

-8 -6 -4 -2 0 2 4 6 8-1.5

-1

-0.5

0

0.5

1

1.5transformer model magnetizing curve

current(A)

flux(w

eb)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-400

-300

-200

-100

0

100

200

300

400

time(s)

ma

gn

itu

de

(vo

lt)

Vs=120 v rms

capacitore voltage

supply voltage

transformer voltage


Figure 7: 3

rdsubharmonicferroresonance – current and transformer flux waveforms

The waveforms show perfect symmetry and hence the absence of any even harmonics. The

graphs show that the transformer saturates twice on the positive side followed by twice on the negative

side. After transient state (0.22 s), pattern repeats every 3 cycles hence the generation of the 3rd

subharmonic fundamental waveform of 16.6 Hz.

When the transformer is being driven into saturation the transformer voltage falls to near zero

causing the source voltage to be applied to the other circuit elements, namely the series LRC. At the

beginning of the cycle, the capacitor voltage is large and negative. The capacitor voltage combined

with the supply voltage result in a high positive transformer voltage peaking at 0.25 cycle time. This

causes a high d

dt

λ which is the cause of the transformer going into saturation.

The transformer voltage falls to low values again when saturation occurs. At this time the

source voltage is effectively applied across the series LRC elements forcing the capacitor with a large

negative voltage to a smaller negative voltage. The change in capacitor voltage is determined by the

natural frequency of the Land C which in this case is 50 Hz. the damping effect of R, the shape of the

transformer (i)λ curve and the time the transformer remains saturated.

When the transformer comes out of saturation, the transformer offers a high impedance to the

circuit and allows minimal current flow. During this part of the cycle the capacitor voltage remains

nearly constant. The small negative capacitor voltage combined with the supply voltage result in a high

positive transformer voltage which forces the transformer into heavy positive saturation.

During the second positive saturation phase, the capacitor charges rapidly and attains a large

positive voltage peaking. The large positive capacitor voltage combined with the supply voltage result

in a large negative transformer voltage which forces the transformer into saturation in the negative

direction.This process repeats over 3 complete 50 Hz cycles and hence the resulting waveform has a

3rd subharmonic fundamental (16.7 Hz) with higher order odd harmonics (e.g. 50 Hz, 83.3 Hz, 116.7

Hz etc.)[22].

The phase plane analysis is a graphical method, in which the time behaviour of a system is

represented by the movement of state variables of the system in state space coordinates. As time

elapses, the states position move on a trajectory. If the trajectory is a single closed line, then the system

is periodic. In a chaotic system, however, the trajectory will never close on itself as the cycles are

completed.Three state variables are required to describe behavior of the series compensated circuits.

These can be selected as:

• line current

• capacitor voltage

• transformer primary current

The transformer current and the line current are almost identical with the small difference being

due to the effect of the 1500 ohm resistor used to represent the transformer no load losses. Based on the

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-3

-2

-1

0

1

2

3

4

time(s)

magnitude(A

,Web)

Vs=120 v rms

line current

flux linkage


assumption that the line and transformer currents are approximately equal, the circuitscan be

represented by two state variables. The state variables are line current and capacitor voltage.

Figure 8: state plane trajectories of circuit, VS=120 v_rms

Figures 8 to 11 show state plane trajectories of circuit for 3rd subharmonicferroresonance that

are useful for understanding of performance of circuit.

Figure 9: state plane trajectories of circuit considerate time for 2s simulation

Figure 10: Three-dimensional diagram contain three-variable capacitor voltage, line current and flux

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-200

-150

-100

-50

0

50

100

150

200state plane trajectories - Vs=120 v rms

capacitor current(A)

capacitor

voltage(v

)

-3-2

-10

12

-200

-100

0

100

2000

0.5

1

1.5

2

line current(A)

state plane trajectories - Vs=120 v rms

capacitor voltage(v)

tim

e(s

)

-2

-1

0

1

2

-200

-100

0

100

200-1.5

-1

-0.5

0

0.5

1

1.5

line current(A)



flu

x(W

eb

)


Figure 11: state plane diagram(flux – current)

Figure 8 shows the phase plane trajectories of circuit at a supply voltage of 0.84per unit. The

shows are the 50Hz trajectory, 3rd subharmonic trajectory from the time domain model.

The ferroresonant circuit also demonstrated the ability to generate 2nd subharmonic voltages,

currents and fluxes. In power systems, the linearity of most system components and the symmetry of

the transformer B-H loops normally dictates that even harmonics cannot occur.

