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IEEE Transaction$ on Power Delivery. Vol. 9. No. 2, April 1994

A TRANSFORMER MODEL FOR WINDING FAULT STUDIES

Pauick BASTARD

Electrical Engineering Dpt. ECOLE SUPERIEURE DELECTRICITE 91190 Gif-sur-Yvette FRANCE

Pierre BERTRAND

Protection and Control Dpt. M ERLIN-C ERIN 38000 Grenoble FRANCE

Michel MEUNIER

Electrical Engineering Dpt. ECOLE SUPERIEURE DELECTRICITE 91 190 Cif-sur-Yvette FRANCE

KEY WORDS Transformer ; Modeling ; Simulation ; Winding faults ; EMTP ; Leakage ABSTRACT This paper deals with a method of modeling intemal faults in a power transformer. The method leads to a model which is entirely compatible with the EMTP software. It enables simulation of faults between any turn and the earth or between any two turns of the transformer windings. Implementation of the proposed method assumes knowledge of how to evaluate the leakage factors between the various coils of the transformer. A very simple method is proposed to evaluate these leakage factors. At last, an experimental validation of the model allows the estimation of its accuracy. INTRODUCTION The development and the validation of algorithms for a digital differential transformer protection require the preliminary determination of a power transformer model [SI. This model must allow to simulate all the situations which will be chosen to study the behaviour of the protection algorithms. In particular, it must allow the simulation of internal faults [4]. This is the aspect of the model that we shall study in this paper. Study of the algorithms implemented in a transformer protection leads us to simulate a large part of the power network, and not only the transformer itself. The upstream circuit with its lines, cables and grounding system, the power transformer itself, current transformers, potential transformers and a part of the downstream circuit with its grounding system and loads must be taken into account. If the aim is not to finally develop a complete simulation software for electrical transients, the transformer model must absolutely be compatible with a commercially available software ensuring the simulation of the "environment" and the computation management. That's why we have develcped a model which is entirely compatible with

EMIT.

I. GENERAL MODELING PRINCIPLES

The basic model used is the one supplied by the BCTRAN routine of the EMTP simulation software. Based on excitation and short-circuit tests, in positive and zero sequences, this routine computes two matrices [RI and [L] modeling the transformer. In the case of a three-phase transformer with two windings, these matrices are of order 6 : see figure 1, where Ri and Li are the resistance and the self inductance of coil i, and M , is the mutual inductance between i coils i and j. Note that B C T R A N does not take magnetic asymmetry into account.

,

R 1 O

0

0

0

0

0

phase I

[RI =

O R 2 0

0 0 0

0

0

0

0 0 0

O R 3 0

0

0

0

O R 4 0

0

0

phase I1

O R 5 0

0 0

>k

igure 1

R6,

phaselll

f

II = L

rimary secondary

"

I

M21

L1

M12 M13 M14 M15

4, M23 M24 MZ5

L3 M34 M35 h ' L4 M45

M31 M32

M41 M42 M43

M51 M52 M53 M54

L5

h! I

M61 M62 M63 M64 M65

93 SM 382-2 PWRD A paper recommended and approved by the IEEE Power System Relaying Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1993 Summer Meeting, Vancouver, B.C., Canada, July 18-22, 1993. Manuscript submitted Apr 27, 1992; made available for printing May 3, 1993. PRINTED IN USA

The BcTRAN routine must clearly be considered as an auxiliary routine of the EMTP itself, and not as a part of the main program. In point of fact, BCTRAN merely computes the elements of the matrices [RI and [L] and makes a file which can be directly read by EMTP. This file [R,L] can then be included in any EMTP input file. The transformer will thus be handled as mutuallycoupled R,L branches [l] [2]. This document does not aim at validating the BCTRAN routine. All the theoretical details relating to this routine are explained in [3]. Let us therefore assume the accuracy of the 6x6 matrices. The principle used to model a fault between a coil turn and the earth or between any two tums is to divide the faulty coil. Figure 2a shows the diagram enabling study of a turn-toearth fault and figure 2b shows the diagram enabling study of a turn-to-turn fault. In the former case, the transformer can be described with two 7x7 matrices [RI, [L] ; in the latter case, two 8x8 matrices are required.

