Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 46

Bifurcation Lines Calculations of Period-1 Ferroresonance

FATHI BEN AMAR, RACHID DHIFAOUI
Networks and Machines Electric research unit, INSAT, Tunisia.
Centre urbain Nord, B.P. N°676, 1080 Tunis Cedex,
TUNISIA
Fathi.benamar@ipeis.rnu.tn

Abstract: - The principal contribution of this paper is to develop a numerical tool to calculate the bifurcation
lines of the fundamental ferroresonance (i.e., period-1). The developed process uses the same methods for
construction of the bifurcation diagrams : Galerkin method and continuation by the pseudo-arclength method.
The obtained Galerkin algebraic equations are nonlinear. The applied iterative method of resolution is that of
Newton-Raphson. The determination of Jacobien of this problem requires computation of the matrix
determinant derivatives. Except for a few simple cases, it is difficult to express this derivation analytically. To
solve this difficulty, a numerical relation is developed.
Applied to a series ferroresonant circuit, describing an opening operation of a circuit-breaker supplying a
voltage transformer, we could continue the bifurcation point in a plan with two parameters. With this type of
curve, it is possible to estimate the associated safety margin, and thus to operate the electrical supply network
with total safety. Several results obtained numerically, starting from a real case, are illustrated and discussed. In
addition, to validate these results, a simplified analytical method is developed.

Key-Words: - Transformer, Nonlinear, Ferroresonance, Bifurcation, Galerkin method, Continuation method,

Bifurcation line.

1 Introduction the study of these dynamic systems is the bifurcation

Ferroresonance is a nonlinear resonance theory [2, 5, 7].
phenomenon that can affect power networks. The We are especially interested here in the study of the
main problem with ferroresonance is that an dynamic system of figure 1, which describes the
overvoltage (as for fundamental ferroresonance), or practical problems of series ferroresonance. Most
an overcurrent (as for subharmonics) is generated. commonly, this type of situation is achieved when a
This is often dangerous for the electrical equipement magnetic voltage transformer (the nonlinear
[1]. These phenomena are not transient and are inductance) is connected to a busbar separated by the
present during normal operation [2, 3, 4]. grading capacitance of an open circuit-breaker (the
The engineer's practical problem is to know whether series capacitance) [4, 6, 8].
these dangerous phenomena may appear in his C R1
circuit. Simple simulation of the representative i=f(ϕ)
equations is not suited to the problem. Indeed, many
parameters are poorly known in a real circuit : losses, R2
saturation curves, switching-in instants, etc. To e(t)
ensure that there is no risk, many variants must be
simulated before any conclusion can be drawn -and
transient states are long to simulate. In addition there Fig.1 : Series, single-phase, nonlinear
is a risk of being at the limit of the dangerous zone ferroresonant circuit.
without knowing it. This is why the engineer wants The physical parameters of this circuit are:
to have an overall view of his circuit's behavior. He E : amplitude of the sinusoidal voltage source
wants to know whether he has a good safety margin e(t) = E sin (100πt),
or not. To have an overall view of the phenomenon, C : equivalent capacitance of the circuit,
the latter must be placed within an appropriate corresponding to the capacitance of the open
mathematical framework. circuit breaker and to all the capacitances to
Modeling this problem results in a system of time earth of the voltage transformer and the
differential, nonlinear equations (known as the connexion
dynamic system) which depends on various physical R1 : series losses of the circuit, and
parameters. The mathematical framework adapted to R2 : parallel losses of the circuit.
Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 47

The magnetic characteristic of nonlinear inductance diagram into three branches : a normal branch from
i(φ), where i is the inductance current and φ, its flux, the origin (0,0) to point PL1, an unstable branch from
is modeled by a polynomial function with an odd PL1 to PL2, and a ferroresonant branch beyond PL2
power: [2, 4, 5].
+
i( ϕ ) = k1ϕ + k nϕ n ; n ∈N , (k1, kn) ∈R
6000
(1)
Ferroresonant branch

Numerical data, corresponding to the parameters of 5000

the circuit illustrated in figure 1, are: * PL2 (E2, Ф 2)

Unstable branch
R1 = 32 kΩ ; R2 = 714 MΩ ; C = 0.4 nF. 4000

i( ϕ ) = 10 −8 ϕ + 2.34 ⋅10 −34 ϕ 9 , corresponding to a * PL1 (E1,Ф1)

