Specific issues of equivalent longitudinal resistances calculation for three windings transformers Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Specific Issues of Equivalent Longitudinal Resistances Calculation for

Three Windings Transformers

Golovanov N., Porumb R., Toader C., Bulac C., Triçtiu I., Seritsan G.

University Politehnica of Bucharest Bucharest, Romania

Abstract. Accomplishing the aspirations of electrical power engineering consists, nowadays, in accurately determining the operating modes. Knowing the parameters of network elements is imperative for accurate solving the power flow regimes. This paper addresses unitarily the calculation of longitudinal resistance of three windings transformers and autotransformers, considering its rated parameters, and discussing the validity of relations demonstrated in detail. Keywords: transformer, three windings, accurately determining, parameters.

Aspecte specifice ale calcularii valorii ecivalente a rezistentei longitudinale a transformnatoarelor cu trei

înfâçurâri

Golovanov N., Porumb R., Toader C., Bulac C., Triçtiu I., Seritan G.

Universitatea Politehnica Bucureçti Bucureçti, România

Rezumat. Realizarea aspiratiile ingineriei electrice consta în zilele noastre în determinarea cu precizie a modurile de operare. Cunoaçterea parametrilor elementelor de retea este un imperativ pentru rezolvarea corecta a problemei determinarii regimurilor de schimb de putere în retea. În aceasta lucrare se examineaza minutios problema determinarii prin calcul a rezistentei longitudinale a transformatoarelor çi autotransformatoare cu trei înfâçurâri, luând în considerare parametrii evaluati; se prezinta rezultatele discutarii valabilitatii relatiilor obtinute de calcul.

Cuvinte cheie: transformator, trei înfaçurari, precizie, parametrii.

Особенности расчета продольного эквивалентного сопротивления трехобмоточных

трансформаторов Голованов Н., Порумб Р., Тоадер К., Булак К., Тристиу И., Серицан Г.

Университет "Политехника Бухарест" Бухарест, Румыния

Аннотация. Достижения поставленных целей при расчете режимов в электрических цепях (сетях) в значительной степени зависит от точности определения параметров элементов этих цепей. Знание значений параметров является обязательным условием для правильного расчета режима обмена мощностью в сетях. В данной работе рассматривается задача определения значения продольного эквивалентного сопротивления трехобмоточных трансформаторов и автотрансформаторов. Выполнен анализ корректности полученных расчетных соотношений. Ключевые слова: трансформатор, три обмотки, точность, параметры.

Introduction

In the current stage of electrical power systems development, transformers and autotransformers with three windings have gained wide spread use. This requires accurate knowledge of their parameters, and deep understanding of the various operating modes of the power system as well [1 - 4].

1. Equivalent scheme

Transformers are passive elements of electric networks and are represented by equivalent schemes with concentrated and constant parameters. (fig. 1. a, b).

Equivalent schemes described below are recommended [2,4] due to the fact that the calculation of the node stores the actual voltage due to the complex processing operators N12, N13.

When calculating the equivalent diagram elements require some necessary tests to determine the parameters of transformers and autotransformers with three windings.

!i i -i

system neutral

Z2 "i2 -1—on—

II 2 ,

III l3

—dZD—®——^

a. with range related to the same voltage

¡¡ra z2

i—C

il 12

!i i -1

system neutral

—13 z

■—Œ—C

YT

3 III 1,

b. with range related to different voltages

Fig.1. Equivalent schemes of three winding transformers

2. Three-winding transformer tests

These tests are described in literature

such as:

■ empty test

Transformers with three windings, the empty test is performed as transformers with two windings and consists of primary winding nominal supply voltage Un1, determining the sizes:

io - current nominal idling percentage (relative to voltage Un1 primary winding and Sn1 nominal apparent power source);

APo - active power losses at nominal load (relative to the primary winding voltage Un1 and Sn1 nominal apparent power source).

