Modeling Ferroresonance in Single-PhaseTransformer Cores with Hysteresis

P. S. Moses, Student Member, IEEE, and M. A. S. Masoum, Senior Member, IEEEpaul.s.moses@gmail.com m.masoum@curtin.edu.au

Department of Electrical and Computer EngineeringCurtin University of Technology, Perth, WA, 6845, Australia

Abstract—Ferroresonance is a highly dynamic and nonlinearpower quality phenomenon caused by nonlinear inductances inferromagnetic materials and power system capacitances. It isnotorious for causing severe damage to power systems. Thispaper carries out an investigation into single-phase transformerferroresonance initiated by switching transients. A hysteretic coremodel is implemented which includes major and minor hysteresisloops for the study of dynamic nonlinear phenomena. Time-domain waveforms of transformer flux, voltage and magnetizingcurrent are computed from the model. Poincare and phase-planeportraits are used to examine the stability domain of observedferroresonance modes.

Index Terms—Ferroresonance, hysteresis, nonlinear models,power quality.

I. INTRODUCTION

FERRORESONANCE in power networks involving non-linear transformers and capacitors has been well re-

searched for nearly a century. However, it is only in recentyears that nonlinear transformer modeling techniques havebegun to approach the level of sophistication required foraccurate ferroresonance studies. To that end, the importance ofhysteresis nonlinearities in dynamic and transient simulationstudies and its impact on the stability domain of ferroresonancemodes has recently been demonstrated [1]–[3].

Ferroresonance can be understood as a complex oscillatoryenergy exchange between magnetic field energy of nonlineartransformer/reactor cores and electric field energy of nearbycapacitances (e.g., series compensated lines or circuit breakergrading capacitors). Without adequate dissipation through nor-mal loads and losses, a substantial amount of energy sloshesback and forth within a power system and manifests as over-voltages and currents exhibiting high levels of distortion. Thishas caused significant equipment damage in several cases andcontinues to be a large safety hazard [4]–[6].

Over the years, research in ferroresonance has concentratedinto three main areas: (1) improving analytical methods andtransformer models, (2) development of transformer protectionand mitigation strategies and (3) case studies of system levelimpacts [7], [8]. Despite the extensive literature available inthis area, ferroresonance continues to be a challenging problemto analyze, predict and understand due to its highly nonlinearand dynamic behavior. Researchers must adopt complex math-ematical notions such as chaos theory to gain insight into thisphenomenon [9]–[11].

The four generally accepted ferroresonance modes whichcan occur are (1) fundamental ferroresonance (period-1),

(2) subharmomic ferroresonance (e.g., period-3), (3) quasi-periodic ferroresonance and (4) chaotic ferroresonance. Thelast two are non-periodic modes. There is also the possibilityof mixed modes or unstable modes where gradual systemvariations or perturbations cause sudden jumps (known asbifurcations) from one mode to another [12], [13]. Froma purely mathematical standpoint, these modes are due tomultiple competing solutions (known as attractors) to a sys-tem of nonlinear differential equations of an electromagneticcircuit. The nonlinearity is due to the magnetic properties offerromagnetic material.

In this paper, a single-phase transformer model includinghysteresis nonlinearity is implemented to study possible fer-roresonance behavior (e.g., subharmonic modes). The modelis used to generate and plot time-domain waveforms includingflux, voltages and magnetizing currents, as well as Poincareand phase-plane diagrams for selected values of series andshunt capacitors initiating ferroresonance conditions.

II. HYSTERESIS CORE MODELING

The modeling of hysteresis has evolved significantly sincethe 1970s when digital nonlinear core models were first beingdeveloped. References [14], [15] provide a thorough historicalreview of progress in hysteresis modeling. Early modelingattempts indirectly incorporated hysteresis by the use of single-value nonlinear inductors in parallel with a resistor represent-ing eddy-current and hysteretic losses. Models evolved to usefamilies of ascending and descending curve functions for theinclusion of major and minor loop effects of ferromagneticmaterial. Some authors ignore minor loops and focus only onmajor loops or make use of scaling factors on major hysteresisloops to derive minor loops. These nonlinear approximationsare typically based on piece-wise, hyperbolic, trigonometric ordifferential equations.

Today, there is still a tendency for transformer power qualitystudies to ignore hysteresis and use anhysteretic approxima-tions of the B − H characteristic because of the modelingcomplexity and computational burden associated with hystere-sis nonlinearities. This simplification can be justified for somestudies because transformer design has improved over theyears and hysteresis loop widths have narrowed significantly toappear almost anhysteretic. Therefore, for steady-state powerquality studies such as harmonic power flow in nonlinear trans-formers, single-valued anhysteretic functions are consideredacceptable.