0885-8977/94/$04.00 Q 1993 IEEE

69 1

1. Consistency If, when using the 7x7 matrix, coils a and b are supplied in series without any fault, the same results must be found as when the 6x6 matrix is used. Let us thus assume that coils a and b are supplied in series and that a current i flows through them : see figure 3. All other coils are on no-load.

XJ-

Figure 2a

figure 2b

In this way :

Our aim in this paper is to describe a method for the &termination of these 7x7 or 8x8 matrices using, firstly, the 6x6 matrices [R],[L] computed by BCTRAN and, secondly, the leakage factors between the various coils. The main advantage of this method lies in the fact that it requires no other test results than those required by the BCTRAN module to determine the 6x6 matrices. In point of fact, the various leakage factors can be determined from the geometrical data of the transformer and the position of the fault : see section IV. In a first step, we shall assume that the various leakage factors are known. These factors will thus be considered as parameters.

and These relations lead to the well-known expression of L3 , considered as two inductances in series :

II. TURN-TO-EARTHFAULT

If coil 3 has to be divided, as shown in figure 2a. the position of the fault point is defined by the following data : na = number of turns of "sub-coil" a nb = number of turns of "sub-coil" b The 7x7 matrix [RI will be determined with the help of the following relations : Ra=%R3 "3

;

La, Mab and I+, are elements of the 7x7 matrix. L3 is an element of the 6x6 matrix.

2. Leakage Taking into account a leakage factor between coils a and b is essential since the fault current will largely depend on the leakage. The leakage factor is :

R =R b: 3

P I

oh= 1 --

The main difficulty is to determine the new 7x7 matrix [L], as written in the next column. The elements in italics of this matrix are unknown. The other ones are determined by the routine BCTRAN and can thus be considered as known data. The first step is to determine the elements relating to the faulty coil : L , I+ and Mab. This purpose will be achieved, , . according to three rules : consistency, leakage and proportionality.

f

3. Proportionality To determine the three unknowns (La , L b and Mab), we must add a third equation to relations (1) and (2) :

2

G=( ) :

Letusnote: k = % .

nb

L

(3)

L1

M21

M12 M l a M l b M14 M15 M16 L2

M 2 a M 2 b M24 M25 M26 La M a b M a 4 M a 5 Ma6 Lb M b 4 M b 5 Mb6

L4 M45 M46 L5 M56 L6

M a l Ma2

[LI =

M b l Mb2 Mba

M41 M42 M 4 a M 4 b

Relation (3) is approximate. It expresses that k is the voltage ratio between coils a and b. It is strictly true only if there are no leakage ((Tab = 0). However, it represents an excellent numerical approximation when (Tab is close to zero and is very widely used. L3 is given by the 6x6 matrix computed by B C T R A N . k characterizes the fault position along the coil. In this way, if

M51 M52 M 5 a M 5 b M54 ,M61

M62 M 6 a M 6 b M64 M65

692

we consider Q as a parameter which can be computed, 3 equations are obtained for 3 unknown quantities : La, L , and + Mab. The resolution leads to the following relations :

(7) and (11) lead to :

(4)

I

I

Lb=

L3 k2 + 2kd=+

I

1

1

M3i is given by the 6x6 matrix computed by BCTRAN.

We have now to determine the mutual inductances between the coil a and any other coil i except for b (as well as those between the coil b and any other coil i except for a). Generally speaking, the consistency principle leads to : M3i = Ma, + Mbi,

(7)

U . TURN-TO-TURNFAULT I

Let us consider the turn-to-turn fault shown in figure 2b. As in part II, the matrix [RI can be easily determined with the help of the following relations :

A second equation must then be determined enabling Ma, and Mbi to be computed.