Ф transfo (Wb)
real voltage transformer 400/20 kV. 3000

This paper aims at a global response to the

ferroresonance phenomenon in permanent state. For 2000

a specific circuit, this answer is given by the Normal branch

continuation of the bifurcation points when two or 1000

E2 E1
several parameters vary simultaneously. The Enom

obtained curve is called bifurcation line. Then it will 0

0 2 4 6 8 10 12 14
be possible to study the limits of the existence zones E source (V)
x 10
5

of singular phenomena, to take a safety margin,

between the studied case and the close dangerous Fig.2 : S-curve bifurcation diagram of flux amplitude
solutions, and to operate the network under totally versus applied voltage for the fundamental
safe conditions. ferroresonant phenomenon.
In this study, our effort is mainly devoted to the Computation of voltages E1 and E2 is easily
development of a numerical tool, based on the performed if the harmonics of flux are neglected and
Galerkin method and the pseudo-arclength only the fundamental component is considered.
continuation method. It is a tool for computation of These voltages correspond to a null derivative of the
bifurcation lines of the fundamental series source voltage in relation to the flux, i.e.,
ferroresonance case (i.e., period-1), in plans with two dE
parameters. Our objective is to determine the critical =0 (2)
dϕ
borders between the ferroresonant and normal zones
in the two plans (losses, applied voltage) and We propose to express these jump voltages by
(capacitance, applied voltage). considering the sinusoidal flux of the form :
ϕ (t ) = φ cos(ω t − θ ) (3)
2 Construction of the bifurcation lines
The fundamental ferroresonance is a periodic The differential equation governing the study circuit
phenomenon of the same frequency as the source. (figure 1) as follows :
That is why a model equation is adopted by the 1 dϕ
C∫
Galerkin method [2]. To validate the results of this R1i1 + i1dt + = E sin(ωt ) (4)
dt
method, a simplified analytical approach is
developed. After derivation of equation (4), with suppression of
the current, the following equation (5) is obtained :
2.1 Analytical method ⎛ R1 ⎞ d 2ϕ ⎡ 1 ⎤ dϕ
Fundamental ferroresonance is essentially ⎜ + 1 ⎟ 2 +⎢ + R1 ( k1 + nknϕ n −1 ) ⎥ +
characterized by discontinuous variations of the flux ⎝ R2 ⎠ d t ⎣ R2 C ⎦ dt (5)
1
amplitude when the source voltage is constantly
modified. These variations - commonly called C
( k1ϕ + knϕ n ) = Eω cos(ωt )
“ferroresonnant jumps” - occur at different source
which gives the following equation (6), after
voltages according to the variation direction (figure
replacement φ by its expression (3) and keeping only
2).
the fundamental terms (which suppresses time t) :
On this figure, one sees that the values E1 and E2 are
limiting values or critical values of the supply ⎛ Bφ ⎞
2

E 2 = ( Aφ ) + ⎜
2
voltage parameter. The two corresponding solutions ⎟ (6)
⎝ ω ⎠
PL1 and PL2 are called bifurcation points of the limit
or turning point type. These points divide the where A and B are functions of φ n −1 :
Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 48

R2
+ R1 ( k1 + αφ n −1 )
5
x 10
A= 12
C
1 ⎛ R ⎞
B = ( k1 + αφ n −1 ) − ω 2 ⎜ 1 + 1 ⎟ 10
C ⎝ R2 ⎠
and α is a constant of value: 8

2n 1
Cn(n−−11) 2 .

E source (V)
α= kn
n + 1 2n −1 6

Bifurcations occur when the relation (2) is verified,

i.e. 4

dE a) E = f (R1)
=0 (7) 2
dφ
0
which gives for equation (8): 0 2 4 6 8 10 12
R1 (Ω) 6
x 10
aφ 2 n − 2 + bφ n −1 + d = 0 (8)
6
x 10
2.5
where a, b and d are constants depending on the
circuit’s parameters and on the coefficients of the
non-linear element : 2
b) E = f (R2)
⎛ 1 ⎞
a = α 2 n ⎜ R12ω 2 + 2 ⎟
E source (V)

⎝ C ⎠ 1.5

⎡ ⎛ 1 ⎞ ω ⎛ R1 ⎞ k1 ⎤
2
b = α ( n + 1) ⎢ R1ω 2 ⎜ R1k1 + ⎟− ⎜1 + ⎟ + ⎥
⎣⎢ ⎝ R2 C ⎠ C ⎝ R2 ⎠ C ⎦⎥ 1
2 2
k12 ⎛ 1 ⎞ 4⎛ R1 ⎞ 2k1ω 2 ⎛ R1 ⎞
d= 2
+ ω 2 ⎜ R1k1 + ⎟ + ω ⎜1 + ⎟ − ⎜1 + ⎟
C ⎝ R2 C ⎠ ⎝ R2 ⎠ C ⎝ R2 ⎠
0.5

