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8/9/2019 Ferroresonance in Power Systems
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8/9/2019 Ferroresonance in Power Systems
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closed-form solution is then used to draw contour maps
of the system, showing the safe and ferroresonant regions
as a function of the circuits parameters.
To verify the validity of the analytical results, the
circuit was solved numerically with a Runge-Kutta inte-
gration routine. Excellent agreement was found between
the analytically-predicted steady-state values and the
numerical solution.
An important observation derived from this study is
the effect of the losses in determining the value of CcririCol
(proportional to the feeders length) beyond which ferro-
resonance can
occur.
The methods of analysis presented are being extended
for the analytical prediction of harmonics and sub-
harmonics in steady state solutions and for the character-
isation of jump and chaotic phenomena in certain regions
of the parametric maps.
2 Problem description
The operation of a power transformer under ferro-
resonance can be illustrated with the diagrams of Fig. 2,
which show the relationship between the fundamental
frequency components of voltages and currents in the
circuit of Fig.
1
ignoring the
losses R
for simplicity. In the
development of the general case,
losses
are shown to be
an important consideration in the determination of the
critical feeder length. Upper case letters are used to indi-
cate fundamental frequency phasor quantities.
In
Fig. 2a, the intersection of the straight-line E,
V
(representing the voltage applied to the transformer coil)
with the transformers magnetisation characteristic is
possible at three points: points A and C are stable oper-
ating points. Point
B
is unstable. The instability of point
B can be seen, for instance, by increasing the source
voltage by a small amount. In Fig. 2a, increasing E, dis-
places the line
E, V
upwards, parallel to itself. When
this happens, the current at intersection points
A
and C
also increases, as expected. For point 8 the current
decreases, which is not physically possible. Point
A
corre-
sponds to normal operation in the linear region, with flux
and excitation current within the design limits. Point C
corresponds to the ferroresonant condition, characterised
by saturated flux and large excitation current. Phasor
diagrams for these operating conditions are shown in Fig.
2b.
Mathematically, both points A and C are equally valid
solutions to the circuit of Fig.
1.
Which operating point
0.4
I . . . . .
0.01
0.2 0.4
0 6
a8 1.0
L
the system settles at depends on he initial conditions and
on the trajectory towards the final state. During normal
service, the transformer is supplied through inductive
lines and operation is only possible in the lower flux
region (line
E, V
in Fig.
2a
becomes E,
-
AVfedrr
with
a negative slope). Topological abnormalities in the
network, as for example, the disconnection of one
or
more of the supply conductors, can result in the trans-
former being fed through the coupling capacitance from
adjacent lines
or
conductors. When these topological
changes occur, whether the operating point stays
in
the
linear region
or
jumps into the saturated region is a
random event which depends on the specific character-
istics of the transient.
3 Transformer modelling
Fig. 3shows the magnetisation curves of the
25
MVA,
110kV power autotransformer of Reference
1
for differ-
ent ranges of magnetising current. During ferroresonance,
the operating point of the transformer is located in the
saturated region. An accurate description of the condi-
tions of operation in this region involves the flux and
current ranges shown in Fig. 3c. At this scale, for the core
materials used in modern high-voltage power trans-
formers, hysteresis loops are not significant and a single-
valued curve can be assumed.
For the analytical description of the circuit in Fig. 1,
it is convenient to fit the
4(i)
characteristic of the
transformer by a polynomial. The notation
4
is used to
indicate total
flux
linkages. A good fit of the Hi) charac-
teristic at the scale Fig.
3c
can be accomplished by the
following two-term polynomial
(1)
= a+ + b4
The first term in eqn.
1
corresponds to the linear region
of the magnetisation curve. The higher-order term
approximates the saturation region. There is a strong
decoupling between the linear and nonlinear terms of
eqn. 1 because of the high order of the second term. The
coefficient
a
of the linear term corresponds closely to the
unsaturated magnetising inductance (a
= l/L).
Coefficient
b and the exponent of the nonlinear term are chosen to
provide a best fit of the saturation region.
Fig. 4shows the detail of the saturation region for dif-
ferent exponents of the nonlinear term in eqn.
1.
As can
be seen in the Figure, polynomials of order seven
or
less
0.4
t y.p.u.
0
Fig. 3 Magnetisation characteristic of power transformer
322
IEE PROCEEDINGS-C, Vol.138,N o .
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Ignoring the harmonics the term +w:a {. } in eqn.
klw:Wsin(w,t+O)
10can be expressed (with eqn.
