# EVALUATION OF THE LIFE TIME OF POWER TRANSFORMER WINDING INSULATION USING RAYLEIGH REPARTITION - Ubiquitous Computing and Communication Journal by tabindah

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```									      EVALUATION OF THE LIFE TIME OF POWER TRANSFORMER
WINDING INSULATION
USING RAYLEIGH REPARTITION

Adrian Munteanu, Laura Lelutiu, Lucian Iulian
Transilvania University of Brasov, Faculty of Electrical Engineering and Computer Science, Brasov, Romania

ABSTRACT
Smart grid technology is a new approach for promoting the energy efficiency and for accomplishing
the environment objectives. In this context, the lifetime of transformers have a great impact on the
quality of the delivered energy. The paper deals with a mathematical statistic method for estimation
of winding insulation lifetime of the power transformers, based on Rayleigh distribution. The
concept of the winding equivalent temperature is introduced.

Keywords: charge factor, normal repartition, Rayleigh repartition, equivalent temperature, life time.

1     INTRODUCTION                                           operation. There are divergent points of views
regarding the main parameters characterizing
The continuous diversification of using                  the degradation state and the evaluation manner
conditions and increasing of electrical,                     of the lifetime reserve. Many conferences and
mechanical,      thermal    and    environmental             international symposiums (IEEE, CIGRE,
solicitations of the electrical equipments                   INSUCON etc.) and specialized reviews keep
requires accurate knowledge of materials                     « the diagnosis of insulation state » as
characteristics, and especially the evolution of             preferred subject.
these characteristics considering the nature,                    The simplest method for life prediction is
intensity and duration of solicitations. On the              assuming that the total amount of water in
other hand, the users of this electrical                     electro     technical    equipment     increases
equipment want to know exactly the state of                  continuously and linearly, neither paper
the equipment components and eventually to                   humidity nor dissipation factor, nor specific
estimate their lifetime reserve.                             resistance will have a linear relationship with
For power transformers the most sensible                 temperature, which makes the calculation of
components      to    electrical   and    thermal            residual life directly from these parameters
solicitations are insulation systems. Although               impossible [1].
the number of dry transformers is in continuous
development, the energy system in all countries                              G end -  G
is still based on traditional transformers with                rres  100                                       (1)
oil-paper insulation. Due to the fact that these
 G end   Gbeg
transformers have a relatively extended
operating lifetime, knowing the insulation state             where: G is the water content, the time period,
and furthermore, diagnosing this state is of                 G beg is the total amount of water at starting
great importance in order to avoid an
unplanned and undesirable placing out of                     point of measurement, G end is the total amount
operation of this type of transformers. This is              of water at the end of measurement.
the reason why all manufacturers and users of                   In years the relation became:
power transformers constantly record the
information       regarding     permanent      and
accidental electrical and thermal solicitations                rres  
 G end -  Gbeg                    (2)
of insulation systems of the transformers in use                           G   Gbeg
and try to estimate their state and their lifetime
reserve.                                                         If the initial value G beg is not known the
Even though the paper oil insulation system              residual life time is:
has been studied for about a century, the
diagnosis of these states is still a difficult
G    end   - G   τ2
In order to establish the equivalent temperature
rres  τ                                    (3)     of the transformer winding the differential equation
G     τ2    G   τ1
of winding heating is considered:
where: τ 2 and τ1 and τ is the interval of
dT
years between measurements.                                C      +T = P.                                 (8)
dt
Generally the end of life is due by a
combination of factors but in industry the
principal factor for end of life relates only to         where: C - caloric capacity of the winding [J/K],  -
the transformers thermal factor. A classical             thermal conductivity of the winding representing the
method of calculating the remaining life of a            total transferable heat by conduction, convection and
transformer has been the Arhennius-Dakin                 radiation from the winding insulation in the time unit
formula.                                                 p, when the difference between the environmental
Remaining life t is: [2]                             temperature and the winding insulation temperature
is 1 K [W/K], T- winding temperature, [K].
B                                             The solution of the equation (8) is:
t  Aexp                                   (4)
T 
   P
where A is the initial life; B is a constant              T = C1 exp  - t   .                           (9)
 C  
depending on the material properties; and T is
the temperature in K .                                   where C1 is integration constant.
Other method [3] shows that is possible to
calculate the remaining value for each of                   For t=0 results T  Ta where Ta represent the
electrical parameters (tan δ of insulation,              environmental temperature.
insulation resistance, electric strength of oil,
tan δ of oil, conductivity oil, etc.). If the limits        From relation (2) results the expression of
of these parameters are designated as r 0 and r 1        constant C1:
the current value r is between these limits.                           P
Useful values of any parameters are given in               C1 =T a      .                                (10)

percentage by:
From relations (9) and (10) results the expression
r  r0
Ruse  100                                   (5)     of the winding temperature:
r1  r0
P      P       
T=       T   exp  - t  .                  (11)
For residual value:                                         a         C 

r1  r                                    The second term from relation (11) is likely to be
R res  100                                  (6)     neglected because C<<Λ and
r1  r0

