Electric Power Systems Research 76 (2006) 873–879

Investigation of subharmonic ferroresonantoscillations in power systems

K.S. Preetham, R. Saravanaselvan ∗, R. RamanujamDepartment of Electrical and Electronics Engineering, Anna University, Chennai 600 025, India

Received 4 May 2005; accepted 16 November 2005Available online 18 January 2006

Abstract

Steady state bifurcation diagram based on continuation technique is a versatile tool for analyzing the behaviour of ferroresonance circuit, sinceit gives the visualization of multiple solutions. Stability analysis of the solutions obtained by steady state bifurcation diagram [K. Al-Anbarri, R.Ramanujam, T. Keerthiga, K. Kuppusamy, Analysis of non-linear phenomenon in MOV connected transformer, IEE Proc. Generation TransmissionDistrib.148 (6) (2001) 562–566; K. Al-Anbarri, R. Ramanujam, R. Saravanaselvan, K. Kuppusamy, Effect of iron core loss nonlinearity on the chaoticferroresonance in power transformers, Electric Power Syst. Res. 65 (2003) 1–12; K. Al-Anbarri, R. Ramanujam, Ch. Subba Rao, K. Kuppusamy,EsRDtK1astsc©

K

1

mqomeit

(

0d

ffect of circuit configuration on chaotic ferroresonance in a power transformer, Electric Power Components Syst. 30 (2002) 1015–1031] demarcatestable fundamental ferroresonant solution, unstable fold and flip segments. Flip segment in steady state bifurcation diagram [K. Al-Anbarri, R.amanujam, T. Keerthiga, K. Kuppusamy, Analysis of nonlinear phenomenon in MOV connected transformer, IEE Proc. Generation Transmissionistrib.148 (6) (2001) 562–566; K. Al-Anbarri, R. Ramanujam, R. Saravanaselvan, K. Kuppusamy, Effect of iron core loss nonlinearity on

he chaotic ferroresonance in power transformers, Electric Power Syst. Res. 65 (2003) 1–12; K. Al-Anbarri, R. Ramanujam, Ch. Subba Rao,. Kuppusamy, Effect of circuit configuration on chaotic ferroresonance in a power transformer, Electric Power Components Syst. 30 (2002)015–1031] implies unstable fundamental solution, but does not reveal the range, order and stability of subharmonics. This paper proposes anpproach to explore the unstable flip segment further and unearth the hidden subharmonics. As a consequence of this approach, continuum of stableolutions of well-defined domains can be identified. It is found that the existence of subharmonic solutions is sensitive to various factors such asransformer saturation index, non-linearity in the core loss and presence of an arrester. Exhaustive studies reveal that in the presence of an arrester,ubharmonic mode exists for wider range of bifurcation parameter value at a high value of core saturation index, there by reducing the range ofhaotic attractors.

2005 Elsevier B.V. All rights reserved.

eywords: Ferroresonance; Subharmonics; Period doubling; Evolving Poincare map; Steady state bifurcation diagram

. Introduction

Existence of different periodic and non-periodic steady stateodes viz, fundamental low and high voltage, subharmonics,

uasi periodic and chaotic is the main characteristic of ferrores-nance. Methods based on theory of bifurcation and chaos areore suitable for analysing ferroresonance. In Ref. [4] Mozaffari

t al. established the period doubling route to chaos by construct-ng bifurcation diagrams. These diagrams are generated withhe Poincare sampling of state variables at the exciting source

∗ Corresponding author. Tel.: +91 44 22203349.E-mail addresses: nandhu sara@yahoo.com, nandhu varna@yahoo.com

R. Saravanaselvan).

frequency. The shortcomings of these bifurcation diagrams areinformation gap during a jump in the solution and an extendedperiod of integration for lightly damped system.

The application of bifurcation diagram for actual ferrores-onant solution is well reported in literature. The steady statebifurcation diagram can be generated using a continuation tech-nique [1–3] and pseudo arc length method [5]. Here the formerapproach is extended to unearth the subharmonics in the flipsegments.

The flip segments in the steady state bifurcation diagram cancontain subharmonic and/or chaotic attractor. The subharmonicsexhibit loop phenomena characterized by period doubling andperiod halving. The later has been referred to as “reverse bifur-cation” [6]. The analysis using the technique mentioned above

378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2005.11.005

874 K.S. Preetham et al. / Electric Power Systems Research 76 (2006) 873–879

besides detecting the fundamental solutions, identifies subhar-monic and chaotic modes. The system under study represents

a case of ferroresonance that occurred on 1100 kV system ofBonneville Power Administration [7,8].