Equation (8) is a quadratic equation in φ . Both n −1

solutions, if available, are the values φ1 and φ2 of the 0

0 1 2 3 4 5 6 7 8 9 10
R2 (Ω) 7
flux at the turning points. The corresponding x 10

voltages E1 and E2 are given by relation (6). x 10

6

To specify the critical borders between the two zones 2.5

in which the network behavior is either

ferroresonnant or normal, it is necessary to trace the 2
bifurcation lines in spaces at two parameters. That
enables us to specify the values domains of different
E source (V)

1.5
parameters controlling the physical model, where the
solution will have the desired behavior.
We apply this method to the study of the circuit of 1
the figure1; it has enabled us to draw certain
bifurcation lines (figures 3). c) E = f (C)
0.5

2.2 Galerkin method

0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
2.2.1 Model and equations C (F)
x 10
-7

This method consists in finding an approximate

Fig.3 : Bifurcation lines – Analytical method.
periodic solution of the nonlinear differential
equation (4) by minimizing the error associated with
For that, a network modeling is adpted, without the
this solution . The idea is to seek this solution in the
nonlinear element, based on the equivalent Thévenin
form of Fourier series [2]. Here the method is
model (figure 4).
exposed only for the case of the first harmonic, but it
can be generalized for an unspecified harmonic rate.
Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 49

Zth i 2.2.2 Bifurcation lines – Galerkin method

To determine a fundamental ferroresonnant state,
system (12) has to be solved; this can be written as
v i(ϕ) follows :
Eth
ξ (φ , E , P) = 0 (15)
Fig.4 : Equivalent study circuit. where :
φ , state variable : vector formed by the two Fourier
The complex equation of circuit (figure 4) for the components of the flux ;
fundamental one is as follows : E, first parameter : scalar representing the supply
jωφ1 + Z1 I1 − E1 = 0 (9) voltage ;
P, second parameter : scalar representing for
where ω is the pulsation at 50 Hz of the excitation, example the capacitance of the circuit or the losses,
φ1 , E1, Z1 and I1 represent respectively the complex etc.
components at this pulsation of the flux in the Using the pseudo-arclength continuation method [9],
nonlinear element, of the supply voltage, the curve φ =f(E) will be followed, where E is variable
equivalent impedance of Thevenin and the current and P is constant.
traversing the circuit . If we assume that the conditions of derivability are
Flux φ(t) is supposed sinusoidal (3) and by adopting satisfied, by derivation of system (15), we get :
the following complex notations :
∂ξ ∂ξ
φ1 = φ1c − jφ1s ⋅ dφ + ⋅ dE = 0 (16)
∂φ ∂E
I1 = I1c − jI1s
(10) Like at the limit point (figure 2), there are dE = 0 ,
E1 = E1c − jE1s
whereas dφ ≠ 0 , the Jacobien J is singular when its
Z1 = R1c + jφ X 1s
determinant is null, i.e.:
equation (9) is converted into a nonlinear algebraic
⎡ ∂ξ ⎤
system of 2 equations, as follows : det ⎢ (φ , E , P) ⎥ = 0 (17)
⎣ ∂φ1 ⎦
ωφ1s + R1 I1c + X 1 I1s − E1c ≡ ξ1c = 0
(11) Thus, when this condition is met, the local solution is
ωφ1c + X 1 I1c + R1 I1s + E1s ≡ ξ1s = 0
not unique; and this marks the emergence of a
Since we know the nonlinear characteristic i(φ), it is bifurcation.
possible to compute the harmonic components of the To determine all bifurcation points for each value of
current in terms of flux components. The system of P, it is possible to solve the new system (18) with
equations to be solved is thus as follows : three variables ( φ , E) :
ξ (φ , E ) = 0 (12) ξ (φ , E , P) = 0
where φ is the unknown vector formed by the ⎡ ∂ξ ⎤ (18)
det ⎢ (φ , E , P ) ⎥ ≡ ξ d = 0
fundamental component of flux and E is the ∂
⎣ 1φ ⎦
amplitude of the supply voltage.
To solve this system (12), the Newton-Raphson by the Newton-Rapson method which requires the
method is used which requires the computation of computation of the new Jacobien J1 :
the Jacobien J of the system, i.e., ∂ξ1c ⎤
⎡⎡ ⎤ ⎤ ⎡⎡ ⎤
⎡ ∂ξ1c ∂ξ1c ⎤ ⎢⎢ ⎥ ∂ξ ⎥ ⎢⎢ ⎥ ∂E ⎥
⎢⎢ J ⎥ ⎢ ⎢ J ⎥ ⎥
⎢ ∂φ ∂φ1s ⎥ ⎥ ∂E ∂ξ1s ⎥
J =⎢
1c ⎥ (13) J1 = ⎢ ⎢⎣ ⎥⎦ ⎥ = ⎢ ⎢⎣ ⎥⎦ (19)
⎢ ⎥ ⎢ ∂E ⎥
⎢ ∂ξ1s ∂ξ1s ⎥
⎢ ⎥ ⎢ ∂ξ d ∂ξ d ⎥ ⎢ ∂ξ d ∂ξ d ⎥
∂ξ d ⎥
⎣ ∂φ1c ∂φ1c ⎦ ⎢ ∂φ ∂E ⎥ ⎢
⎣ ⎦ ⎢⎣ ∂φ1c ∂φ1s ∂E ⎥⎦
of which the elements are expressed in a general way
by : All elements of the last column, representing
derivatives in relation to unknown E, are simple to
∂ξ (φ , I ) ∂ξ ∂ξ ∂I calculate. To express J1 in its entirety, the elements
= + ⋅ (14)
∂φ ∂φ ∂I ∂φ of the last line must be computed.
Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 50

