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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 email@example.com 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 . 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:  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  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  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  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  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 ββ 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 βT 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 gT 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  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  we consider the integral To establish the coefficients a and b of module. equation (39) was used . 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 ab 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. 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