This sample measurement schemes are consistent technical rules in force, depending on the number of phases of transformers and are unitary power transformer manufacturers. The relations are associated calculation unit sizes and APo him.

■ Short-circuit(s) test

These samples can be achieved in two ways:

a) pairs of windings, third winding being empty;

b) simultaneous charging of three windings without exceeding their nominal charge.

Short variants of questionable evidence in the following ways:

- The value of testing costs;

- Duration of the tests;

- Accuracy in calculating transformer parameters using experimental data of test results for short.

The variant is preferred by most manufacturers a) pairs of windings for reasons of greater precision in determining the longitudinal parameters of transformer windings three.

When this preference is a need to make three different short samples [5 to 10].

Admittedly, three-phase transformer windings are noted by (i, j, k) as follows:

• 1 - primary winding (high voltage) with a rated voltage Un1 apparent power Sn1;

• 2 - the secondary coil (MV), rated voltage Un2 apparent power Sn2;

• 3 - tertiary winding (low voltage), nominal voltage Un3 and apparent power Sn3.

The sample circuit denoted by 1-2 proceed as follows: bypasses winding 1 (3 remaining empty winding) and apply a voltage called short circuit voltage defined by the expression usc, Usc 1-2 = usc 1-2 / (100-Un 1) at the terminals of the winding 1, so that the intensity of the current in the windings 1 and 2 (the short-circuit) does not exceed its nominal value.

Similarly proceed and short samples 1-3 and 2-3 to which the short-circuit voltage Usc 1-3 and Usc 2-3 as defined in the test circuit 1-2.

The three-phase system, they can cause either between phases or between phase and neutral winding system.

i. Short-circuit on windings pairs test - a)

Making a short sample - pair of coils - is determined:

usc i_2 - Percentage of short circuit rated voltage windings pair 1-2, in%;

usc 1-3 - Percentage of short circuit rated voltage windings pair 1-3, in%;

usc 1-3 - Percentage of short circuit rated voltage windings pair 2-3, in%;

APsc 1-2 - short nominal losses for the pair of coils 1-2, in W.

APsc 1-3 - short nominal losses for the pair of coils 1-2, in W.

APsc 2-3 - short nominal losses for the pair of coils 2-3, in W.

The experimental values listed and defined above relate to the rated voltage of winding 1, Un1 and the nominal apparent power of the winding 1

Sn1.

After this sequence of values such reports can be obtained catalog sizes that can be used to calculate the transformer with three windings parameters:

u'sc i_2 - rated voltage short circuit percentage (relative) per pair of coils 1-2, in%;

u'sc 1-3 - nominal voltage short circuit percentage (relative) per pair of coils 1-3, in%;

u'sc 2-3 - rated voltage short circuit percentage (relative) per pair of coils 2-3, in%;

AP'sc 1-2 - rated short-circuit losses (reported) on the pair of coils 1-2, in W.

AP'sc 3-1 - nominal short-circuit losses (reported) on the pair of coils 1-3, in W.

AP'sc 2-3 - short nominal losses (reported) on the pair of coils 2-3, in W.

It points out that it is strictly necessary to unify these parameters by reference to which depends on the ratio of windings apparent power proved both impedance voltage and short-circuit active losses. In addition impedance voltage depends on the ratio of nominal voltage windings are shorted

ii. Short-circuit test with simultaneous windings loading - b)

This option is rarely used, as required explicitly by the beneficiary of the three windings transformer; these maximum active power losses are measured.

It stresses that the short-circuit voltage of the three winding transformer is given only to the first variant testing and recorded in order of decreasing the rated value of voltages:

- windings 1-2: high voltage - medium voltage;

- windings 1-3: high voltage - low voltage;

- windings 2-3: medium voltage - low voltage, stating the power at which the calculation were made.

The short-circuit losses are given in the same order as the impedance voltage with the same mention over power to which they were determined.

It is stated for maximum losses, short version simultaneously tested windings, provided it is defined according to the relationship that defines just prior assumptions losses in short everything APsc rated maximum.