2.2.3 Computation of the last line of matrix J1 2.2.4 Results obtained

The elements of the last line of the matrix J1, This method is applied to draw the branches of the
representing the derivatives of the Jacobien J bifurcation lines of the circuit, illustrated in figure 1,
determinant in relation to the unknown parameters of in the plan (E, R1). Figures 5 represent the obtained
the problem, are expressed numerically by the results. Classically these branches meet and they
following theorem: coincide perfectly with those obtained by the
Theorem: Let us suppose that the elements of analytical method (figure 3).
5

matrix J (φ ) = ⎡⎣ J ij (φ ) ⎤⎦ (square of order m) are 12

x 10

derivable functions of the unknown variable φ , then *C

the derivative in relation to φ of det [ J (φ ) ] , that is to

10

d
say
dφ
( det [ J (φ )]) , is the sum of m determinants 8

E source (V)
obtained by replacing in all possible ways the 6

elements of the one of the lines (columns) of

det [ J (φ ) ] by their derivatives in relation to φ . 4 a) E1 = f (R1) : upper branch

Based this theorem, the elements of the last line are 2

expressed in a general way by:
m 0
d
dφ
( det [ J (φ ) ]) = ∑ det ⎡⎣ J i' (φ ) ⎤⎦ (20) 0 2 4 6
R1 (Ω)
8 10 12
x 10
6
i =1

5
where J i' (φ ) is the matrix obtained starting from J by 12
x 10

replacing each element J ij (φ ) of line i by its *C

10
d
derivative
dφ
( J ij (φ ) ) .
8
By using the development of Laplace according to
E source (V)

line i, the expression of det ⎡⎣ J i' (φ ) ⎤⎦ is given by the 6

scalar product of line i of J (φ ) by the vector of the

i
'

corresponding cofactors: 4
b) E2 = f (R1) : lower branch

m
⎡d ⎤
det ⎡⎣ J i' (φ ) ⎤⎦ = ∑ ⎢ ( J ij (φ ) ) ⋅ cofactor ( J ij (φ ) ) ⎥ (21) 2

j =1 ⎣ d φ ⎦
0
with cofactor ( J ij (φ ) ) = (−1)i + j ⋅ M ij ( J i' (φ ) ) 0 2 4 6
R1 (Ω)
8 10 12
6
x 10