Proceedings of the 8th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING

ISSN: 1790-2769 246 ISBN: 978-960-474-131-1

On the other hand, for the study of dynamic, transientand nonsinusoidal power system behavior, the representationof minor hysteresis loop trajectories becomes important asadditional operating points are created by the dynamic ex-citation of a nonlinear hysteretic core. This is especially truein ferroresonance where major and minor loop trajectories canpotentially generate more ferroresonant operating points [3].For this paper, a suitable nonlinear hysteretic core model ofa single-phase transformer is implemented to study ferroreso-nance.

III. SINGLE-PHASE TRANSFORMER CORE MODELINCLUDING HYSTERESIS NONLINEARITY

A PSPICE computer model was developed for a single-phase transformer with hysteresis nonlinearity (Fig. 2). Ascalar hysteresis model based on [16] is implemented forthis work due to its simplicity in implementation for circuitsimulation. The magnetic flux density b and field intensityh are the main parameters for this model. However, for thispaper, the hysteresis equations are modified such that fluxlinkages and magnetizing currents are computed instead ofb and h (i.e., b→ λ and h→ im). These parameters are moreaccessible in PSPICE and easily measured from laboratorytests. The modified hysteresis equations are as follows:

dimdt

=dλ

dt· 1

L0 + λ−(im)−λ(im)λ−(im)−λ+(im)

(dλ+(im)dim

− L0

)for

dλ

dt≥ 0

(1a)dimdt

=dλ

dt· 1

L0 + λ(im)−λ+(im)λ−(im)−λ+(im)

(dλ−(im)dim

− L0

)for

dλ

dt< 0 (1b)

where λ+ and λ− are the limiting ascending and descendingcurve functions dependent on magnetizing current im. L0

is the inductance or slope in the saturated region along thelimiting hysteresis curves. As noted from (1), the basis forthis model is the use of slope functions for λ+ and λ−to compute magnetization processes. This is derived fromthe assumption that domain wall motion density (Barkhausenjumps) are proportional to the growth of domain regions whichincrease with field strength [16].

The flux linkage and current relationships for the hysteresismodel are

e (t) =dλ (t)dt

(2)

i (t) = im (t) + ic (t) (3)

Before equation (1) can be computed, the ascending anddescending limiting hysteresis loop segments must be spec-ified. This paper proposes nonlinear analytical expressions todescribe the ascending (λ+) and descending (λ−) limitinghysteresis loop segments. The proposed nonlinear function (4)can accurately be fitted to measured λ− i characteristics of areal nonlinear transformer.

f (im) = sgn (im) · α loge (β |im|+ 1) (4)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5

−4

−3

−2

−1

0

1

2

3

4

5

Magnetizing Current im (t) [A]

Cor

eFl

uxLi

nkag

eλ

(t)

[V.s

]

LimitingDescendingBranch λ− (im)

LimitingAscendingBranch λ+ (im)

Fig. 1. The developed hysteresis model is shown under ramped sinusoidalexcitation to demonstrate the capability of the model to form major and minorhysteresis loops constrained by ascending (λ+) and descending (λ−) limitinghysteresis curve functions.

The fitting parameters α and β control the vertical andhorizontal scaling. The ascending limiting loop segment isderived from (4) by shifting the function f (im) to the right byan increment of σ and similarly the descending loop functionis obtained by shifting the function to the left by σ (eqs. (5)-(6), Fig. 1). Effectively, σ controls the width of the limitinghysteresis loop.

λ+ (im) = −f (im − σ) (5)λ− (im) = f (im + σ) (6)

The slope of the above ascending and descending functionsmust be computed before the modified hysteresis equations (1)can be used. The ascending and descending slope functionsdλ±(im)dim

are derived through differentiation of (5)-(6) withrespect to magnetizing current.

dλ± (im)dim

=αβ

β |im ∓ σ|+ 1(7)

Equation (1) is then computed from substitution of (2), (5), (6)and (7). The next step is for the program to select one of thetwo equations (1) for dim

dt based on whether the magnetizationis increasing or decreasing. An IF statement in PSPICE isused to select one of the two equations in (1) based on thesign of the induced voltage (2). The magnetizing current imis evaluated by integrating (1).