During test of idling and short-circuit is necessary to give indications about the alleged incidents, which may prevent subsequent failure of the transformers (transformers with two windings).

Also during simultaneous short-circuit test of windings further incidents may occur due to the synchronous operation of the three windings, which can be detected and remedied without destroying the transformer [11 - 16].

The figure and other characteristics of the transformer, namely:

• nominal voltages(U1n,U2n,U3n);

• nominal apparent power on each

wrapper (S1n, S2n, S3n);

• their reports against the greatest power N12, N13, according to Table 1.

Parameters must be considered for calculation, that in all cases Sn1 = 100% and nominal power density of secondary windings 2 and 3 relative to the primary winding 1 are represented by the general ratio:

100%/sn 2%/sn 3%,

(1)

and which may be found most commonly as in table 1:

Table I. Three - windings transformers, listed by winding loads

Transformer type Rated power of each winding, % from sn 1 of winding 1*)

sn 1** c -> 2 sn 3****)

I 100 100 100

II 100 100 66,(6)

100 66,(6) 100

III 100 66,(6) 66,(6)

IV 100 66,(6) 33,(3)

TID*****) 100 50 50

*maximum power winding (usually, high voltage) **) high voltage winding- hv ***) medium voltage winding- mv **** low voltage winding- lv *****) TID divided windings transformer (with equal rated voltages)

Thus are being defined a multitude of three-winding transformers.

3. Windings resistances with known maximum short-circuit losses

Total active losses [14^19] in short-circuit APsc tot are determined with the following equation:

APt

R • S2 + R2 * + R33 * S3

sctot

U

(2)

n1

with respect to the primary.

Consider the balance of apparent power on three windings transformers:

S -(S2 + S3 ) = 0 (3)

Given the Lagrange multipliers method to find the optimum position (2) the optimized condition of equality (3), yield the following equivalent function

transformer type I

I Sn1 = 100 k = 1 ^ | ; k2 = 0 1 K 2 = 100 2

® = APœtot -( + S3)]

(4)

Extreme conditions are found when cancels the first-order derivatives of the function (4) in relation to the variables Si, S2, S3. By performing these operations and putting windings are subjected to corresponding nominal loads, it follows that:

R ■ Sn1 = R2 • Sn 2 = R2 • S,

n 2

Jn 3

(5)

Interpretation of relations (5) follows from the principle of operation of the transformer, the existence of a single voltage value on a single spiral, for all windings.

For fulfilling the condition of extreme conditioning is required that between windings load the apparent power must be satisfied requesting windings in this sample to be as different from each other, some may be nominal charge, as follow:

N H S2IH S3

(6)

If the load coefficients are associated to the 2 and 3 windings, thus

S

k =k2 = ^

S3

S,

S,

(7)

The values of nominal coefficients defined in equation (7), are

k = n 2 . k = n 3

n1 = 100 ; n 2 = 100

(8)

Equation (6) is replaced in (3) and it results:

ki + k2 = 1 (9)

The maximum active losses in short-circuit APsc tot max, are obtained using equation (9) in the condition (6) for transformers in table i, obtaining:

maximum losses in copper are obtained by charging i and 2 windings at nominal values and leaving empty the 3 winding;

k1 = 0; k2 = 1 ^ -

(sn1 =100 Un 3 =100

(11)

or when windings i and 3 are loaded at nominal values and leaving empty the 2 winding;

sn 2 = 100 ; sn 3 = 100

(12)

or when windings 2 and 3 are loaded at nominal values and leaving empty the i winding;

transformer type II a

r Sn1 = 100

kl = 1 ^ | ; k2 = 0

1 |sn 2 = 100 2

(13)

maximum losses in copper are obtained by charging windings 1 and 2 at rated values and leaving empty the winding 3;

transformer type II - b

r sn1 = 100

ki = 1 ^ ■

|sn 2 = 100

; k2 = 0

(14)

maximum losses in copper are obtained by charging windings 1 and 3 at rated values and leaving empty the winding 2;

