Chaotic Ferroresonance In Voltage Transformer Biology Essay

Abstract- Ferroresonance or nonlinear resonance is a complex electrical phenomenon, which may do over electromotive forces and over currents in the electrical power system which endangers the system dependability and uninterrupted safe operating. This paper surveies the consequence of circuit ledgeman shunt opposition on the control of helter-skelter ferroresonance in a electromotive force transformer. It is expected that this opposition by and large can do ferroresonance ‘dropout ‘ . For verification this aspect Simulation has been done on a one stage electromotive force transformer rated 100VA, 275kV. The magnetisation feature of the transformer is modeled by a single-value two-term multinomial with q=7. The simulation consequences reveal that sing the shunt opposition on the circuit ledgeman, exhibits a great mitigating consequence on ferroresonance over voltages.. Significant consequence on the oncoming of pandemonium, the scope of parametric quantity values that may take to chaos along with ferroresonance electromotive forces has been obtained and presented.

Index Terms- Circuit surfs Shunt Resistance, Chaos, Bifurcation, Ferroresonance, Voltage Transformers

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Ferroresonance over electromotive force on electrical power systems were recognized and studied every bit early as 1930s. Kieny foremost suggested using pandemonium to the survey of ferroresonance in electric power circuits [ 1 ] . He studied the possibility of ferroresonance in power system, peculiarly in the presence of long capacitive lines as highlighted by happenings in France in 1982, and produced a bifurcation diagram bespeaking stable and unstable countries of operation. Then the combination of nonlinear Fe nucleus inductance with series capacitance has been investigated and shown that this nucleus is the most possible instance for happening ferroresonance in the power system. These electrical capacities can be due to figure of elements, such as the line-to-line electrical capacity, parallel lines, music director to earth electrical capacity and circuit ledgeman rating electrical capacity. A Particular Ferroresonance Phenomena on 3-phase66kV VT-generation of 20Hz zero sequence uninterrupted electromotive force is given in [ 2 ] . Typical instances of ferroresonance are reported in [ 3 ] , [ 4 ] , in these documents power transformer and VTs has been investigated due to ferroresonance over electromotive forces. Digital simulation of transient in power system has been done in [ 5 ] . Application of nonlinear kineticss and pandemonium to ferroresonance in the distribution systems can be found in [ 6 ] . The susceptibleness of a ferroresonance circuit to a quasi periodic and frequence locked oscillations has been presented in [ 7 ] , in this instance, probe of ferroresonance has been done upon the new subdivision of pandemonium theory that is quasi periodic oscillation in the power system and eventually ferroresonance appears by this path. Modeling Fe nucleus nonlinearities has been illustrated in [ 8 ] . Mozaffari has been investigated the ferroresonance in power transformer and consequence of initial status on this phenomena, he analyzed status of happening pandemonium in the transformer and suggested the decreased tantamount circuit for power system including power switch and trans [ 9 ] , [ 10 ] .

The extenuating consequence of transformer connected in analogue to a MOV arrester has been illustrated in [ 11 ] . Analysis of ferroresonance in electromotive force transformer has been investigated by Zahawi in [ 12 ] and [ 13 ] . Analysis of Ferroresonance Phenomena in the Power Transformers Including Neutral Resistance Effect has been reported in [ 14 ] . Ferroresonance Conditions Associated with a 13 kilovolt Voltage Regulator During Back-feed Conditions is given in [ 15 ] . Performance of Various Magnetic Core Models in Comparison with the Laboratory Test Results of a Ferroresonance Test on a 33 kilovolt Voltage Transformer investigated in [ 16 ] . Extenuating Ferroresonance in Voltage Transformers in Ungrounded MV Networks has been reported in [ 17 ] . An Approach for Determining the Subsystem Experiencing and Producing a Bifurcation in a Power System Dynamic Model has been reported in [ 18 ] .