where M ij ( J (φ ) )
'
represents the minor of the
i
Fig.5 : Bifurcation Lines – Galerkin method.
coefficient J ij (φ ) .
That makes it possible to conclude, by using the When the determinant of the Jacobien J1 becomes
relation (21), that the elements of the last line of the null (corresponding to the limiting value of the
matrix J 2 are given by the following relation: parameter R1), the Galerkin method does not
m m
converge. Bifurcation point "C" corresponding to
d d
dφ
( det [ J (φ ) ]) = ∑∑ ( J ij (φ ) ) ⋅ cofactor ( J ij (φ ) ) (22) this value is called " Cusp ".
i =1 j =1 d φ It is possible to overcome the difficulties due to the
To draw a bifurcation line, we need simply the apply non-inversibility of J1 and, consequently, wholly
a continuation method to system (18). Given solution draw the bifurcation lines. We need simply apply
( φ0 , E0) with value P0 in the 2nd parameter, it is Galerkin and pseudo-arclength methods [2]
possible to seek the solution for P0+∆P by simultaneously. The problem can thus be solved.
initializing with ( φ0 , E0). The parameter P is used as
2.3 Pseudo-arclength method
a continuation parameter.
A continuation principle is still applied. However,
instead of going from a point M to M+1 moving
Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 51

length ∆P along the parameter axis, we move by x 10

5

12
length S on the tangent to point M. Ferroresonance state
The principle of this method is to add an equation to *C
10
system (18) so that the Jacobien of the new system
becomes inversible. In this new system, φ0 and E are
8
no longer the only unknowns, since parameter P is

E source (V)
also unknown. Additional parameter S is used as a 6
continuation parameter.
The principle used to determine the additional 4
equation is similar to that used in [2]. We need
simply express the tangent to the curve in ( φ0 , E0, a) E = f (R1)
2
P0) which is : Normal state

∂ξ ∂ξ ∂ξ 0
0 2 4 6 8 10 12
⋅U + ⋅V + ⋅W = 0 (23) R1 (Ω)
∂φ ∂E ∂P (φ0 , E0 , P0 )
6
x 10
(φ0 , E0 , P0 ) (φ0 , E0 , P0 )
6
x 10
where (U, V, W) is a tangent vector. 2.5

The additional equation is given by :

Ferroresonance state
b) E = f (R2)
U ⋅ (φ − φ0 ) + V ⋅ ( E − E0 ) + W ⋅ ( P − P0 ) − S = 0 (24) 2 *C

The new system to be solved is composed of

1.5
equations (18) and (24), i.e.,
E source (V)

ωφ1s + R1 I1c + X 1 I1s − E1c ≡ ξ1c = 0

1
ωφ1c + X 1 I1c + R1 I1s + E1s ≡ ξ1s = 0
⎡ ∂ξ ⎤ (25)
det ⎢ (φ , E , P) ⎥ ≡ ξ d = 0 0.5 Normal state

⎣ ∂φ1 ⎦
U ⋅ (φ − φ0 ) + V ⋅ ( E − E0 ) + W ⋅ ( P − P0 ) − S ≡ ξ p = 0
0
0 1 2 3 4 5 6 7 8 9 10
for which the Newton-Raphson method will be used, R2 (Ω)
x 10
7

thus requiring computation of the Jacobien J2 of 6

x 10
system (25), i.e., 2.5

Ferroresonance state
⎡⎡ ⎤ ∂ξ1c ⎤ *C

⎡⎡ ∂ξ ⎤ ⎢ ⎢ ⎥ ∂P ⎥ 2
⎤ ⎢ ⎥
⎢⎢ ⎥ ∂P ⎥ ⎢ ⎢ ⎥ ∂ξ1s ⎥
⎢ ⎢ J1 ⎥ ⎥ ⎢⎢ J1 ⎥
⎢⎢ ∂ξ d ⎥ ⎢ ⎥ ∂P ⎥ (26) 1.5
⎦⎥ =⎢ ⎥
E source (V)

J2 = ⎢⎣
∂P ⎥ ⎢ ⎢ ⎥ ∂ξ d ⎥
⎢ ⎥ ⎢ ⎥⎦
⎢ ∂ξ p ∂ξ p ∂ξ p ⎥ ⎢ ⎣ ∂P ⎥
⎢⎣ ∂φ ⎢ ⎥ 1
∂E ∂P ⎥⎦ ⎢ ∂ξ p ∂ξ p ∂ξ p ∂ξ p ⎥
⎢⎣ ∂φ1c ∂φ1s ∂E ∂P ⎥⎦ c) E = f (C)
0.5