- transformer type III

(15)

3Sn3 = 3

maximum losses in copper are obtained by charging windings 1 and 2 at rated values and winding 3 being charged at 50 % of the rated power

2 sn1 = 100 1

=--> < 200 k2 s3

3 Sn 2 =~ 3

100 200

k2 =--> <

2 3

sn1 = 100

sn 3 = ■

200 ; kl =~ ^ s2 =

(16)

100

3

- sn 2 =

200

3

maximum losses in copper are obtained by charging windings 1 and 3 at rated values and winding 2 being charged at 50 % of the rated power

- transformer type IV

type Il-b transformer

2 , APsctotmax 'Un1 R1 =~ R2 = R3 =-2-

3 2 • Sn

n1

type III transformer

u 2 2 6 •AP.sctotmax Un1 R1 = — R = — R3 =---

1 3 2 3 3 11 s 2

n1

type TIDtransformer

(23)

k =-->

sn 1 = 100

1

200 ; k2 =~ ^

Sn2 = ~ 3

sn1 = 100

100 (17)

5n 3

R1 =

R2 = R3 = APsctot max Un1

2 = 2 = 2 • S2.

n1

(24)

maximum losses in copper are obtained by charging at the rated values all the windings. - V type transformer - TID

ki =—

sn 1

sn 2

= 100 1

; k2 =--> <

= 50 2 2

sn1

sn 3

= 100 = 50

(18)

maximum losses in copper are obtained by charging all of the windings to their rated values.

Maximum losses in the copper are obtained by replacing equations (5), (7) and taking into account (8) in the (2) expression and results:

AP.

sctot max

f - . M

R1 • S«1• 1+100• k1 +

K sn 2 sn 3 j

U

(19)

n 1

4. Calculation of winding resistance in case in which the short circuit losses on windings are known

If the short circuit losses [10^14] are not reported to the rated power of primary winding, windings resistance are calculated from the following equations:

P.c 1-2 = (( + R2)

i S \2 Sn 2

P.c 1-3 =(( + R3)

VSn1/ f S ^

Sn 3

Sn1 V ni

Sn 2 - Sn 1

Sn 3 - Sn1

(25)

P.c 2-3 = (2 + R3)

( S Sn 3

Sn1 V ni

Sn 3 - Sn 2

In conclusion, it results that maximum short-circuit losses are obtained when the power in primary winding is maximum, and its distribution on secondary and tertiary winding is accomplished as unequal.

Taking into account the above computation results the relations for winding resistance, for each type of transformers: ■ type I transformer

R1 = R2 = R3 = ■

' sctot max

2 • S,

n 1

type II-a transformer

t AP

1 2 , ^ sctot max R1 = R2 = — • R3 ="

U2n1

2 • Sh

(20)

In case of S3 > S2 the last equations (25) becomes:

P.sc 2-3 = (R2+ R3)

( c A2 Sn 2

; Sn2 - Sn3

(26)

Solving the system formed by the equations (25) in relation to R1, R'2 and R'3 are determined the general solution system:

■

■

■

■

R -

AP

f S A2

Sn1

sc 1-2 ■

V Sn 2 ,

+ APs

sc1-3 '

f S A2

Sn 1

Sn 3 V y

-AP

f S A2

Sn1

sc 2-3 ■

Sn 3 V y

u2

2 • S%1

R2 -

ap,

f o A2

Sn1

sc 1-2

s2

+ AP

f O A2 Sn1

sc 2-3

f S, A2

-APs

sc 1-3

n1

U2

2 • S;

n1

R3 =

AP

f S A2

Sn1

sc 1-3 '

Sn 3 V y

+ AP

f S A2

Sn1

sc 2-3 ■

Sn 3

V 'ID y

-APs

sc 1-2 '

S2 Sn1

V Sn 2 y

Un2

2 • Sn21

Or with notations:

APsc1-2 - APsc 1-2 •

f S \2 Sn1

V Sn 2 y

2

APs'c 1-3 - APsc 1-3

APsc 2-3 -APsc 2-3

S

n1

(28)

f S A2

n1

Sn

equations system (27) is written with equations (28).