7)
as
= k,w$V - ' [ ( @os 0)sin w,
+
(asin 0)cos w,
]
=
klw: V -l(aXinos +ay os w, )
Substituting into eqn.
10,
the resulting equation can be
separated into sine and cosine terms and the following
relationships can be established:
(1
1)
0
[ - of
-
W@
+
k1~:W -']@,
-
Qr
=
0
RC
[ - of
- w@ +
k,w: @ -']ay5 ax
=
w,
(12)
with
az= @ + a;
Eqns.
11-13
determine the values ofmX and a,, and thus
the solution 4(t) or the transformer flux eqns.5-7) in a
Ritz sense. A similar procedure can be. used to determine
the harmonics and subharmonics of the general solution.
6 Parametric maps of fund amental solu tion
In what follows, eqns.
11-13,
which define the fundamen-
tal component of the transformer flux in the circuit of
Fig. 1, are manipulated into a more convenient form for
the graphical location of the solution points. The effects
of the circuit parameters, e,, C and
R,
in the location of
the borders separating the normal and ferroresonant
regions are investigated.
Eqns.
11
and 12 can be combined by squaring and
adding. After some algebraic manipulations (Appendix),
the following single relationship is obtained:
where
+
P l t -
o =
0
(14)
z
p + l ) / Z
c
= az
(15)
Eqn.
14
can be solved numerically using a root-finding
routine for polynomials. Once the roots t are found, the
amplitude of the transformer flux
fit)
in eqn.
5
is given
by
=
+Jt. The phase angle
8
is given by 0 =
tan-'(@,&.), where ax nd ay re obtained from eqns.
11 and
12
as follows:
in which
A =(wf - wt)
+
k,w:W -'
B = O
RC
Depending on the particular set of circuit parameters,
convergence of a numerical procedure to solve eqn. 14
can be difficult if the initial estimates of the roots are not
close enough to the solution. A single-point numerical
solution does not give any information on he position of
this solution with respect to the overall boundaries
between normal and ferroresonant regions.
A
much better visualisation
of
the margins of safety for
a given solution point can be obtained with graphical
mappings of parametric solutions.
One possible pro-
cedure is to express eqn. 14 as the intersection of a linear
function
I
with a nonlinear function
I
as follows:
1,
= P l c - o
I2 = p 2 p )/2
(21)
(22)
A plot showing a given I curve with several cases of I
curves is shown in Fig.
5.
For curve I l there are three
Fig. 5
Graphical solution of circuit
o =
Jc;
solutions points, A, B and
C.
Solutions
A
and
C
are
stable. Solution B is unstable (a small disturbance at
B
would shift the operating point to either A or
C).
Solu-
tion
C
is the ferroresonant state. Solution
A
is the normal
linear state. Curve
2
of
I
corresponds to the critical case
beyond which ferroresonance is no longer possible.
Ferroresonance is not possible for curve 3.
The effect of the parameters
e,, C
and R in the circuit
of Fig. 1on the possible regions of operation of the
system is investigated with the help of graphical solu-
tions.
7
Of the coefficients
p o ,
p1
and p 2 defining the
1,
and
dz
functions of Fig.
5 ,
only the slope
p1
of I depends
on
he
shunt
losses
(eqns.
21
and
22
with eqns.
16-18).
Fig.
6
illustrates the effect of the shunt losses in terms of
G =
1/R
on the location of the solution points. Fig.
6a
corresponds to Fig. 5 . The plots in Fig.
6
correspond to
the actual parameters of the test system studied with
E
=
1.0pa. Increasing the losses (increasing
G)
will
increase the
slope
of
I
and beyond a certain value of p1
the intersection of
I
and I, will occur only outside the
ferroresonant region.
Figs.
6b
and
c
show a general case of possible varia-
tion of transformer flux with shunt conductance G. In
Fig. 6b, the values G, and G , of shunt conductance
delimit the possible regions of operation of the circuit.
For G < Gl (region 111 in the Figure), operation is only
possible in the ferroresonant region. For G, < G
Gz
region
I),
ferroresonance cannot occur. Fig.
6c
shows the special case in which the numerical solution
Critical values of shu nt
losses
IEE PROCEEDINGS-C, Vol. 138, N o . 4, J U L Y 1991
24
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for G , is less than zero and, therefore, only regions I and
I1
exist.
6 r
G-12 mS
51 / / G = l O m S
- 2 y
0.0
0.5 1.0 1.5 2.0 2.5
5
4
3
2
0
-2
0.0
0.5 1.0 1.5 2.0 2.5
5
0
I
G I G2 62
G G
b
C
Fig.