There are two approaches to realize the above expressions ina PSPICE circuit. The first method is to implement a controlledvoltage source for the induced voltage (2) based on (1). Thesecond approach is more abstract but more numerically stableand is thus the favored approach. Equation (1) is realized byimplementing a circuit loop consisting of an arbitrary capacitorwith inherent current and voltage relationship together with acontrolled current source.

i = CdV

dt= 1 · dλ

dt(8)

Proceedings of the 8th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING

ISSN: 1790-2769 247 ISBN: 978-960-474-131-1

( )e t

( )u t

sR

sL

( )i t ( )ci t

cR

( )mi t

Nonlinear Hysteresis

Core Model

t=0.1s

seriesC

shuntC

Fig. 2. Single-phase ferroresonance circuit with hysteresis core model

The capacitor voltage is in fact the flux linkage and itscurrent is governed by the controlled current source based on(1). A large resistance (1012 Ω) is placed in parallel with thecapacitor to suppress numerical convergence problems.

The developed hysteresis model can form major and minorhysteresis loops based on specified λ+ and λ− functions. Thisis demonstrated in Fig. 1 for linearly increasing sinusoidalexcitation.

IV. SIMULATION RESULTS

The scenario under investigation is an unloaded single-phasetransformer operating under steady-state conditions interruptedby a switch which is opened at t = 0.1s. This case is represen-tative of single-phase fuse or circuit breaker action resultingin a series ferroresonance circuit as shown in Fig 2. This canalso occur for three-phase transformer banks (i.e., 3 single-phase transformers) where one of the phases has developeda fault. The series capacitance could be grading capacitors incircuit breakers, stray capacitances in bus-bars and feeders, orpossibly series reactive power compensation capacitor banks.The impedance values and λ − i hysteresis loop data ofthe circuit are based on a real single-phase 440 V 50 Hztransformer (Rs = 9.4 Ω, Ls = 6.34 mH). The parametersfor the hysteresis model are listed in the appendix. In thefollowing simulations, selected values of Cseries and Cshuntwere chosen to generate different ferroresonance modes.

Fig. 3 demonstrates period-3 type ferroresonance initiatedby the opening of the switch at t = 1.0s when Cseries andCshunt are set to 10 and 38µF , respectively. The flux andvoltage waveforms show sustained harmonic distortions withexcessive magnetizing currents. The phase-plane and Poincarediagrams indicate the system settling to a stable attracting limitcycle.

For Cseries and Cshunt set to 10 and 22µF , harmonicdistortions have increased for flux and voltage waveforms (Fig.4). Furthermore, the transient period from normal operation toferroresonance has a longer duration compared to the previouscase. The group of five dots in the Poincare map indicate this isperiod-5 type ferroresonance. The phase-plane diagram showsthe presence of competing attractors in the stability domain.

Similarly, when Cseries and Cshunt are set to 29 and38µF , respectively, the transformer exhibits extremely dis-torted period-7 ferroresonance voltages (Fig. 5). The phase-

0 0.2 0.4 0.6 0.8 1−1

0

1

v(t

)(p

u)

0 0.2 0.4 0.6 0.8 1−10

0

10

i m(t

)(p

u)

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

Time (sec)

φ(t

)(p

u)

(a)

−2−1

01

2

−10

−5

0

5

100

0.5

1

1.5

φ (t) [pu]dφ(t)

dt

Tim

e[s

ec]

Poincare section

(b)

Fig. 3. Subharmonic ferroresonance (Period-3) occurring at Cseries =10µF and Cshunt = 38µF ; (a) time-domain waveforms indicating dis-torted flux and voltages with large magnetizing current, (b) Poincare andphase-plane diagrams indicating the system settling to stable steady-stateferroresonance after switching transient.

plane portrait indicates the presence of multiple competingattractors in the stability domain.

Finally, to emphasize the sensitivity of this phenomenonto a small change in a system parameter, Cseries is variedby a small amount from the previous case to 28µF and thesimulation is repeated (Fig. 6). The resulting waveforms showunstable ferroresonance modes which appear to be period-3 type lasting for only a few cycles before dampening out.It is interesting to observe that due to the flux and voltagerelationship (2), the flux waveform takes a long time tostabilize compared to the voltage waveform.

In some simulations, the developed PSPICE model had dif-ficulties converging to a solution. This is most likely due to thediscontinuities presented by (1) at the point of magnetizationreversal when the model switches between equations. Prob-lems can be circumvented somewhat by reducing the time-stepin the solver but at the expense of increasing processing time.

