ferroresonance in power systems

8/9/2019 Ferroresonance in Power Systems

1/9


2/9

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 .

4 ,

J U L Y

1991

Authorized licensed use limited to: Universidad de chile. Downloaded on December 1, 2008 at 01:43 from IEEE Xplore. Restrictions apply.


3/9


4/9

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



5/9

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

-

)


6/9

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 = +,/


7/9

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

(@

=

+,/


8/9

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



9/9

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

ferroresonance in power systems

Documents