6
a
Teal
system E =
1.0
p.u.
b
General
cax
e
Gemral case
Effect of losses on circuit solution
The critical values G , and G , in Fig. 6b can be found
From eqn.
14,
solving for GZ
by taking dG/dr
=
0 in eqn. 14 (G = l/R).
Taking
dG/dr = 0
gives
- y](n
+
- U: - 03
(n
-
)
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some algebra
a:[
-
E2)C2
(a[
+ klb[( +')'Z)C
1 2
N r Y
(28)
For a given value of
[
(or of the transformer flux
4 = +,/
8/9/2019 Ferroresonance in Power Systems
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V, =
KO,,,
The transformer core losses are taken as 1
of the rated transformer capacity. With these values, the
parameters in the reduced equivalent circuit of Fig.
10
are C
=
777 nF, E
=
O.15Km,, and
R
= 48.4 kR
1.0-
0 8
0 6
0.4
Fig. 10 Reduced equivalent circuit
E = 0.5 v
c, +
2c.
c =
c,
+ ZC,
10.1Magnetisation curve
The magnetisation curve of the transformer is approx-
imated by the 11th order two-term polynomial of eqn. 1.
The resulting approximation,shown in Fig. 4, is
= 0.28 x 10-'1$
+
0.72 x 10-2Q1'
(30)
for i and Q in per unit. I
=
1 p.u. corresponds to the
transformer's rated current of 131
A
and Q = 1 p.u.
corresponds to the transformer's phase-to-neutral rated
voltage of
V = 63.5
kV.
The coefficients of the approximation in eqn. 30 were
chosen for a best fit of the high current region of the
magnetisation curve of Fig. 4.
702
Fundamental solutions
The solution for the fundamental-frequency component
of the transformer flux can
be
obtained from eqn.
14,
which, for this case study, is given by
111
- 85.6916
+
18451;- 42.98 = 0
(31)
The coefficients of eqn. 31 were evaluated from eqns.
16-18,
with the frequencies in eqns.
4
and
9
given by
os 377 rad/s 60 z)
o0
=
53 ad/s 8.5Hz)
ki =
0.451
o2 85 rad/s (13.5 Hz)
Eqn. 31 was solved using a root-finding numerical
routine. The results obtained are shown in Table 1
(@
=
+,/
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Table 2: Crit ical values of resistance, source voltage and
capacitance
Critical resistances R,= n/a
R ,
=
21.4
kfl
Critical source voltages E, = n/a E , = 1.02P . U .
Critical capacitance
n/a: not applicable to this case (analytical solution is negative)
C,,
=
509
nF
2.4~
C
-6
Y.2
0.01
0.02 0.03
1 0
0.5
1.0 1.5
2.0
2.5
3.0 3.5
4.0
0.0
c.
p.u.
Fig. 14
Test system
Upper inset:detail around
C
critical
Lowerinset:Solution
in
linc r region
Flux against series capacitance
The critical values of the shunt conductance
G
(=
1/R)
were found from eqn. 24 using a root-finding routine.
There was
no
positive solution for GI, and the plot of q5
against
G
is
as
in Fig.
6c.
The plot for the test system is
shown in Fig. 12.For values of R smaller than Rz=
1/G,
=
21.4
kR,
i.e., for
losses
Larger than
2.3%
of the
rated transformer capacity, ferroresonance does not
occur. In the test system, R = 48.4 kR
(losses
= 1 ) and
therefore ferroresonance can occur, as shown in the
results of Table
1.
The critical values for the equivalent source voltage
E
were found from eqn. 26 using a root-finding routine. The
plot of
D
against E is shown in Fig. 13. This plot corre-
sponds to that of Fig. 7c in the general analysis. The @-E
plot indicates that for the parameters of this system fer-
roresonance can always occur,
no
matter how small the
source voltage. For values of E larger than E2=
1.02
P.u., operation is only possible under ferroresonant
conditions. For values of E smaller than 1.02 p.u. ferro-
resonance may
or
may
not
occur. This is the case for the
value of
E =
0.15 p.u. in the test system.
Fig. 15
E = 0.15
P.u.;
C =
777 nF;
R =
48.4 kC2
328
Transient simulation of
circuit
The critical value C, for the capacitance C was read
directly from Fig. 14as C, = 509 nF (0.093 P.u.). This
value of C corresponds for the test system to a line length
of 65 km. For line lengths
less
than 65km, ferro-
resonance cannot occur in this system. For the test case
value of 100 km, ferroresonance can occur. The critical
value of the line length was also calculated for the case
with no losses (line length for Cg in Fig. 8). This value
was only 2.6 km. The large difference in critical line
lengths with and without losses shows the importance of
considering the losses in ferroresonance predictions.