* la case : the coil i is wound onto the same leg as a and b.

than

obi.

If n, 2 nb. aa, will be determined with a greater accuracy In these conditions, we set down : The 8x8 matrix [L] to determine is :

I

We assume that o~ and (33i ,hence E, can be computed. (8) leads to :

L1 M21

M12 M l a M l b M l c M14 M15 M16 L2

M 2 a M 2 b M 2 c M24 M25 M26 La M a b Mac Ma4 M a 5 M a 6 Lb Mbc Mb4 M b 5 M b 6 Lc Mc4 Mc5 Mc6

L4 M45 M46 L5 M56 L6

M a l Ma2 (9)

M b l Mb2 Mba

M c l Mc2 Mca Mcb

Li, L3 and M3i are given by the 6x6 matrix computed by BCTRAN. La is computed by (4).

M41 M42 M 4 a M 4 b M 4 c

M51 M52 M 5 a M 5 b M 5 c M54 ,M61

(7) leads then to :

M62 M 6 a M 6 b M 6 c M64 M65

* 2"dcase : the coil i is not wound on the same leg as a and b.

The magnetic couplings between coils i and a on the one hand and i and b on the other hand are extremely bad since the third leg represents a leakage reluctance. The leakage factors are then very different from zero and it is hard to evaluate them. Since the leakage between these coils is channeled by a magnetic circuit, the corresponding leakage inductances are of the same order of magnitude as the self-inductances and thus, they have no effect on the numerical value of the short-circuit; currents appearing in case of a turn-to-earth fault. A simple proportionality relation is sufficient :

Let us now come back to the 3 "rules" set out above: consistency, leakage and proportionality, in order to determine the parameters relating to the faulty coil : Mab , Mac , Mbc ,L a , Lb and Lc. Let us assume that the three coils a, b and c are supplied in series and that a current i flows through them, as shown in figure 4. All other coils remain on no-load.

693

The resolution of the above system enables the construction of a 8x8 matrix consistent with the 6x6 matrix. This means that if coils a, b and c are supplied in series, the same simulation results are obtained using the 6x6 or the 8x8 matrix. In other words, imposing i, = ib = enables the 8x8 matrix to be reduced to a 6x6 matrix which will be identical to the one computed by BCTRAN. What will happen if two of the three "sub-coils" are grouped together, for example b and c ? As soon as ib = i,, the 8x8 matrix can easily be reduced to a 7x7 matrix. However, nothing enables us to affirm that this matrix is identical to the one obtained by the method set out for the study of the turn-to-earth fault. For this to be the case, the relations (16), (17) and (18) must be modified in the above system, no longer considering leakage factors Gab, and oac but od(b+c)O(a+b)/c and o(a+c)b

1

i

0

figure 4

I

(ki+ M ,

i +Mbc i)

0 = + a + Qb + +c

and Q = L3 i hence :

= (hi+ Mat, i +Mac i)+ (I+ i +Mab i + M h i),

(L3 : value computed by B ~ R A N )

Let us assume that the coils b anc c are supplied in series , as shown in figure 5 .

This relation represents the rule of consistency. By adding to this the rule of proportionality and the expressions of the leakage factors between the various "sub-coils", a system of equations is obtained : see hereunder. Considering the three leakage factors as known parameters (Gab , oacand (Tbc ), we obtain thus six equations for six unknowns : Mat,, Mac, Mb,, b , and Lc. 2 (Remark : for n sub-coils, n + Cn unknowns are thus obtained 2 for 1 + Cn + (n-1) relations ...) $a=M*i+Maci hence : and

7

a

I

+a

= Ma/b+ci

[consistency)

moreover : Mdb+C -LaLb+c

2

and : 2 Mab o a b = l - T s 1 , (16) a b

0.I(pK) = 1

(Tat= 1

-2

thus: [leakage)

Mac a c

2

(17)

I

2

1

%=(%)

2 (19) (proportion .)

