short circuits lecture 8 transformer models part ii

Short-circuits Lecture 8Power systems elements models for short-circuit studies

Transformers part 2

Teaching materials distributed for free.

Transformers part 2

Prof. Dsir Rasolomampionona

Prof. dr hab. Jan Machowski

Outline of the lecture

Power transformers Transformer parameter calculation

Example

Power transformers zero sequence transformer model


Mathematical Models

Power transformers parameter calculation example

Parameters of one of the most frequently used autotransformers in Poland, the symbol

of which is RTdx125/200 are as follows : winding apparent power SrH=160 MVA,

SrT=50 MVA; nominal ratio 230/120/10,5 kV/kV/kV; short-circuit voltage uKHL=10%,

uKHT=33,5%, uKLT=20,8%; power losses are neglected.

The parameters of the equivalent circuit can be determined using the formulas


The parameters of the equivalent circuit can be determined using the formulas

presented in the previous lecture.

Impedances referred to the primary side are given by the following expressions

r

2rLLTK

LTr

2rHHTK

HTr

2rHHLK

HL 100 ;

100 ;

100 S

UuZ

S

UuZ

S

UuZ ===

Mathematical Models


These impedances are related to the short-circuit impedances as follows:

LHHL ZZZ +=

THHT ZZZ +=

TLLT ZZZ +=


Considering the above-mentioned equations as a set of three equations with three

unknown variables, the values of the corresponding variables are determinad e as

follows: ( )LTHTHLH 5,0 ZZZZ +=

( )HTLTHLL 5,0 ZZZZ +=

( )HLLTHTT 5,0 ZZZZ +=

Mathematical Models


=

+=

+= 5,37160

230

1002

8,205,3310

1002

2

r

2rHKLTKHTKHL

Hm S

UuuuX

=

+=

+= 5,4160

230

1002

3,33108,20

1002

2

r

2rHKHTKHLKLT

Lm S

UuuuX


=

+=

+= 2,73160

230

1002

108,205,33

1002

2

r

2rHKHLKLTKHT

TmS

UuuuX

Mathematical Models


The positive sequence circuit of the autotransformer is shown in the figure below. Coreless

and magnetization losses are not presented. In short-circuit calculations these losses are

usually neglected, the values of the equivalent passive elements are not calculated.

a) b)

220 kV 110 kV37,5

73,2

-4,5


10 kV

73,2

= 230 kV/120 kV/10,5 kV = 160 MVAS

r

r

Equivalent circuit (b) of the autotransformer depicted in (a)

It is worth noting that for most of three winding transformers the middle voltage reactance is

negative. This is the result the mutual reaction of the dissipative fluxes from the different

windings.

Mathematical Models


As it was derived for the positive sequence transformer model a T-type circuit is used

for the zero sequence transformer model. Only the impedances are different.

There are two essential characteristics of transformers which decide about the

zero sequence impedance values :

1. The construction of the transstormer core influences the magnetising

impedance (Z0m)


impedance (Z0m)

2. The winding arrangement decides whether the zero sequence component

will flows through the transformer or not.

ad. 1. The transformer core

Fluxes A, B, C produced in the transformer yokes correspond to the currents IL!, IL2, IL3 of different windings.

Mathematical Models


The sum of the three currents IL1 + IL2 + IL3 = 3I0 correspond to a zero sequence

component (0) of the fluxes A, B, C multiplied by three (30 )

This sum is not equal to zero IL1 + IL2 + IL3 = 3I0 0 if 3f0 0 .

A part of fluxes generated in the coils of a three column transformer (a) passes through

the external isolation (oil, air, transformer tank) because could not pass through the

coils (leakage flux).


coils (leakage flux). b)a)

Mathematical Models


This flux path is characterised by a high magnetic reluctance, which is equivalent to a

low magnetising impedance . Then for three column transformers the following

expressions are obtained

( ) 10 006,0004,0 ZZ = ( ) T0 64 ZZ =and

In contrary to the three column transformer in case of a five column transfomer (b

previous slide) or three one phase transformers there is a passage through the iron for


previous slide) or three one phase transformers there is a passage through the iron for

the magnetic flux, then the zero sequence magnetising impedance is equal to the positive

sequence magnetising impedance

10 ZZ T1 1000 ZZ and

ad. 2. The influence of winding arrangement

The winding arrangement decides also whether the zero sequence current will pass

through the transformer or not.

Mathematical Models


Yy configuration with one earthed neutral point. The earthing current (zero

sequence current) flows through the earthing wire at the earthed -Y side. There is

no path for the zero sequence current at the non-earthed Y side.