If the variation of this parameter with time,             
exp   t   0                                (12)
r                                                              C 
, is known,
τ
Results:
r1  r
Rres                                                   P P  Pk
r                                (7)      T=  0                                          (13)
    


where Pk represents the loses in windings and P0
Analyzing and studying these methods we
intend to propose a new method, based on the             the loses in iron (in paper [4] the loses in iron where
statistical processing of measurement records.           neglected.
Relation (13) becomes:
2     EQUIVALENT TEMPERATURE                      OF
WINDING INSULATION                                        Pk  P0
T=            .                                 (14)

The authors of the paper [4] defined the
equivalent temperature of winding insulation as the
supposed constant temperature at which, providing           Noting with Pkn the winding loses in the nominal
the winding insulation was solicited, there would be     regime result the winding loses in the nominal
produced a similar effect of insulation degradation to   anything regime:
the degradation effect produced by the solicitation at
real temperature.
2                              From the daily consumed energy the current
P kn    I 
Pk = I 2           2
=   P kn .
                         (15)     intensity in the 400 kV winding is determined with
I n  In                                  the relation:

The transformer charging factor, noted                                   Wi 6
10
with, is defined β as the ratio between the                      I1         24         ,                                (21)
3 U 1 cos 
current intensity I and the nominal current
intensity I n :
where cos   0,92 is the neutral power factor,
I
β=       .                                         (16)     the transformer being equipped with a control
In
system of the power factor.
The value of the nominal current intensity
For calculus simplification the variation                 in the 440 kV winding results:
with temperature of the winding resistance is
neglected.                                                                              S
I1n                          392,68 A
3 U 2 cos                               (22)
Considering    relations (14) and (15), the
relation (13) becomes:
β 2 Pkn  P0                                              The ratio between the current intensity value in
T=                .                                 (17)
                                                the secondary winding, given by (22) relation and the
nominal current intensity value leads to the charging
If in (17) relation β  0 than T  Ta , and              factor of the transformer, noted with β (relation 14).
relation (17) will be:
3           THE NORMAL REPARTITION                            OF
P0
Ta =      .                                         (18)                THE CHARGE FACTOR

In [1] considers almost the load current and
From(17) and (18)                relations,   noting   implicitly the charge factor, keeps under a
Pkn
2
normal repartition laws with the repartition
with  =     , results:                                      density:
P0

f β  
1
exp 

 ββ 2 

                 (23)
2
T
P0
                   
T  a β 2 Pkn  P0  Ta β 2 δ 2  1 .              (19)
2π σ      

2 β 


where:
or:
n
1
β        β1  β 2  β 3  ......  β n   1   βk    (24)
1 T                                                                n                                    n
β         1  p T  Ta                            (20)                                                         k 1
δ Ta
1
1
2
β 
n

    2
 2
 2
β1 - β  β 2 - β  β n - β  



where: p                    .
δ Ta                                                         n                                              (25)
1
 β k - β 
2

The load current and implicitly the charge                       n
k 1
factor are hazardous variables taking discrete
values.                                                              The reference value for the difference d is given
The study was realized on a transformer
made by ELECTROPUTERE CRAIOVA                                by λ n where n is the series volume, and it is equal
factory, with: 250 MVA power and voltages of                 to 1920. For the reference value, taking in
400/110/20 kV.                                               consideration the level of signification α=0, 05, λ=1,
Daily working regimes of this transformer for             36, results the value of 59, 59221. The Table 1
1996-2009 periods are characterized by daily active          presents the evaluation procedure in concordance
consumed electric energy (Wi).                               with the nominal repartition.
The authors considered the active electrical                 In the Table 1 it is observed that for 4 from 12 classes
energy records for this transformer, recording daily         (marked with *) the maximum values of the distance are
the active electrical energy on the for 400 kV side          exceeded, so the normal repartition it is refused.
throughout a period of 13 years (1997-2003), in
MWh.
Table 1: The verification of concordance with
a normal repartition