2

e=(?)

(20)

Both these relations can advantageously replace two of the three relations (16), (17) and (18). (Td(p+,)and 6(a+b)/c will be actually computed in the same 1 way as the leakage factor Gab of part 1 which enables the computation of every 7x7 matrix obtained by dividing the coil no longer in three (a,b,c) but in two (a+b,c) or (a,b+c). In this way, by using the two coefficients (Td(b+c) and (T(a+b)/c. the 8x8 matrices generated in this case will be consistent with the 7x7 matrices generated to model a turn-to-earth fault. The third relation out of (16), (17) and (18) will be replaced by :

694

We have now t determine the mutual inductances of one o of the three windings a, b or c with any other winding i. The consistency principle results in

The 8x8 matrix will thus be characterized by only one "specific" parameter : Q(,+,.~. This parameter represents the leakage between coils (a+c, supplied in series) and (b). It should be noted, moreover, that the coils (a+b) and (b+c) allow a more accurate evaluation of the leakage factors than the coils a, b and c taken separately.

As in part 1 , two cases must then be considered in order to 1 establish the two complementary equations enabling the determination of M a , Mbi and Mci.

* 1st case : i is wound on the same leg as a, b and c.

The two windings with the largest number of turns will then be considered out of a, b and c. Let us assume for example : n a h C and nb2nc. Two equations are then determined as in part 1 : 1

(25

In short, the system to be solved is the one described hereunder : see next column. It should be observed that this 6 equation system with 6 unknowns is not linear and cannot lead to an explicit expression of the various inductances and mutuals. In practice, a numerical resolution method must therefore be used. If such a method requires knowledge of an approximate solution to initialize its iterations, the following will be taken :

2

We assume we know how to compute oa, o b i and (T3i hence E1 and E2 . Equations (25) lead to

2

2

(consistency with the 6x6 matrix)

M3i and L3 are computed by BCTRAN; and Lb are computed with the help of the equations (15) and (19) to (23). Equation (24) then yields (leakage consistent with every 7x7 matrix)

.

* 2nd case : i is not wound on the same legs a, b and c. For the reasons already laid out in part 1 in a similar case, 1 relation (24)is completed by two proportionality relations :

2

(leakage specific to the 8x8 matrix)

The various mutual inductances can then be expressed :

2

1

(proportion.

2

The numerical method we used to solve this system is one of the mathematical routines of the Fortran library IMSL.

695

IV. CALCULATION OF THE LEAKAGE FACTORS

1. General principles

I t has already been shown by several engineers [6] that leakage inductances between two concentric coils in a power transformer could be easily approximated, using the following assumptions : no saturation occurs (pmre pair), >> current density is constant in the windings, field field

where v1 ,v12 and v2 are the volumes of the internal winding, the inter-winding space and the external winding. With the rotation-symmetry assumption, equation (35) leads to :

R+a,+a,,+a, x=R+a, x=R+a,+a,,

8 is parallel to the axis of the magnetic core, 8 is symmetric in relation to this axis.

The complete calculation leads to the following relation : W = po .nf .f(h,R,al ,al2.a2).i? (37)

Let us assume that one of the windings is supplied with a :urrent il and the other one is short-circuited.

' axis of symmetry

where f(h,R,al,a12,a2) is a function of the geometric quantities of the transformer. W can also be considered as the energy stored in Lccl, the total leakage inductance,reduced to winding 1 :

With the help of (37) and (38). it is then possible to determine Lccl :