The sensitivity of subharmonic solutions has been investi-gated for various levels of core saturation in the presence of anarrester. In addition the non-linearity in core loss is taken into anaccount. The effect of non-linearity in core loss on chaotic solu-tions has been analysed in Ref. [2]. The performance of metaloxide arrester exposed to ferroresonant conditions in padmounttransformers are analysed in Ref. [9]. In Ref. [1], the influenceo

2

a

b

i

w

dc

i

wte

x

wfl

The resulting state equations with the non-linearity in coreloss are

px1 = x2, px2 = eth(t) − x3 − x2 − R2(ax1 + bxq1 + h0 + h1x2 + h2x

22 + h3x

32) − L2(ax2 + qbx

q−11 x2)

L2(h1 + 2h2x2 + 3h3x22)

,

px3 = ax1 + bxq1 + h0 + h1x2 + h2x

22 + h3x

32

C(2.3)

The state equations with the inclusion of arrester are given by

px1 = x2, px2 = eth(t) − x3 − x2 − R2(ax1 + bxq1 + h0 + h1x2 + h2x

22 + h3x

32) − L2(ax2 + qbx

q−11 x2)

L2(h1 + 2h2x2 + 3h3x22)

,

px3 = ax1 + bxq1 + h0 + h1x2 + h2x

22 + h3x

32 + ((eth(t) − x3)/k)αsign of(eth(t) − x3)

C(2.4)

3. Steady state bifurcation diagram

The qualitative change in state variables when a parameteris varied gradually can be easily visualized by constructing abifurcation diagram. The continuation algorithm starts from theknown solution and uses a predictor-corrector scheme to findsubsequent solutions at different bifurcation parameter values.The construction of steady state bifurcation diagram involvesfour steps:

(

(

3

b

f arrester parameters on chaotic solutions has been reported.

. System description and modelling

The equivalent circuit for ferroresonance investigation isdopted from Ref. [2], and shown in Fig. 1.

The saturation characteristics of the transformer is modelledy a two-term polynomial [10]:

Lm = aφ + bφq (2.1)

here q = 7, 11, . . ..The core loss non-linearity is modelled as per the method

escribed in Ref. [11]. A third-order polynomial given below isonsidered adequate to match the non-linear characteristics [2].

Rm = h0 + h1vm + h2v2m + h3v

3m (2.2)

here vm is the voltage across the transformer. In general,opological approach can be used to formulate the state spacequations [12,13].

The state variables chosen are

1 = φ, x2 = pφ, x3 = vc

here vc is the voltage across the capacitor; φ is the transformerux linkage; pφ = vm is the voltage across the transformer.

(i) Calculation of initial steady state solution by Newton–Raphson method.

(ii) Prediction of next solution along the tangent path using alocally parameterized continuation technique.

iii) Correction of predicted solution using a Newton–Raphsontechnique.

iv) Stability analysis of the corrected solution by computingcharacteristic multipliers.

.1. Calculation of steady state solution

Assume that a non-linear circuit in Fig. 1 can be describedy the following system of first-order differential equation:

∂x∂t

= f(x, µ, t) (3.1)

Fig. 1. Ferroresonant circuit.

K.S. Preetham et al. / Electric Power Systems Research 76 (2006) 873–879 875

where f is the vector of non-linear functions; x the vector of statevariables and µ is the bifurcation parameter.

The complete response of a circuit, described by the stateequation (3.1) consists of two components, namely, the transientand steady state. There are one or more values of the initial statevector x0, for which the complete response to the state equation(3.1) will not contain the transient component.

Let x(t) be the solution vector to the state equation (3.1) withinitial steady state x(0) = x0. Then

x(t) =∫ t

0f(x(t), µ, t) dt + x0 (3.2)

the above equation can be written as

x(t) = F(x0, µ, t) (3.3)

The solution of Eq. (3.3) at any particular value of the parameterµ and time t depends on the initial state vector x0. The statevector x0 is varied systematically until the value of the functionF(x0, µ, t) at t = T is the same as at t = 0.