Class                  X          N_c       Z

(0; 0,072]              0,072           78      -1,45
(0,072; 0,144]             0,144          213      -1,07
(0,144; 0,216]             0,216          450      -0,69
(0,216; 0,288]             0,288          861      -0,31
(0,288;0,360]             0,360          1197      0,06
(0,360; 0,432]             0,432          1374      0,44
(0,432;0,504]              0,504          1494      0,81       Figure 1: Charging factor histogram
(0,504;0,576]              0,576          1638      1,19
(0,576; 0,648]             0,648          1722      1,57        The concordance criteria between theoretical
(0,648; 0,720]             0,720          1839      1,95     and        empirical       repartition      is:

(0,720; 0,792]             0,792          1875      2,33     d max      d  0,031.
n
(0,792; 0,864]             0,864          1920      2,71
Table 2: The concordance with a Rayleigh
Class                 F_(0)         n*F_(0)     D       repartition
(0; 0,072]              0,073         141,120   *63,12
(0,072; 0,144]             0,144         277,632   *60,21           Class          N         nC         F
(0,144; 0,216]             0,245          0,2451   *86,2          (0; 0,072]      78         78      0,0406
(0,216; 0,288]             0,382          1867,3   31,84       (0,072; 0,144]     135       213      0,0703
(0,288;0,360]             0,523          1900,4   25,41       (0,144; 0,216]     237       450      0,1234
(0,360; 0,432]             0,670          1913,2    6,72       (0,216; 0,288]     411       861      0,2140
(0,432; 0,504]             0,788          1513,1   24,72        (0,288;0,360]     336       1197     0,175
(0,504; 0,576]             0,975          1873,1   57,36       (0,360; 0,432]     177       1374     0,0921
(0,576; 0,648]             0,941          1808,2   *86,2       (0,432; 0,504]     120       1494     0,0625
(0,648; 0,720]             0,972          1867,3   31,84       (0,504; 0,576]     144       1638     0,075
(0,720; 0,792]             0,989          1900,4   25,41       (0,576; 0,648]      84       1722     0,0437
(0,792; 0,864]             0,996          1913,2    6,72       (0,648; 0,720]     117       1839     0,0609
(0,720; 0,792]      36       1875     0,0187
4 THE RAYLEIGH REPARTITION OF                                      (0,792; 0,864]      45       1920        5
0,0234
THE CHARGE FACTOR                                                                                         3
It is considered for the charging factor a Rayleigh                 Class         F_a       F_x         D
repartition, with the density of repartition give by                  (0; 0,072]     0,0406   0,03295    0,0076
relation:
(0,072; 0,144]   0,1109   0,12545    0,0145
 β     2                              (0,144; 0,216]   0,2343   0,26038    0,0260
2β
f β              exp           .              (26)        (0,216; 0,288]   0,4484   0,41504    0,0303
η   2
 η         
                                      (0,288; 0,360]   0,6234   0,56735    *0,056
Where η is the Rayleigh repartition parameter,                      (0,360; 0,432]   0,7156   0,70075    0,0148
equal by 0, 3933 determined by the graphic                          (0,432; 0,504]   0,7781   0,80643    0,0283
method and by the least squares theory. This                        (0,504; 0,576]   0,8531   0,88291    0,0297
repartition was proposed considering the                            (0,576; 0,648]   0,8968   0,93376    0,0308
histogram of the charging factor presented in                       (0,648; 0,720]   0,9578   0,96496    0,0071
Figure 1, as realised after the repartition in classes of that
1920 values.                                                        (0,720; 0,792]   0,9765   0,98266    0,0061
(0,792; 0,864]      1     0,99198    0,0080
From the Table 2 it is observed that only                                                    If note:
the 5th value is over the imposed value, so it is
possible to consider the charge factor verify a                                                    p
c
Rayleigh repartition.                                                                              η

5       ESTIMATION    OF   THE                                     WINDING                       Results:
INSULATION LIFE TIME

The temperature repartition density is:

g T   4c 2 exp  c 2 T  Ta      
(35)
 A exp c T      2

dβ                                dβ
g T   f βT        f β1 T   f β 2 T 
dT                                dT                (27)                where:

A  4c 2 exp c 2Ta                                   (36)
From (16) relation results:
Verifying if (26) relation is a repartition
density is calculated:
β1  p T  Ta                      ;      β    p T  Ta                 (28)
                         

dβ1      p                                dβ 2       p

Ta
 
g T dT  4c 2 exp  c 2 Ta  T  dT  4
Ta
       (37)
                           ;                                     (29)
dT    2 T  Ta                            dT      2 T  Ta