With running and repeating of simulation in over a range of supply voltages, the 2rd

subharmonic mode is sustainable only over a range of supply voltages from 1.4 to 1.68 p.u. Figures 12

and 13 show the Time Domain Modelled voltage, current and flux waveforms at a supply voltage of

1.45 p.u.

Figure: 122

rdsubharmonicferroresonance – voltage waveforms

The key feature of the 2nd subharmonicferroresonant state is that the transformer goes into

"light" saturation twice in one direction followed by one "heavy" saturation in the opposite direction.

This process repeats over 2 complete 50 Hz cycles and hence the resulting waveform has a 2nd

subharmonic fundamental (25 Hz) with higher order harmonics both odd and even (e.g. 50 Hz, 75 Hz,

100 Hz etc.)[23].

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5state plane trajectories - Vs=120 v rms

line current(A)

flux(W

eb)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-800

-600

-400

-200

0

200

400

600

time(s)

magnitude(v

olt)

Vs=215 v rms

capacitore voltage

supply voltage

transformer voltage


Figure: 132

rdsubharmonicferroresonance – current and transformer flux waveforms

It should be noted that in the 2nd subharmonicferroresonant state, the currents generated are in

the order of full load current of the transformer and the capacitor voltage is approximately twice the

level expected at full load. These are extremely high levels of current and voltage in a circuit with no

connected load.

Simulations show that with increasing in voltage magnitude of supply,current of circuit

increases but subharmonic voltage and current increase until a point then reduce that display a negative

incremental impedance occurs in circuit. This circuit have linear RLC elements with the following

characteristics:

X/R= 8.25, Frequency ratio= 0.998

With reference to Table 1 "Predictor ofFerroresonace" this circuit falls into to the category of

fr≈1 and X/R >>lin which it is correctly predicted that ferroresonance is possible.

Figures 12 to 17 show state plane trajectories of circuit for 2rd subharmonicferroresonance that

are useful for understanding of performance of circuit.

Figure 14: state plane trajectories of circuit,VS=215 v_rms

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-8

-6

-4

-2

0

2

4

6

time(s)

magnitude(A

,Web)

Vs=215 v rms

line current

flux linkage

-8 -6 -4 -2 0 2 4 6-300

-200

-100

0

100

200

300state plane trajectories - Vs=215 v rms

line current(A)

capacitor

voltage(v

)


Figure 15: state plane trajectories of circuit considerate time for 1s simulation

Figure 16: Three-dimensional diagram contain three-variable capacitor voltage, line current and flux

Figure 17: state plane diagram(flux – current)

-10

-5

0

5

-400

-200

0

200

4000

0.2

0.4

0.6

0.8

1

line current(A)



tim

e(s

)

-10

-5

0

5

-400

-200

0

200

400-1.5

-1

-0.5

0

0.5

1

1.5

line current(A)



flux(W

eb)

-8 -6 -4 -2 0 2 4 6-1.5

-1

-0.5

0

0.5

1

1.5state plane trajectories - Vs=215 v rms

line current(A)

flux(W

eb)


Figure 18 shows the variation in capacitor voltage over a range of applied source voltages

i.e.showsthe impact ofchanges inthesupply voltageontheFerroresonance.With running and repeating of

simulation in over a range of supply voltages, the 3rd subharmonic mode is sustainable only over a

range of supply voltages from 0.84 to 1.3 p.u and the 2rd subharmonic mode is sustainable only over a

range of supply voltages from 1.4 to 1.68 p.u.

Figure 18: Bifurcation diagram Ferroresonant capacitor voltage

B. Impact of Core Hysteresis of Transformer on the Ferroresonance in Series Capacitors

Magnetic hysteresis phenomenon also plays an important role in the behavior of a transformer

ferroresonance, however, most transformermodels available for analysis Ferroresonance, hysteresis

effects ignore or consider the main loop. This modeling not reflect real physical behavior of the

magnetic core with respect Ferroresonancephenomenon[24].

Recent studiesshow that providing non-linear areas offerro magnetic iron core such as

saturation, hysteresis, eddycurrentsare importantin Ferroresonance phenomenon. Structure of

hysteresis significantly in Ferroresonance stability especially for chaosand sub-harmonics modes

inmodels of transformer core has an effect. Common approximation to approximate the core non-linear

area with single-valued function without hysteresis like a piece of linear, polynomial, and exponential

has proven to be in adequate for studies about the Ferroresonance in transformers[25].