In all old surveies, possibility of happening ferroresonance and nonlinear phenomena in power system had been studied and control of this unwanted phenomena has non been studied, besides the effects of circuit ledgeman shunt opposition on VT ferroresonance in the deeper instance has non been investigated. Current paper surveies the consequence of circuit ledgeman shunt opposition on the control of ferroresonance over electromotive forces in VT. It is shown that by sing this opposition, the behaviour of system has been changed and ferroresonance bead out.


During Voltage Transformer ( VT ) ferroresonance an oscillation occurs between the nonlinear Fe nucleus induction of the VT and bing electrical capacities of web. In this instance, energy is coupled to the nonlinear nucleus of the electromotive force transformer via the unfastened circuit ledgeman rating electrical capacity or system electrical capacity to prolong the resonance. The consequence may be impregnation in the VT nucleus and really high electromotive force up to 4p.u can theoretically gained in worst instance conditions. The magnetising feature of a typical 100VA VTs can be presented by 7 order multinomial [ 12 ] .

These VTs fed through circuit ledgeman rating electrical capacity, and studied utilizing nonlinear kineticss analysis and bundles such as Rung kutta Fehlberg algorithm and Matlab Simulink. Fig.1 shows the individual line diagram of the most normally encountered system agreement that can give rise to VT ferroresonance [ 13 ] . Ferroresonance can happen upon gap of disconnector 3 with circuit ledgeman unfastened and either disconnector 1 or 2 closed. Alternatively it can besides happen upon closing of both disconnector 1 or 2 with circuit ledgeman and disconnector 3 unfastened.

Fig..System one line diagram agreement ensuing to VT Ferroresonance

The system agreement shown in Fig. 1 can efficaciously be reduced to an tantamount circuit as shown in Fig. 2.

Fig.2. Basic reduced tantamount ferroresonance circuit [ 13 ]

In Fig. 2, E is the rms supply phase electromotive force, Cseries is the circuit ledgeman rating electrical capacity and Cshunt is the entire phase-to-earth electrical capacity of the agreement. The resistance R represents a electromotive force transformer nucleus loss that has been found to be an of import factor in the induction of ferroresonance. In the peak current scope for steady-state operation, the flux-current linkage can be approximated by a additive characteristic such as where the coefficient of the additive term ( a ) corresponds closely to the reciprocal of the induction. However, for really high currents the Fe nucleus might be driven into impregnation and the flux-current characteristic becomes extremely nonlinear, here the feature of the electromotive force transformer is modeled as in [ 9 ] by the multinomial

( 1 )


The multinomial of the order seven and the coefficient B of equation ( 1 ) are chosen for the best tantrum of the impregnation part that was obtained by the comparing between different estimates of the impregnation parts against the true magnetisation feature that was obtain by gumshoe and Watson [ 5 ] . It was found that for equal representation of the impregnation features of a electromotive force transformer nucleus, the exponent Q may get value 7 [ 10 ] . Fig.3 shows simulation of these Fe nucleus feature for q=5, 7, 11. The basic electromotive force transformer ferroresonance circuit of Fig.2 can be presented by a differential equation. Because of the nonlinear nature of the transformer magnetizing features, the behaviour of the system is highly sensitive to alter in system parametric quantity and initial conditions. A little alteration in the value of system electromotive force, electrical capacity or losingss may take to dramatic alteration in the behaviour of it. A more suited mathematical linguistic communication for analyzing ferroresonance and other nonlinear systems is provided by nonlinear dynamic methods. Mathematical tools that are used in this analysis are phase program plot, clip sphere simulation and bifurcation diagram.

Fig.3. Nonlinear features of transformer nucleus with different values of Q


Mathematical analysis of tantamount circuit by using KVL and KCL Torahs has been done and Equations of system can be presented as below:

( 2 )

( 3 )

( 4 )

( 5 )

( 6 )

( 7 )

( 8 )

( 9 )

( 10 )

Where is supply frequence, and E is the rms supply phase electromotive force, Cseries is the circuit ledgeman rating electrical capacity and Cshunt is the entire phase-to-earth electrical capacity of the agreement and in equation ( 1 ) a=3.4 and b=0.41 are the seven order multinomial sufficient [ 13 ] .