Determination of the equivalent circuit for transformators with Yy winding

configuration. a) the flow of zero sequence current from the Y connection point.

b) characteristic elements of the zero-sequence model c) equivalent circuit

Mathematical Models


If the transformer is a four or five column core the impedance is

quite high and can be neglected. If the transformer is a three column core the

impedance is smaller and must be considered in the

calculations. Then the following relation should be assumed. In

such a situation a voltage resulting from the induction of identical emfs in

each phase will appear at the non-earthed Y neutral point .

10 ZZ

( ) 10 006,0004,0 ZZ == 00T ZZ

03U


Yy configuration with two earthed neutral points. The earthing current (zero

sequence current) flows through the earthing wire at both earthed -Y sides.

As for the previous case if the transformer is a four or five column core the

impedance is quite high and can be neglected.10 ZZ

Mathematical Models



Determination of the equivalent circuit for transformators with Yy winding

configuration. a) the flow of zero sequence current from the Y connection point.

b) characteristic elements of the zero-sequence model c) equivalent circuit

Mathematical Models


Yd configuration with earthed neutral point. The zero sequence component

path is at the Y side. A cophase emf is inducted in each phase of the secondary

winding. As a result if this induction a voltage and a current appear in

each phase of the transformer delta side. According to Kirchchoffs law the delta

side current cannot flow out of the delta ( ) (Fig (a)) and this is shown in the

equivalent circuit by a break (a) and (c). The current closed in the delta loop is

shown as a connection of the equivalent impedance to the reference node.

03U 0I

Z

0I

0I


shown as a connection of the equivalent impedance to the reference node.

c)b)a)

LH

I

ZI

I0

0

0 I0

I0

Z

0ZI0

LH LH

0TZ

Determination of the equivalent circuit for transformators with Yd winding

configuration. a) the flow of zero sequence current from the Y connection point. b)

characteristic elements of the zero-sequence model c) equivalent circuit

Mathematical Models


Z

If the transformer is a four or five column core the impedance is

quite high and can be neglected. If the transformer is a three column core the

impedance should be connected in parallel with . Then the following

expression is obtained .

10 ZZ

0Z

( )00T0 / ZZZZZ +=

Yz configuration. The transformer windings are connected in zig zag when



obtaining a small zero sequence impedance value is essential. If measurements

of zero sequence impedance are performed at the star side (Y), emfs are inducted

at half length of the transformer windings, however considering the currents

flowing in the inverse direction. The zero sequence impedance measured is equal

to the open-circuit impedance though the transformer is short-circuited

(magnetising impedance). (Fig. a).

Mathematical Models



obtaining a small zero sequence impedance value is essential. If measurements

of zero sequence impedance are performed at the star side (Y), emfs are inducted

at half length of the transformer windings, however considering the currents

flowing in the inverse direction. The zero sequence impedance measured is equal

to the open-circuit impedance though the transformer is short-circuited




c)

0Z

a)I0

I0

I0

b)I0Z

I0Z

I0Z

Z0 0,5ZT Z = Z0,10Z T

Mathematical Models


If the transformer is supplied by a zero sequence voltage at the zigzag side, the

fluxes generated in the different winding halves (Fig b) are mutually

compensated, and the series reactance of the transformer is very small a few

times smaller than XT (in practice it is convenient to assume that Z0Z = 0,1ZT).

There are no emfs in the windings situated at the wye side (because the

resulting flux is equal to zero) therefore there is no importance whether they are

short-circuited for measurement or if the neutral point is earthed or not.


short-circuited for measurement or if the neutral point is earthed or not.

c)

0Z

a)I0

I0

I0

b)I0Z

I0Z

I0Z

Z0 0,5ZT Z = Z0,10Z T

Mathematical Models


In case of wye-zigzag configuration the zero sequence currents at each side are

completely independent one form another (Fig. c).

a)I0

I0

I0

b)I0Z

I0Z

I0Z


c)

0Z

Z0 0,5ZT Z = Z0,10Z T

Transformer in wye-zigzag configuration a) determination of the zero sequence impedance at the

zigzag side b) determination of the zero sequence impedance at the wye side; c) zero sequence

model of the transformer

Mathematical Models


Three winding autotransformers and transformers. They are usually Yyd connected,

and the wye point is earthed. Their equivalent circuit are modelled the same way as it was

for two winding transformers.

a)

H L

b)

Z

H LZ H Z L

Z T

c)

Z

H LZ H Z L


Zero sequence models for high power (auto) transformers

TZ 0 T

Z TZ 0T

Machowski J, Kacejko P. Zwarcia w systemach elektroenergetycznych

(Power System Short-Circuits in Polish), Wydawnictwo Naukowo-

Techniczne, Warszawa 2002

Bibliography


short circuits lecture 8 transformer models part ii

Documents