For the case of the system formed by equations (25), the solution is

R1 -

f S A2

AP

sc 1-2

n 1

s2

f S A2

+ AP

sc 2-3

n1

-APs

f S A2

Sn1

sc 2-3 ■

V Sn 2 y

U2

2 • s2?1

R2-

AP

f o A2

Sn 1

sc 1-2

Sn

+ AP

f o A2

Sn1

sc 2-3

(29)

f S A2

-APs

sc 1-3

n1

Sn 3

V y

Un2

2 • S;

n 1

R3-

APs

f S A2

Sn1

sc 1-3

+ AP

f S A2

Sn 1

sc 2-3

-AP

2

Sn1

sc 1-2

V Sn 2 y

U2

2 • S

n 1

— (27) and with notations (30) from below

APsc 1-2 - APsc 1-2

APs'c 1-3 - APsc1-3 •

2

Sn1

Sn

f S A2

n 1

Sn 3

V y

(30)

APsc 2-3 - APsc 2-3 •

f S A2

Sn1

V Sn 2 y

the system defined by equations (29) becomes:

R -

APsc 1-2 + APsc 1-3 - APsc 2-3

f U A2

Un1

R2 -

APsc 1-2 +APs'c 2-3 -APsc1-3

Sn1

V nl y

2

U

n1

R3 -■

APs'c 1-3 +APs'c 2-3 -APsc1-2

Sn1

V nl y

f U, A2

(31)

n1

Particular types of solutions corresponding to transformer types listed in Table 1 are determined from (27) or (29) through particular mathematical developments. In case of short circuit losses on windings relative to the rated power of the primary - winding, winding resistance calculation will be performed with equations (27) or (29), by putting the following condition:

Sn1 - Sn 2 - Sn 3

(32)

These calculation elements are necessary for calculating proper evaluation of longitudinal resistance to determine both the permanent arrangements for the emergency and the associated electrical substations in which they are components.

5. Case studies

Table 2 shows the resistance values of three windings transformers class type TID. For

6. Conclusions

These calculations were required because transformer construction companies give their characteristics in one of the situations mentioned above, especially up to 100%, (however, up to 80%) as short-circuit loss of windings pairs unreported rated power of the transformer primary side, the rest being of TID-type transformer. These transformers are increasingly being used in the electrical power system in case of renewables.

References

[1] Boyajian, A., Resolution of Transformer Reactance. Into Primary of Secondary Reactances. Transaction AIEE, June 22-26, pp. 805-820, 1925;

[2] Denisov, P.K. Asymmetrical Loading of Three-Phase Three-Winding Transformer Banks. Transaction, January 1944, volume 63, pp 27-29

[3] Potolea, E., Calculul regimului permanent al sistemelor electrice. Editura Tehnica, Bucure§ti, 1967.

[4] Bercovici, M., Arie, A., Poeata, Al. -Retele electrice. Calculul electric. Editura Tehnica, Bucure§ti, 1974.

[5] Iacdbescu, Gh., Iordarescu, I., Tenovici, R., Retsle electrice. EdituraDidactica ¡jiPedagogica, Bucuregi, 1975.

[6] Potolea, E., Calculul regimurilor de functionare ale sistemelor electroenergetice. Editura Tehnica, Bucure§ti, 1977.

resistance calculation. For resistances calculating, relations from the preceding paragraph were used.

[7] Iacobescu, Gh., §a, Retele electrice. Editura Didactica §i Pedagógica, Bucure§ti. 1982.

[8] Arie, A., §a, Transportul §i Distributia Energiei Electrice, Editura Didactica §i Pedagógica, Bucure§ti. 1981.