In this instance, system under survey is similar with the instance above, but the theoretical account of circuit ledgeman has been changed. Equivalent Thevenin circuit of this instance has been illustrated in fig.4.

Fig.4. Basic reduced tantamount ferroresonance circuit

Nonlinear Equation of this circuit is as below:

( 11 )

( 12 )

( 13 )

( 14 )

( 15 )

( 16 )

( 17 )

In this theoretical account of circuit, is paralleled with the Cseries and its value is. Other parametric quantities of system are similar with the instance 1.

In the undermentioned analysis, alternatively of utilizing existent values of circuit parametric quantities etc. , system equations are made dimensionless by utilizing per unit values, equation ( 10 ) may be written as

( 18 )

Where g and Q are the driving force amplitude and muffling parametric quantity, severally, given by

( 19 )

( 20 )

And equation ( 17 ) can be written as:

( 21 )

Simulation RESUILTS

Eq. ( 18 ) contains a nonlinear term and does non hold simple analytical solution. So the equations were solved numerically utilizing an embedded Runge-Kutta-Fehlberg algorithm with adaptative measure size control. Valuess of E and were fixed at 1p.u, matching to AC supply electromotive force and frequence. C series is the C.B scaling electrical capacity and its value evidently depends on the type of circuit ledgeman. In this analysis C series is fixed at 0.5nF and C shunt vary between 0.1nF and are obtained for initial values of at t=0, stand foring circuit ledgeman operation at maximal electromotive force. This corresponds to a lightly damped, lightly goaded system in which q=47.24 and g=0.02. In this province, system for both instances, with and without circuit ledgeman shunt opposition has been simulated for E=1, 3p.u. it shows that the system under survey has a periodic behaviour for E=1p.u and helter-skelter behaviour for E=3p.u while in the instance of using shunt opposition, system behavior remain periodic for E=1,3p.u Corresponding stage program diagrams has been shown the clearance consequence of using the shunt opposition to the system and it is shown in figs.9, 10 for E=1p.u and figs.11 and 12 for E=3p.u.

Fig.5. Time domain simulation for periodic gesture without C.B shunt opposition shunt opposition, E=1p.u

Fig.6. Time domain simulation for periodic gesture with C.B shunt opposition, E=1p.u

Fig.5, 6 show Time sphere simulation for these two instances that represent the sinusoidal moving ridge with a frequence equal to the system frequence, i.e. 50 rhythms per second for E=1p.u, but comparing the simulation consequence for E=3p.u in the instance of sing shunt opposition it has been shown the consequence of shunt opposition on system behaviour this is presented in figs.7 and 8.

Fig.7. Time domain simulation for helter-skelter gesture without C.B shunt opposition, E=4p.u

Fig.8. Time domain simulation for clamping the helter-skelter gesture with C.B shunt opposition, E=4p.u

In fig 7 System behaviour has been simulated without sing shunt opposition, clip sphere simulation is wholly helter-skelter and ferroresonance over electromotive forces reaches up to 10p.u, in the equal status, by using shunt opposition, this over electromotive forces has been damped and behaviour of system goes to linear part, harmonizing to the fig 8 system frequence is equal the periodic status and electromotive force of transformer has been fixed to 1.8p.u.

Fig.9. Phase program diagram for periodic gesture without C.B shunt opposition, E=1p.u

Fig 9 shows the corresponding stage program diagram when there is no shunt opposition consequence and electromotive force of system is 1p.u. in this secret plan, flux linkage has been simulated versus electromotive force of transformer, it is shown that in the instance of normal status system behaviour is periodic and there is no Ferroresonance phenomenon in it.

In the following province, by sing the shunt opposition consequence, it is shown that the electromotive force of transformer range to 0.16p.u with the frequence equal with the input frequence that has been shown in fig 10.