A 2 1

I I I

Note that most of the authors who have worked on the method described above propose a correction factor - based on experimental results - which is supposed to increase the precision of (39). Unfortunately, this coefficient depends on the type of transformer which has been used to obtain experimental results. In order to determine the correction factor characteristic of the transformer in question, we propose to use the 6x6 matrix computed by the routine BCTRAN. With the notations described on figure 1, the leakage inductance between windings 1 and 2, reduced to twinding 1, is :

igure 6

Because of the f i s t hvmthesis. the magnitude of field is close to zero everywherg h the core itself.-Moreover, because of the last two hypothesis, the magnitude H at any point in the air depends only on the distance x between the axis of the core and this point . Using the Amp&re's law applied to the 3 paths drawn on figure 6, it is possible to determine the following values of H(x) : for x<R : for R+al<x<R+al+alZ : for x>R+al+alz +a2 : H(x) = 0 nl il H(x) = -= h H(x) = 0 (32) H, (33) (34)

8

If we know the geometric quantities of the transformer, this leakage inductance between the primary and the secondary windings of phase I can also be computed with (39). The correction factor to be used whenever we calculate a leakage inductance between two sub-windings of the transformer with (39) can then be estimated once for all :

LEI between prim. and sec. windings computed with(40) k, = LW1between prim. and sec. windings computed with(39)

Equation (39) becomes :

If we assume that H(x) is linear for R<x<R+al and for R+al+a12<x<R+al+a12 +a2 , the curve H(x) has the shape drawn on figure 6, and the equations Hl(x) and Hz(x) of the lines AI and A2 are easy to determine. It is then possible to calculate the energy W stored in the windings :

If the geometric method used to compute the leakage inductances is good, k should be close to 1. That's what we have noticed in the example described below.

696

2. Applications

The problem is to use (41)to determine all the leakage factors oijused in relations ( ) (5), (6), and (21), (22), (23). This 4. is quite simple to do. The only difficulty is to choose correctly the windings and the shape of H(x), in order to compute the function f(h,R,al,al2,a2) specific to each situation. Let US describe some examples :

* calculation of Gab, used in (4) to simulate the case of fig. h

The situation to be considered is described on figure 7. The maximum values of H(x) are :

The method described above allows to determine Lcc,. It is then possible to calculate (Tab :

Note that L + b is the inductance of the primary winding. It is computed by BCTRAN. N,, Nb, n, and nb depend only on the position of the fault.

n,t

I

I I

I*

I I I

l l l

l l l

I

I

I

, "

I I I

axis of symmetry

Once again, the method described in IV.l allows to determine &(a=). It is then possible to calculate O(a+c)/b :

O(a+c)/b

=

Note that L + b + c is the inductance of the primary winding. It i computed by BcTRAN. s

3. Special cases

TH

I

I I

r' l

I

A121

When calculating a leakage factor, one of the sub-windings may not be long enough to cover the height of the core. The shapes of the field lines are then very distorted and the third hypothesis of section IV.1 is not valid anymore. What we propose in such a case is to use correction factors kh, which increase the leakage inductance calculated with the method described in section IV.l and applied in section IV.2. The formulas which allow calculation of the correction factors have been determined by Maurice DENIS-PAPIN [ 6 ] . They are based on many experimental results relating to various transformers.

* calculation of ~(,+,p,, in (23) to simulate the case of used figure 2b

The situation to be considered is described on figure 8. Windings a and c are fed in series with a current i, whereas winding b is short-circuited. The maximum values of H(x) are :

For the two cases shown on figure 9, the correction factor is determined as follows :

01 + 0 2

697 The short circuits have been made by closing a contactor between two taps. The test transformer was supplied at low voltage by an autotransformer wired in series with an insulation transformer. Some of the results of these experiments are described in [7].

4 %

figure 9

I

U

4

2. Calculation of the leakage factors

812 a 2

For the two cases shown on figure 10, the correction factor is determined as follows : (47)

Whenever an internal fault has to be simulated, some leakage factors must be determined with the method described in section IV. Below two curves are drawn which relate to the test transformer.

* l& example :turn-to-earth fault

as defined in (2) to Figure 12 shows the variations of describe the case of figure 2a. for a turn-toearth fault when the position of the fault moves on one of the HV windings,. The location of the fault is defined as follows : location = 2 1556

4 %

figure 1 0 Both in (46) and (47), the coefficients k l and k2 are defined as follows :

1.6

I

0.018

LEAKAGE FACTOR 161

0.015

ki=l+O,l(!)