Proceedings of the 8th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING

ISSN: 1790-2769 248 ISBN: 978-960-474-131-1

0 0.5 1 1.5 2

−1

0

1v

(t)

(pu)

0 0.5 1 1.5 2

−5

0

5

i m(t

)(p

u)

0 0.5 1 1.5 2−4

−2

0

2

4

Time (sec)

φ(t

)(p

u)

(a)

−2−1

01

2

−10

−5

0

5

100

0.5

1

1.5

φ (t) [pu]dφ(t)

dt

Tim

e[s

ec]

Poincare section

(b)

Fig. 4. Subharmonic ferroresonance (Period-5) occurring at Cseries =10µF and Cshunt = 22µF ; (a) time-domain waveforms for flux, magne-tizing current and voltage, (b) Poincare and phase-plane diagrams showingresulting steady-state ferroresonance with competing cyclical attractors.

V. CONCLUSION

This paper demonstrates a single-phase transformer modelincluding hysteresis nonlinearity applied to the study of fer-roresonance. The model is used to compute a number ofuseful outputs such as phase-plane trajectories and bifurcationresponse, as well as, fluxes, voltages and current waveforms. Itis observed that the seemingly innocuous switching action canlead to many different ferroresonance modes causing havoc ina power system. A method for defining and modeling limitinghysteresis loop segments for incorporation into an existinghysteresis model exhibiting major and minor loop effects isproposed and simulated in PSPICE.

APPENDIXHYSTERESIS MODEL PARAMETERS

α = 0.7, β = 1000, σ = 0.06, L0 = 0.07

0 0.5 1 1.5 2−1

0

1

v(t

)(p

u)

0 0.5 1 1.5 2−10

0

10

i m(t

)(p

u)

0 0.5 1 1.5 2

−2

0

2

Time (sec)

φ(t

)(p

u)

(a)

−1.5−1

−0.50

0.51

1.5

−10

−5

0

5

100

0.5

1

1.5

φ (t) [pu]dφ(t)

dt

Tim

e[s

ec]

Poincare section

(b)

Fig. 5. Subharmonic ferroresonance (Period-7) occurring at Cseries =29µF and Cshunt = 38µF ; (a) time-domain waveforms indicating extremevoltage and flux distortions, (b) Poincare and phase-plane diagrams showingresulting steady-state ferroresonance with multiple competing attractors.

0 0.5 1 1.5 2−1

0

1

v(t

)(p

u)

0 0.5 1 1.5 2−10

0

10

i m(t

)(p

u)

0 0.5 1 1.5 2−4

−2

0

2

4

Time (sec)

φ(t

)(p

u)

Fig. 6. Temporary ferroresonance condition for Cseries = 28µF andCshunt = 38µF . The waveforms exhibit Period-3 type ferroresonance forthe first few cycles after switch is opened before dampening out.

Proceedings of the 8th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING

ISSN: 1790-2769 249 ISBN: 978-960-474-131-1

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[3] A. Rezaei-Zare, R. Iravani, and M. Sanaye-Pasand, “Impacts of trans-former core hysteresis formation on stability domain of ferroresonancemodes,” IEEE Trans. Power Del., vol. 24, no. 1, pp. 177–186, Jan. 2009.

[4] R. C. Dugan, “Examples of ferroresonance in distribution,” in Proc.IEEE Power Engineering Society General Meeting, vol. 2, 13–17 July2003.

[5] D. A. N. Jacobson, “Examples of ferroresonance in a high voltage powersystem,” in Proc. IEEE Power Engineering Society General Meeting,vol. 2, 13–17 July 2003.

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Paul S. Moses (S’09) received his B.Eng. and B.Sc. degrees in ElectricalEngineering and Physics in 2006 from Curtin University of Technology, Perth,Australia. He was awarded an Australian Postgraduate Award scholarship in2009 and is presently working towards a PhD degree in Electrical Engineering.His interests include nonlinear electromagnetic phenomena, power qualityand protection, renewable energy systems and power electronics. He hasalso performed scientific research for the Defence Science and TechnologyOrganisation, Department of Defence, Australia.

Mohammad A.S. Masoum (SM’05) received his B.S., M.S. and Ph.D.degrees in Electrical and Computer Engineering in 1983, 1985, and 1991,respectively, from the University of Colorado at Boulder, USA. Dr. Masoum’sresearch interests include optimization, power quality and stability of powersystems/electric machines and distributed generation. Currently, he is anAssociate Professor at the Electrical and Computer Engineering Department,Curtin University of Technology, Perth, Australia and a senior member ofIEEE.

Proceedings of the 8th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING

ISSN: 1790-2769 250 ISBN: 978-960-474-131-1