In the circuitof figure 4 was used from themagnetic curve of figure 5 tosimulate the behavior

ofthe transformer core. In the following is used formfigure 19 (characteristic a)formodeling the

behavior ofthetransformer corethatis closertoreality.

After the simulation and observe wave form sand frequency spectrum can understand that the

hysteresis loop has positive role for damping of Ferroresonance. Figures 20 and 21 show thatsub-

harmonic Ferroresonance completely is eliminated that depending on the width of the loop andit's slope

will be differentthe amount of limiting and capacitor currentand voltage significantly are decreased.

If we use from figure 22 (characteristic b) for transformer characteristicas is observable in

figures 23 and 24 (state plane trajectories and frequency spectrum respectively), due to increase in the

slope of the characteristicand to become thinner hysteresis loop in to characteristic (a), Ferroreson

anceoccurs.



Figure 20


Figure 19:

20: current and transformer flux waveforms considerate hysteresis loop (a)

Figure 21: state plane trajectoriesconsiderate hysteresis loop (a)

0-4

-3

-2

-1

0

1

2

3

4

magnitude(A

,Web)

-3-50

-40

-30

-20

-10

0

10

20

30

40

50

cap

acit

or

volt

age

(v)


: transformer magnetizing curve (characteristic a)

current and transformer flux waveforms considerate hysteresis loop (a)

state plane trajectoriesconsiderate hysteresis loop (a)

0.05 0.1 0.15

-2 -1


Initiation of Ferroresonant Oscillations in Series Capacitors of Power System

transformer magnetizing curve (characteristic a)



0.2 0.25 0.3

time(s)

Vs=160 v rms

-1 0


line current(A)





0.3 0.35 0.4

line current

flux linkage

1 2






0.45 0.5

line current

flux linkage

3


359

360

IV. In this paper over voltage and over curren

model of hysteresis properties of the core, a nonlinear polynomial function is used for modeling core

losses. Since ferroresonant phenomenon is a kind of chaotic dynamics,

behavi

these values cause harmonic oscillation, sub

compensation offers great potential for electricity supply authorities. Effective series compensation can

360

IV. ConclusionIn this paper over voltage and over curren



behavior in power system are time

System behavior for diffe


Increasing the power carrying capacity of transmission and distribution lines by series


Fig

Figure

Fig

Conclusion In this paper over voltage and over curren



or in power system are time

System behavior for diffe


easing the power carrying capacity of transmission and distribution lines by series


Figure 22: transformer magnetizing curve (characteristic b)

ure 23: state plane trajectories

Figure 24: Frequency response of circuit (characteristic b)

In this paper over voltage and over curren



or in power system are time-domain waveform, phase plane diagram and bifurcation diagram.

System behavior for different values of voltage




-8-200

-150

-100

-50

0

50

100

150

200

capacitor

voltage(v

)

ransformer magnetizing curve (characteristic b)

state plane trajectories

Frequency response of circuit (characteristic b)

In this paper over voltage and over current due to ferro



domain waveform, phase plane diagram and bifurcation diagram.

rent values of voltage

these values cause harmonic oscillation, sub-harmonics in the system.



-6 -4 -2



state plane trajectories considerate hysteresis loop (b)


t due to ferroresonance




rent values of voltage was investigated and revealed that increase

harmonics in the system.



0 2 4


line current(A)

Ataollah Abbasi and S. Hamid Fathi


considerate hysteresis loop (b)


resonance discussed. In order to have a better




was investigated and revealed that increase

harmonics in the system.



4 6 8




considerate hysteresis loop (b)


discussed. In order to have a better


losses. Since ferroresonant phenomenon is a kind of chaotic dynamics, the tools used to analyze this








he tools used to analyze this








he tools used to analyze this






reduce voltage regulation in distribution and transmission systems and provide an effective

countermeasure to voltage dips caused by load fluctuations.

A thorough understanding of the possible ferroresonant modes of bahavior is essential when

considering series compensation of distribution and subtransmission lines.

Simulationsshowedthatthehysteresisloop has positive effectonreducing theFerroresonancebutit

should bethat if theloopiswiderandless steep, it increaseinfluence,butwideloopincreases loss andwarmin

the transformerandcreate cost problem in generation.

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