For the temperature density repartition is
For the temperature repartition density                                              considered the expression:
results the expression:
g T 
g1 T  
4

 c 2 exp  c 2 T  Ta    (38)
g T   f1 T   f 2 T f 3 T                                         (30)
6        THE REPARTITION DENSITY OF THE
where:                                                                                    LIFE TIME

The life time law expression τ is:
2
2 p T  Ta       p T T                         
a
f1 T                 exp                                 
(31)       τ  aexp bT                                         (39)
η2              η                           
                                 
Than:
2
  p T T                                  1 τ
 2 p T  Ta                                                               T   ln                                               (40)
exp                   
a
f 2 T                                                                                     b a
η   2
    η              (32)
                    
1
dT          dτ                                       (41)
bτ
p                        p
f 3 T                                                                    (33)
2 T  Ta                    2 T  Ta                                      For temperature repartition density we have
the expression:
Results:
dT
h τ   g1 T τ        
2                                          dτ
 2 p 2 T  Ta                                      
gT                                    exp   p T  Ta                                               c 2  τ  1
       η                                   η                   (34)                          
 c 2 exp c 2Ta exp   ln         
(42)
                                      
                    
                                       b  a   b τ
          
2
 c  τ  1
 Aexp   ln 
 b  a   b τ
        
Verifying if (42) is the expression of a                                                                   
 c2           1
repartition density is calculated:                                                                    τ  A exp  τ1  exp τ1  a dτ1 
 b            b
τa       
                   
c2        τ  1                                              Aa

  c2       
    h τ dτ       A exp b  ln a  bτ dτ                                                                   exp  τ1     1   dτ 1 
τa                  τa
 
          

(43)    
b        
- bTa      

 b     


τ                             Aa 1          c2      
Changing the variable ln                                             τ1 results:                      exp  τ1     1        
b c2              b    
a                                  1
 
             - bTa
b                                             (50)
τ  a exp τ1                                                                           (44)                                          
 Aa    c2      
 2   exp  τ1     1       
c b           b    
dτ  a exp τ1 dτ1                                                                                         
             -bTa

(45)

 Aa 1           c 2      
Relation (42) becomes:                                                                                  exp       1  τ1     
b c  2           b      

            - bTa

                                                                                                         1
 c 2  a exp τ1                                                                  b
  h τ dτ  A exp  τ1 

 b  ba exp τ  dτ1                                                          Aa
τa             τ1a                 1                                                            2
c b
 
exp  Ta c 2  b          
           2
A
b
c2
   exp cb τ  dτ
exp c 2Ta        


                 1        1   

τ1a                                                        (46)    Conforming to [21] we consider the integral
                                    module (46), than:
c2 b                c2 

b c2
 
exp c 2Ta exp  τ1  
 b                                                                              Aa
     τ1a                                                              τ     2
 
exp  Ta c 2  b                  (51)
 
2
 exp c Ta 0  exp  c Ta  1          2
                                                  c b

Conforming to [20] we consider the integral                                                          To establish the coefficients a and b of
module.                                                                                             equation (39) was used [3].
In equation (45) limits or integration are:                                                     It is obtained a curve presented in Figure 2.
This curve has the equation:
τ a  a exp bTa  ;
(47)     ln τ  30,01297  0,05379T                          (52)
τ   a exp b    0

7         THE CALCULATION OF THE LIFE TIME
AVERAGE VALUE

The life time average value of the winding
insulation is given by relation:


τ  A τ h τ dτ                                                                    (48)
τ
a

or:
Figure 2: Life time curve
                            
 c2 τ  1                                                          The approximation of the of the life time curve
τ  A τ h τ dτ  A τ exp ln  dτ 
               b a  bτ    
      
with a line is very good, the correlation coefficient,
τ              τ
a                            a
(49)   the indicator r2 having the value of 0, 99681.
                2
c  τ 1
 A exp ln  dτ

 b ab                                                                              8 CONCLUSIONS
τ        
a

The average value of winding insulation life time of
τ                                                                       a transformer depends by statistical parameter of
Changing the variable ln  τ1 it is obtained
a                                                                       Rayleigh distributions, by the ratio between copper
the average value of the winding insulation life                                                    loses (at short-circuit test) and iron losses (at without
time:                                                                                               load test) also by the medium temperature.
The obtained results permit the estimation of           [10] A. Munteanu, “Life-time prognosis of power
life time reserve of the transformer insulation                 transformer insulation” (Prognoza duratei de
and also the estimation of the equivalent                       viaţă a izolanţilor transformatoarelor electrice).
temperature of the winding insulation.                          Doctoral thesis, Politehnica University of
Bucharest, 2001.
[11] A. Munteanu, E. Helerea, St. Szabo,
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