J1 is already determined. The last line of J2 Normal state

represents the components of the tangent vector, i.e. 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
(U, V, W). Finally, computation of the elements of C (F) -7
x 10
the last column of J2 does not involve any difficulty.
Fig.6 : Bifurcation lines – Pseudo-arclength method.
∂ξ ∂ξ ∂Z
= ⋅ (27)
∂P ∂Z ∂P 2.4 Interpretation of results
Matrix J2 is then fully determined. The results obtained by these various methods
Applied to the study circuit of figure 1, the proposed (analytical, Galerkin and pseudo-arclength
method gives good results. Indeed, the bifurcation continuation) perfectly coincide with each other.
lines obtained (figure 6) perfectly coincide with With the pseudo-arclength method, we have a tool
those of the analytical method. well suited to the study of ferroresonance in the
Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 52

electrical networks, in particular, the plotting of We notice that the bifurcation lines in plan (E, C) are
bifurcation lines. These lines make it possible to isolates and have no intersection with banal
obtain a more global view of the system's behavior. solutions.
They correspond to state stability limits. They also Figure 7a shows that as the lower threshold values of
provide the existence of various zones in the the voltage for the occurrence of fundamental
parameters' plan where diverse states can occur. In ferroresonance get smaller the circuit's series losses
the case of fundamental ferroresonance, these lines get weaker. We also observe that, for a given series
actually show the zones corresponding to a normal resistance, the ferroresonance phenomenon
state and a ferroresonnant one. We can observe disappears altogether when the circuit capacitance
(figure 6) that, beyond certain values of the applied exceeds a certain value (C>66.3 nF for R1=64 kΩ
voltage, the state shows fundamental ferroresonance and C>132.6 nF for R1=32 kΩ).
and that, below certain values, the state is normal. Iron losses of the nonlinear element have little
There is also an intermediate zone where the state is influence on the existence limits of ferroresonance.
either normal or fundamental ferroresonance. The Figure 7b shows that, for a given capacitance value,
occurrence of one or the other depends on the initial the lower voltage thresholds are driven up to values
conditions. which become higher as R2 is smaller (i.e. for larger
Taking into account the great sensitivity of the losses).
phenomenon of the circuit parameters, it is
interesting to release a third parameter to see how 3 Conclusion
bifurcation lines evolve in a plan, so as to anticipate To study a ferroresonant circuit, simple temporal
ferroresonance risks with a wider safety margin. The simulation is not enough to understand the general
result of this parametric study is shown by figures 7. behavior of the circuit. The phenomena of jumps, the
multiplicity of solutions for a given set of
6
x 10
2.5
parameters, the sensitivity to initial conditions, etc.
make it difficult to apply such a method or , at least,
2
lead to excessive computation.
The mathematical framework which must be applied
1.5 to understand ferroresonance is the bifurcation
E source(V)

R1=32 kΩ theory.
The answers to concrete problems faced by the
1 R1=64kΩ
system operator can be obtained with diagrams and,
above all, with bifurcation lines. The numerical
0.5 methods described (Galerkin's method and the
a) E = f (C) pseudo-arclength continuation method) here permit
efficient construction of these curves. For the
0
-2 0 2 4 6 8 10 12 14 16 determination of the Jacobien of this problem, a
C (F) -8
x 10
numerical relation is developed allowing the
6
x 10
calculation of derived from a determinant of a
2.5
matrix.
Using these lines, it is possible to learn the values of
2 parameters which guarantee the non-occurrence of
dangerous phenomena. A safety margin is chosen in
relation to these values to operate the network with
1.5
total safety.
E source(V)

R2=7MΩ The application of the methods presented in this

1 R2= 714 MΩ
paper covers fundamental ferroresonance with
satisfactory results. The extension of this study to
more complex cases of ferroresonance (subharmonic,
0.5 b) E = f (C)
harmonic ferroresonance) is currently being studied.

0
-2 0 2 4 6 8 10 12 14 16
References:
C (F)
x 10
-8 [1] S. R. Sagardia, A. Morched : « Potentiel
transformer failure due to ferroresonance », in
Fig.7 : Evolution of the bifurcation lines Procs. 2001 IPST, Rio de Janeiro.
in relation to circuit parameters.
Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 53

[2] F. Ben Amar, A. Sbai, R. Dhifaoui,

« Diagrammes de bifurcation d’un système
ferrorésonnant parallèle », CIFA2002, Ecole
Centrale de Nantes - France, juillet 2002.
[3] M. Val Escudero, I. Dudurych. M. A. Redfern,
« Characterization of ferroresonant mode in HV
substation with CB grading capacitors »,
Intenational Conference on Power Systems
Transients (IPST’05), Montreal - Canada, June
19-23, 2005.
[4] D. A. N. Jacobson, P. W. Lehn, R.W.Menzies,
« Stability domain calculations of period-1
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