11 Conclusions
An analytical technique for obtaining steady-state fer-
roresonant conditions in iron core transformers supplied
through capacitive coupling was presented. The analysis
proposed is general for the type
of
circuit considered and
includes the effect of the system losses.
The solution to the circuits nonlinear differential
equation is based on Ritzs method
of
harmonic balance.
To apply this technique, the transformer saturation curve
is approximated by a two-term 11th-order polynomial.
This 11th order polynomial provides a much better fit to
the saturation region than can be obtained with seventh
and lower order approximations. Differences of about
20% in the magnitude of the transformer flux were
observed in comparisons of the 11th order against
seventh order approximations.
Graphical solutions were used to map the boundaries
between normal and ferroresonant regions and to locate
the possible operating points as a function of the circuits
equivalent source voltage, series capacitance and shunt
losses.
A practical example of a system under ferroresonant
conditions was presented. The solution points as well as
parametric maps were calculated for this system. One
interesting observation from this study is the importance
of considering the transformer core losses in determining
the margin of safety before fenoresonance can occur.
This margin went from less than 2.6 km with no
losses
to
65 km when losses were considered. Another interesting
observation is that for certain combinations of circuit
parameters, ferroresonance can occur even for very small
values
(E
+0)of the source voltage.
12 References
1 DICK, E.P., and WATSON, W.:
Transforma
models for transient
studies
based
on
field measurements, IEEE
Trans.,
981, PAS-100,
2 GERMAY, N.,MASTERO, S., and VROMAN, J.: Review of
ferro-
resonance phenomena in high-voltage power systems and pmen-
tation of a voltage transformer model for predetermining the.
CIGRE, International Conference on Large High Voltage Electric
Systems, 21-29 August 1974.
3 WLAN, EJ, GILLIES, D.A., and KIMBARK, E.W.: Fer-
roresonance in a transformer switched with an EHV line, JEEE
Trans., 1972,PAS-91 p. 12731280
4 PRUSTY, S., and PANDA, M.: Predetermination of lateral length to
prevent overvoltage problems due to open conductors in three-phase
systems, IEE Proc C 1985,132,
(I),
pp. 4S55
5 SWIFT G.W.: An analytical approach to ferroresonance, IEEE
Trans., 1969,
PAS-88,
(l),pp. 42-46
6 PRUSTY, S., and RAO, M.V.S.: New method for determination of
true saturation characteristics
of
transformem and nonlinear reac-
tors,IEE Proc. C, 1980 127, (2), pp. 106-110
7 CUNNINGHAM, W.J.: Introduction to nonlinear analysis
(McGraw-Hil l, New York, 1958 , pp. 157-168
8
TEAPE, J.W.. SLATER, R.D., SIMPSON, R.R.S., and WOOD,
W.S.: Hysteresis effects in transformers, including ferroresonana,
JEE Proc., 1976,123, (2),pp. 153158
(1).
PP. 409-417
JEE PROCEEDINGS-C, Vol. J38 ,
N o.
4, JULY
1991
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13 Appendix
Derivation of eqn.
14
for transformer flux
Eqns.
1 1
and
12 for
the fundamental-frequency solution
of
the transform
flux
components(QX,my can be written,
simplifying the notation, a s
ux - y =
0
a y + bx = H
where
x
=
Y
=
a = -CO,' -0;) k l o ; w - l ]
b
=
o J ( R C )
IEE PRO CEED IN G S -C,
Vol. 138,
N O .
4, J U L Y 1991
(32)
(33)
and
H = o , E
Squaring eqns. 32 and 33, and adding
a2(x2+ y 2 )
+
b2(x2
+
y 2 )
=
H 2
(34)
with eqn. 13,
(x2
+
y 2 )
= (@ +
@
=
cPZ,
and with the
original variables
[(U: - U;)'
+
k :o :@2 -2
- Yo: - 03
x
k l o g
W - '
+
O , ' / ( R C ) ~ ] @ ~
U:
E
=
0
Rearranging
(k:o:)@'
(of - @k,w:@ +'
+ [(U,'
-
+ o . Z / ( R C ) ~ ] @ ~0,
=
0
which with
5 =@
gives eqn.
14
in the text.
329
I I
r
I