Defining

g(x0) = F(x0, µ, T ) =∫ T

0f(x(t), µ, t) dt + x0 (3.4)

gTas

x

E

g

Af

x

w

[

TtTb{wb

To compute the partial derivative, Eq. (3.4) have to be numer-ically integrated twice from t = 0 to t = T, first with x(k)

0 as the

initial state and then with x(k)0 + �x0 as the perturbed initial

state. Then using Eqs. (3.4) and (3.9), expression for the partialderivative turns out to be

∂gi(x0)

∂x0j

= xi(T, [x0](k) + [�x0]) − xi(T, [x0](k))

�x0j

(3.10)

where xi(T, [x0](k) + [�x0]) is the result of second integration,xi(T, [x0](k)) is the result of first integration and �x0j is theperturbation value of x0j. It should be noted that �x0j is the onlynon-zero component of the perturbation vector �x0. Successiveapplication of Eq. (3.7) will yield a converged solution for agiven value of the parameter µ.

The next solution along the expected solution path is pre-dicted, by applying a continuation procedure.

3.2. Prediction of next solution

To trace the possible solution path as the bifurcation param-eter value varies, the later has to be considered as an additionalvariable. Rewriting Eq. (3.4) as

g

w

g

taEt

w

r

uepebceh

(x0) is a function of only x0 for a given parameter value µ sinceis constant. Then the initial state vector x0 which give rise tosteady state solution x(t) with no transient component must

atisfy the following equation:

0 = g(x0) (3.5)

q. (3.5) can be rewritten as

ˆ(x0, µ) = x0 − g(x0, µ) = 0 (3.6)

pplying Newton–Raphson algorithm to Eq. (3.6) and solvingor (k + 1)th iterate of x0,

(k+1)0 = x(k)

0 − [J(k)(x0)]

−1[g(x0)(k)] (3.7)

here [J(k)(x0)] is the n × n Jacobian matrix and is given by

J(k)(x0)] =

[1 −

{∂g(x0)

∂x0

}(k)]

(3.8)

he Jacobian elements in Eq. (3.8) are evaluated by computinghe partial derivatives by numerical differentiation procedure.he partial derivative occurring in the ijth element of the Jaco-ian is

∂gi(x0)

∂x0j

}(k)

= gi(x(k)0 + �x0) − gi(x

(k)0 )

�x0j

(3.9)

here �x0 is a small perturbation vector with non-zero pertur-ation for the variable x0j.

(x0, µ) = F(x0, µ, T ) =T∫0

f(x(t), µ, t) dt + x0

hich is a function of x0 and µ, we get

(x0, µ) = x0 − g(x0, µ) = 0 (3.11)

In order to predict an appropriately sized step in a directionangent to the solution path, the tangent vector has to be evalu-ted. The tangent is evaluated by taking the first differential ofq. (3.11) with respect to the variables x0, µ and setting equal

o zero:

(3.12)

here J(k)(x0) = ∂g([x0], µ)/∂x0 and [J(k)

(µ)] = ∂g([x0], µ)/∂µ.

The dimension of sub matrices J(k)(x0), J(k)

(µ) are n × n and n × 1,espectively. The dimension of the tangent vector is (n + 1) × 1.

The set of Eq. (3.12) are not solvable because the number ofnknowns is greater than the number of equations. One morequation, which describes the rate of change of one of the com-onent of tangent vector has to be included to make the system ofquations solvable. Initially, the bifurcation parameter µ woulde taken as the continuation parameter with a positive rate ofhange. For subsequent prediction steps the continuation param-ter should be chosen as that component of tangent vector, whichas the maximum magnitude, along with its sign. The augmented

876 K.S. Preetham et al. / Electric Power Systems Research 76 (2006) 873–879

form of Eq. (3.12) can be written as:

(3.13)

where Ui is 1 × (n + 1) row vector with all elements equal zeroexcept the ith element, which is equal to unity. The tangent vectoris evaluated by solving Eq. (3.13).

The predicted solution is given by

(3.14)

where “*” stands for the predicted solution, and σ is the stepsize.

3.3. Correction based on local parameterisation

By using the predicted values of the variables, Eq.(3.11) is solved simultaneously with Eq. (3.15) by applyingNewton–Raphson algorithm.

x

wx

wJi

3

poIwpotb

3

cmwoim

4. Steady state bifurcation diagram with subharmonicsolutions

Fig. 2 shows the steady state bifurcation diagram for period 1solution [2]. The unstable period 1 flip segments do not conveyany information about subharmonics.

They are further explored and superimposed on the solutionsin Fig. 2 and is shown in Fig. 3. The solution segments A, B andC shown in Fig. 3 correspond to stable subharmonics.