; i=1;2

with: p = a 1 2 3 m +

(49)

6

1

1

FAULT LOCATION

V. EXPERIMENTAL VERIFICATION 1. Description of the test transformer

figure 12

The leakage factor increases rapidly when the fault occurs on an outer layer. Moreover, there is a slight decrease in leakage when the 2 winding parts (a and b) are overlapped.

A special three-phase transformer has been manufactered to validate the model described above. This transformer has the following characteristics :

rated power : 100 kVA rated voltages : 5500 V I 410 V coupling : Dyn short-circuit voltage : 3,96 % LV winding : 67 turns wound in 2 layers HV winding : 1556 turns wound in 8 layers The external coil (HV) of each phase has been fitted with taps as shown in figure 11.

nb o turns f nb of turns I

* 2nd example :turn-to-turn fault

Figure 13 shows the variations of as defined in (2) to describe the case of figure 2b for a fault between two turns, half a layer apart, when the position of the first defective turn V moves on one of the H windings,. The location of the fault is defined by the location of the first defective tum: location = 2 1556

LEAKAGE FACTOR IS]

I

I

FAULT LOCATION

1

igure 13

698

Figure 14 shows the location of the faults relating to points 1, 2, 3, ... of figure 13.

I

f axis of symmetry

r. . . .

:.

.

;

a.

-5.

:

'

-1

, %; ,

..

I

d'>*'f

..

,

/I-\,

. 111.' . ....................................................

,

'$

.

X I '

'Lf , I

,

I\

>'

I

In the cases 1, 3, 5, .... the sub-windings (a+c) and b are much more overlaped than in the cases 2, 4, 6, 8, ... That's why the leakage factors are smaller.

3 Comparison between calculation and measurements .

By reproducing 16 turn-to-earth faults and several dozens of turn-to-turn faults involving at least half a layer, we have noticed a difference between the calculated currents and the recorded currents which never exceeded 10 % in modulus and 10' in phase. The correlation between tests and simulation is even better when the fault involves only windings covering at least once the full winding height. Two examples are given below. turn-to-earth fault The quantities measured are shown on figure 15.

I .

turn-to-turn fault The quantities measured are shown on figure 16.

t

The results of a fault between the 1167th tum and the 1266" turn (half a layer) on the HV winding of phase A are shown below : The results of a fault between the 5831h tum and the earth on the HV winding of phase A are shown below :

I.. 5 .

i

8

-5.

I

-18.

..... ........ . ........

# U

. r

n

TIHB (a.)

1 . 0

68 be

49

z

-S. -18.

8

# 8.81 # U

21

C

.

-am

-I -6a

8

8.01

#.U

8.84

8.85

886

8.17

#.I8

nun

8 #I

(a.)

a.83

8.84

8.85

# 86

8.87

0.80

TlMB b) .

699 REFERENCES EMTP DEVELOPMENT COORDINATION GROUP ; ELECTRIC POWER RESEARCH INSTITUTE EMTP Revised Rule Book, Version 2.0 ; Volume 2 : Auxiliary Routines , Section 6. LEUVEN EMTP CENTER ATP Rule Book ; July 1987 ; Section 19C. .. V.BRABDWAJN ;H.W. DOMMEL ;11 DOMMEL "Matrix Representation of Three-phase N-Winding Transformers" IEEE TRANSACTIONS ON P.A.S ; pp 1369-1378; june 1982 J.L. BINARD ;J.C. MAUN "Power Transformer Simulation including Inrush Currents and Intemal Faults" JemeConfkrence Internationale IUACS-TCI '90, Nancy, FRANCE ; septembre 1990 A.G. PHADKE, J.S. THORP Computer Relaying f o r Power Systems Research Studies Press LTD ; pp 166-176 ; 1988 M. DENIS-PAPIN

La Pratique Industrielle de Transfomateurs

fiditions Albin Michel ; PARIS ; 1951 P.BERTRAND ;A. DEVALLAND ;P. BASTARD "A Simulation Model for Transformer Intemal Faults, Base for the Study of Protection and Monitoring Systems" 1 2 t h International Conference on Electricity Distribution; CIRED ; Birmingham ; UK ; May 1993

The good results obtained during the experimental validation allow us to use the model to simulate winding faults in large power transformers. A worked example has shown in [7] how to use this transformer model to design protection systems.