Fig.10. Phase program diagram for periodic gesture with C.B shunt opposition, E=1p.u

Due to the unnatural status such as exchanging action or other instances that may do transient phenomena, When input electromotive force of power system goes up to 3p.u, in the instance of without sing shunt opposition consequence, ferroresonance over electromotive force on electromotive force transformer reach up to 9p.u, this province has been shown by stage program diagram in fig 11.

Fig.11. Phase program diagram for helter-skelter gesture without C.B shunt opposition, E=4p.u

By using shunt opposition consequence to the system while the input electromotive force is 3p.u, it is shown in fig 12 that ferroresonance over electromotive forces clamp to 1.6p.u and it is presented in fig12.

Fig.12. Phase program diagram for clamping the helter-skelter gesture with C.B shunt opposition, E=4p.u

It evidently shows that circuit ledgeman shunt opposition clamps the ferroresonance overvoltage and keeps it in E=2p.u. System parameters that sing for these instance of simulation are as below:

, , , E=1,3p.

Another tool that was used for work outing the nonlinear equation of studied system is bifurcation diagram. In this paper, it is shown the consequence of fluctuation in the electromotive force of system on the Ferroresonance overvoltage in the VT, and eventually the consequence of using circuit ledgeman shunt opposition on this overvoltage by the bifurcation diagrams.

Fig.13.Bifurcation diagram for electromotive force of transformer versus electromotive force of system, without C.B shunt opposition


Fig.13. clearly shows the ferroresonance over electromotive forces on VT when the electromotive force of system increases up to 4p.u. Parameters value of the system in this instance are as below:

In fig.13 when E=0.25p.u, electromotive force of VT has a period1 behaviour and system plants under normal status, in E=0.57p.u that has been shown by point1, its behaviour is still period1 and after this electromotive force, all of a sudden crisis takes topographic point and system behaviour goes to the helter-skelter part, after that, when the input electromotive force range to 1.2p.u, system comes out of helter-skelter part, once more in the point2 and 3, bifurcation takes topographic point, by this path system behaviour goes to chaos, eventually between point4 and 5, system remains in ferroresonance oscillation. After point5, system comes out of pandemonium once more and so behaves linearly. It is shown that system behaviour has period duplicating bifurcation logic and there are many resonances in the system behaviour. Bifurcation diagram with the same parametric quantity in the instance of using C.B shunt opposition analogue to the VT is shown in fig.14.

Fig.14. Bifurcation diagram for electromotive force of transformer versus electromotive force of system, corresponded by fig.13 with using C.B shunt opposition

It is shown that by using this opposition, system behaviours coming out of helter-skelter part and C.B shunt opposition can clamp over electromotive forces from 8p.u to 1.8p.u, in this instance there is a leap in the electromotive force of transformer when the input electromotive force reaches up to 1.8p.u which is indicated by point1. In the existent systems, upper limit over electromotive force that VT can stand is 4p.u. If over electromotive force additions more, it can precisely do VT failure. By reiterating simulation for broad scope of parametric quantity value, tabular array ( 1 ) and ( 2 ) has been obtained and is shown in the appendix portion.


Low capacity Voltage Transformers fed through circuit ledgeman rating electrical capacity have been shown to exhibit cardinal frequence and helter-skelter ferroresonance conditions similar to high capacity power transformers fed via capacitive matching from neighbouring beginnings. Repeated simulation of the system ‘s nonlinear differential equation has shown that a alteration in the value of the tantamount circuit electrical capacity to Earth, perchance as a consequence of a alteration in system constellation, can give rise to different types of ferroresonance overvoltage. It has besides been shown that helter-skelter ferroresonance provinces are non likely to happen under practical site conditions. C.B shunt opposition successfully can do ferroresonance bead out and can command it, in the instance of using C.B shunt opposition system shows less sensitiveness to initial status and fluctuation in system parametric quantities. A comprehensive apprehension of the possibilities that exist for ferroresonance is really desirable for applied scientists so that they can run their systems outside unsafe parts and can be after the enlargement of systems without heightening the possibility of ferroresonance.


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