5. Simulation results and discussion

Typical parameter values of the core saturation characteristicare given: a = 0.000375, b = 7.3824E−24 for q = 7; a = 0.000375,b = 1.5648E−37 for q = 11.

The coefficients of the core loss non-linear func-tion are as follows: h0 = −3.5213E−03; h1 = 5.7869E−07;h2 = −1.4167E−12; h3 = 1.21105E−18.

Fl

Fig. 3. Steady state bifurcation diagram for period 1 solution superimposed withsubharmonics: q = 7, linear core loss model without arrester.

i = η (3.15)

here η is the appropriate value of the continuation parameteri.

Eq. (3.15) represents the local parameterisation process,hich identifies each solution along the path being traced. The

acobian matrix for the correction process is evaluated by numer-cal differentiation as in the prediction stage.

.4. Stability of the converged solution

The stability of the converged solution is evaluated by com-uting the eigenvalue of the Jacobian matrix. If the modulusf all eigenvalues is less than ‘1’ the solution point is stable.f the modulus of any one of the eigenvalue is greater than ‘1’ith negative real part the solution point will be on the unstableath bounded by flip bifurcation points. If the modulus of anyne of the eigenvalue is greater than ‘1’ with positive real parthe solution point will be on the unstable path bounded by foldifurcation points.

.5. Steady state bifurcation diagram for subharmonics

Initially, unstable period 1 segments are identified using theontinuation method described earlier. To obtain the subhar-onic solution path the algorithm is repeated with time t = nT,here ‘n’ is a positive integer (n = 2, 4, 8, . . .) indicating therder of subharmonics. The flip bifurcation points provide thenitial steady state solution for tracing the subsequent subhar-

onic solution path.

ig. 2. Steady state bifurcation diagram for period 1 solution: q = 7, linear coreoss model without arrester.

K.S. Preetham et al. / Electric Power Systems Research 76 (2006) 873–879 877

Table 1Case matrix

q Linear core loss Non-linear core losswithout arrester

Non-linear core losswith arrester

7 A.1 (Fig. 3) A.2 (Fig. 8) A.3 (Fig. 11)11 B.1 (Fig. 7) B.2 (Fig. 10) B.3 (Fig. 12)

Fig. 4. Magnification of segment A in Fig. 3: Case A.1.

The electrical parameters of the circuit shown in Fig. 1 aregiven below

L2 = 0.4032 H; R2 = 11.3 �; Rm = 4.49 M�;

C = 0.02616 �F

The arrester parameters are α = 26; K = 1 266 812.Several cases and combinations have been investigated and

only the representative ones are presented here. Table 1 containsthe cases considered with the reference for results.

F

Fig. 6. Conventional Poincare map of period 8 solution in Fig. 4 at Eth = 3.1250.

Fig. 7. Steady state bifurcation diagram for period 1 solution superimposed withsubharmonics: Case B.1.

Fig. 8. Steady state bifurcation diagram for period 1 solution superimposed withsubharmonics: Case A.2.

ig. 5. Evolving Poincare map of period 4 solution in Fig. 4 at Eth = 3.089.

878 K.S. Preetham et al. / Electric Power Systems Research 76 (2006) 873–879

Fig. 9. Magnification of segment A in Fig. 8.

Fig. 10. Steady state bifurcation diagram for period 1 solution superimposedwith subharmonics: Case B.2.

Fig. 11. Steady state bifurcation diagram for period 1 solution superimposedwith subharmonics: Case A.3.

Fig. 12. Steady state bifurcation diagram for period 1 solution superimposedwith subharmonics: Case B.3.

Fig. 4 shows the magnified view of segment A in Fig. 3. Herethe period doubling cascade is traced up to period 8. Beyondperiod 8 the algorithm virtually becomes “motion less”. Thispoint corresponds to an extremely small increment in changein bifurcation parameter value. Typically subsequent solutionsindicate chaotic behaviour as can be verified by time domainsimulation. Similar period doubling cascades are obtained forsegments B and C.

Figs. 5 and 6 confirm the periodicity of period 4, period 8solution, respectively.

From Figs. 3 and 7 it can be inferred that the range of Ethfor which stable subharmonics exists decreases with increase insaturation index of the transformer. For higher value of saturationindex, the algorithm is able to detect up to subharmonics oforder 4.

Fig. 13. Magnification of region C1 in Fig. 12.

K.S. Preetham et al. / Electric Power Systems Research 76 (2006) 873–879 879

Fig. 14. Phase plot of period 8 solution in Fig. 13 at Eth = 6.101513.