CONCLUSION Using the results supplied by the auxiliary routine of EMTP called B f f R A N , we have shown that every turn-to-earth or tum-toturn fault could be modelled in a power transformer as soon as the leakage factors could be evaluated. From the matrix of inductances computed by BCTRAN, we have developped a process to compute a new matrix which allows the simulation of any kind of internal fault. This computation is based on the determination of the leakage factors. We have also developed a simple method in order to determine these factors without using specific test results since such tests relating to the transformer in a fault situation will never be available. Comparisons between internal fault tests conducted on an experimental 100 kVA transformer and the corresponding simulations have been made. They have shown the accuracy of the model. The next step of the work presented in this paper could be the evaluation of the leakage factors with a finite-element method, in order to know if the accuracy could be increased in this way.

ck BASwas born in Pont-Audemer, FRANCE, in 1966. He graduated from the BCOLE SUP~RIEURED'BLECTRICITB (Gif-sur-Yvette, France) in 1988. From 1989 to 1992, he worked for the MERLIN GERIN company as a research engineer in the Protection and Control Department. In 1992, he received a Ph.D. Degree in electrical engineering from the university of PARIS (Orsay). He is now a researcher in the BCOLE SUP~RIEURE

D'ELECTRId.

.

BIOGRAPHIES

was born in Thionville, FRANCE, in 1956. He graduated in 1979 from BCOLE SUPBRIEURE D'INGmEURS fiLECrrUCIEN.9 DE GRENOBLE, France. He joined the MERLIN GERJN company in 1983, first as a research engineer in the Power Network Studies Department. He is presently responsible for the Electrical Engineering Section of the Protection and Control Department. He is a member of the S.E.E. (SociCtC des Electriciens et des Electroniciens).

0

Michel MEUNIER was born in Merdrignac, FRANCE, in 1945. He graduated from the BCOLE SUPBRIEURE D'BLECTRICITB (Cif-sur-Yvette, France) in 1968. He has been working in the Ecole SupCrieure d'Electricit6" since 1968. He is presently professor at the BCOLE SUPBUEURE D'BLECTRICITB where he manages a research group on power networks. He is a member of the S.E.E. (SociCtC des Electriciens et des Electroniciens).

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*transformer*; Achieves failure analysis; provide the*fault*simulation*model*of ...(16) na Presents*for*the turns of*winding**a*, nb is*for*the turns of...

- 电力变压器励磁涌流和故障电流仿真研究_许超英.pdf
*A**transformer**model**for**winding**fault**studies*[ J] . IEEE T ransactions on Power Delivery , 1998, 13( 3) : 738 - 744. [ 8] [ 9] LEUVEN EMTP...

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- Meunier, “
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*Model**for*Internal*Fault*of*Transformers*王 , 王增平 雪 ( 华北电力大学 ,...*a*Trans2 former*Model**for**Winding**Fault**Studies*[J ] . IEEE Transactions...

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- A New STATCOM Model for Power Flows Using_图文.pdf
*A*New STATCOM*Model**for*Power Flows Using_电力/水利_工程科技_专业资料。...*transformer*whose primary*winding*is shunt-connected with the ac power ...

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*A*Full Procedure to*Model*High Frequency*Transformer**Windings*_信息与通信_工程....' This*model*(*for**a*two*winding*magnetic component) can be seen in ...

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- Power Transformer Windings Partial Discharge_图文.pdf
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