[9] Kendrew T. S.: Improved Methods for Distribution Loss Evaluation. Electric Power Research Institute, California 1983.

[10] Toader, C., Tenovici, R., Dumitriu, C., Unele aspect privind calculul rezistentelor echivalente la transformatoarele §i autotrans-formatoarele cu trei rnfa§urári, Conferinta de Nationala de Energetica, Bucure§ti, 1988.

[11] Chiping Sun, Nasser H Kutkut, D.W. Novotny, D.M. Divan. General Echivalent Circuit of a Multi-Winding Co-Axial Winding Transformer. 1995 IEEE, pp 2507-2515.

[12] Albert, Hermina, Mihailescu, Anca, Pierderi de putere §i energie rn retelele electrice. Editura Tehnica. Bucure§ti. 1997

[13] Georgescu Gh., Gavrila§, M., Rada§anu, D. - Calculul §i reducerea pierderilor de putere §i energie rn retelele electrice. Editura Spectrum, Ia§i, 1997.

[14] Xnikov, A.B. Teoria i rascet mnogoobmotocnih transformatorov. Mockva Colon Press, 2003

[15] *** Distribution system loss evaluation manual. American Public Power association. 1994.

[16] *** Switchgear Manual - ABB 10 th Edition - 1999.

[17] John, G. Hayes, Neil O'Donovan, Michael G. Egan. The Extended T Model of the

Table 2. TID-type transformers resistances

Un Sn Si S2 S3 Use AP sc max APo io Ri R2 R3

kV MVA % % % % kW kW % a a a

20 25 100 50 50 9.5 125 29 0.7 0.04 0.02 0.02

20 32 100 50 50 11.5 180 33 0.7 0.04 0.02 0.02

20 40 100 50 50 8.5 180 39 0.65 0.02 0.01 0.01

20 40 100 50 50 14 180 39 0.65 0.02 0.01 0.01

20 63 100 50 50 11.5 280 55 0.6 0.01 0.01 0.01

20 80 100 50 50 9.5 330 65 0.6 0.01 0.01 0.01

20 80 100 50 50 14 330 65 0.6 0.01 0.01 0.01

110 25 100 50 50 10.5 120 30 0.7 1.16 0.58 0.58

110 32 100 50 50 10.5 145 40 0.7 0.86 0.43 0.43

110 40 100 50 50 10.5 150 50 0.65 0.57 0.28 0.28

110 40 100 50 50 12 180 40 0.65 0.68 0.34 0.34

110 63 100 50 50 10.5 245 70 0.6 0.37 0.19 0.19

110 80 100 50 50 10.5 310 85 0.6 0.29 0.15 0.15

110 125 100 50 50 10.5 400 120 0.55 0.15 0.08 0.08

Multiwinding Transformer. 2003, 35th Annual IEEE Power Electronics Specialists Conference, Aachen, Germany, 2004, pp 1812-1817. [18] Eremia, M., Electric Power Systems. Editura Academiei Romane, Bucure§ti, 2006.

[19] Francisco de Leon, Juan A. Martinez. Dual Three-Winding Transformer Equivalent Circuit Matching Leakage Measurements. IEEE Transactions on Power Delivery vol 24, nr 1, January 2009, pp 160-168.

About authors:

Nicolae Golovanov. Profesor, dr.inginer la catedra Sisteme Electroenergetice, facultatea Energetica, Universitatea Politehnica din Bucure§ti, specialist in domeniul utilizarii eficiente a energiei electrice §i in domeniul calitatii energiei electrice, autor, prim autor sau coautor la un numar de peste 30 carti sau tratate. Are peste 100 lucrari ¡jtiintifice publicate in reviste de specialitate din tara §i strainatat.

Radu Porumb. Profesor, dr.ing., Universitatea Politehnica Bucureçti. Experienta profesionalâ a fost axatâ, în principal, pe sisteme electrice, producerea distribuitâ, de planificare a resurselor regenerabile çi dezvoltarea. E-mail:raduporumb@yahoo.com