Fig. 15. Phase plot of period 16 solution in Fig. 13 at Eth = 6.106013.

However a small flip segment (A in Fig. 8) is identified atlower value of bifurcation parameter. Exhaustive analysis of thisflip segment results in interrupted period doubling cascade fol-lowed by reverse bifurcation as shown in Fig. 9.

For higher values of saturation index, the effect of non-linearity in core loss is pronounced. Some of the chaotic solu-tions are converted into periodic solutions, Fig. 10.

The inclusion of arrester and non-linearity in core loss prac-tically eliminates subharmonics for low saturation index. Thisis illustrated in Figs. 8 and 11.

For higher value of saturation index, range of bifurcationparameter for which subharmonics occur is widened as can beobserved from Figs. 10 and 12.

Fig. 13 shows that the period doubling is more significantfor higher values of saturation index with all non-linearitiesincluded.

Figs. 14 and 15 confirm the periodicity mentioned in Fig. 13.

6. Conclusion

An approach to unearth the subharmonic solutions in unsta-ble flip segment, detected by steady state bifurcation diagramof fundamental solution is presented. Sensitivity analysis showsthat the range of bifurcation parameter for which stable subhar-monic solutions exist decrease with increase in core saturation.Inclusion of non-linearity in the core loss improves the occur-rence of subharmonics at higher value of bifurcation parameter.In general the arrester is beneficial for lower value of saturationindex. At high values arrester transforms most of the chaoticsolutions to subharmonic solutions.

References

[1] K. Al-Anbarri, R. Ramanujam, T. Keerthiga, K. Kuppusamy, Analysisof nonlinear phenomenon in MOV connected transformer, IEE Proc.Generation Transmission Distrib. 148 (6) (2001) 562–566.

[2] K. Al-Anbarri, R. Ramanujam, R. Saravanaselvan, K. Kuppusamy, Effectof iron core loss nonlinearity on the chaotic ferroresonance in powertransformers, Electric Power Syst. Res. 65 (2003) 1–12.

[3] K. Al-Anbarri, R. Ramanujam, Ch. Subba Rao, K. Kuppusamy, Effect of

[

[

[

[

circuit configuration on chaotic ferroresonance in a power transformer,Electric Power Components Syst. 30 (2002) 1015–1031.

[4] S. Mozaffari, S. Henschel, A.C. Soudack, Chaotic ferroresonance inpower transformers, IEE Proc. Generation Transmission Distrib. 142 (3)(1995) 247–250.

[5] C. Kieny, G. Le Roy, A. Sbai, Ferroresonance study using Galerkinmethod with pseudo-arclength continuation method, IEEE Trans. PowerDeliv. 6 (4) (1991) 1841–1847.

[6] J.M.T. Thompson, H.B. Stewart, Non Linear Dynamics and Chaos, JohnWiley and sons, 1987, pp. 126–128.

[7] E.J. Dolan, D.A. Gillies, E.W. Kimbark, Ferroresonance in a transformerswitched with an EHV line, IEEE Trans. Power Apparatus Syst. PAS-91(1972) 1273–1280.

[8] H.W. Dommel, A. Yan, R.J.O. De Marcano, A.B. Miliani, in: H.P. Kin-cha (Ed.), Tutorial Course on Digital Simulation of Transients in PowerSystems, IISc, Bangalore, 1983, pp. 17–38 (Chapter 14).

[9] R.A. Walling, R.K. Hartana, R.M. Reckard, M.P. Sampat, T.R. Balgie,Performance of metal oxide. Arresters exposed to ferroresonance in pad-mount transformers, IEEE Trans. Power Deliv. 9 (2) (1994) 788–795.

10] S. Mozaffari, M. Sameti, A.C. Soudack, Effect of initial conditionson chaotic ferroresonance in power transformers, IEE Proc. GenerationTransmission Distrib. 144 (5) (1997) 456–460.

11] W.L.A. Neves, H. Dommel, On modelling iron core nonlinearities, IEEETrans. Power Syst. 8 (1993) 417–425.

12] L.O. Chua, P.M. Lin, Computer-Aided Analysis of Electronic Circuits,Prentice-Hall, Inc., 1975.

13] K. Al-Anbarri, Some investigation into occurrence of chaotic ferrores-onance in power system, Ph.D. Thesis, Department of Electrical &Electronic Engineering, Anna University, Chennai, India, March 2004.