1
Thermal model of a power transformer Samuel Marguet, Joseph El Hayek, Member, IEEE, and Philippe De Werra
Abstract—In order to predict the hot spots in a transformer winding, a thermal model of the latter is presented. Equations system is implemented in Matlab environment. The temperatures obtained from this model are compared to those measured by optical fiber wound within a real winding. A first approach based on a simplified model gives acceptable results. Several ways to improve this model are presented including investigations on fluid and heat transfer physical phenomena. Index Terms— Fluid mechanics, modeling, temperatures, heat transfer, power transformer. I.
Ac
NOMENCLATURE
Conductor cross section
cp
Specific heat capacity
I
Electrical current Characteristic length Mass flow rate Pressure Heat flux
L
m P q Q
( c −o )
Heat transfer rate generated by the conductor
Q ( o −o )
Heat transfer rate absorbed by the oil
R Tcond
Electrical resistance Conductor temperature
Toil α φ η v ρ ξ ∆T
Oil temperature Heat transfer coefficient Shape coefficient Dynamic viscosity Fluid velocity Density Loss coefficient Fluid temperature difference
This work was supported by the Centre of Excellence ‘Energies’ of the University of Applied Sciences of Western Switzerland under Project n° HESSO-.02-00 Energies. S. Marguet is with the Departement of Electrical Engineering, University of Applied Sciences of Western Switzerland, 1705 Fribourg, Switzerland (email :
[email protected] J. El Hayek is with the Department of Electrical Engineering, University of Applied Sciences of Western Switzerland, 1705 Fribourg, Switzerland (email:
[email protected]). Ph. De Werra is with the Departement of Mechanical Engineering, University of Applied Sciences of Western Switzerland, 1705 Fribourg, Switzerland (e-mail :
[email protected])
II. INTRODUCTION
P
OWER transformers are essential elements in the electrical power transmission and distribution chain. A bad operation of one of them could lead to important technical and financial damages. The liberalization of the energy market pushes the utilities to reduce maintenance costs and increase material life cycle. Taking into account the price and the relatively long life of the transformers, on line monitoring systems are developed and used more and more all over the world. Temperature is a key parameter that influences the ageing of the insulation material, and thus the life cycle of the transformer. The winding’s temperature can be measured accurately by the means of optical fibers, this solution is however cost and labour consuming. That’s why many attempts have been made for several years to model the thermal behavior of the transformer, allowing so a good, and low cost, prediction of the temperature profile. The more this model can be used for any transformer type, the more the investment is cost effective. The aim of the present work is to create a Matlab based computer program, which allows to predict temperature values at different spots of an oil cooled power transformer winding (example in Fig. 1.). The simulation model should take into account the type of oil used, the transformer dimensions, the windings arrangement as well as the different mechanical elements allowing to direct the oil into the winding. The program results will be compared to winding temperature measured on a transformer built by ABB Sécheron Ltd (Geneva, Switzerland). Cooler
Return pipe
Disc
Oil guiding ring Fig. 1. Example of a disc winding with its cooling circuit
2
III. TYPES OF TRANSFORMER WINDINGS Classical oil cooled power transformers have two possible layouts depending on the winding type used (Fig. 2). These are the disc or the layer winding transformers. Sometimes one transformer can have a combination of both winding types. The program should deal with this type difference. Physical laws are the same, but equations used are different according to the winding layout (disc or layer).
The principal physical phenomena under study are of two categories. In the first one, we treat the phenomena using the laws of the fluid mechanics; here one can calculate the fluid velocities in the different transformer ducts. The second category encompasses the phenomena dealing with the heat transfer. Dividing them so, helps us to put the system into equations easily. V. DETERMINATION OF THE OIL VELOCITIES
Fig. 2. Disc winding (left) and Layer winding (right)
The different oil velocities depend on the cooling ducts dimensions, the arrangement of the oil guiding rings within the winding, and the number of discs (or layers in the layer winding type). The calculation of the oil velocities is based on a variable module having oil guiding rings as spatial boundaries. The model described in Fig. 4 represents oil ducts surrounding two discs and comprised between two guiding rings. A transformer winding is divided in several such modules that are serial connected. The oil input and output allow linking the modules constituting the winding to each other. Equations are written on this basis for each module. They take into account the interdependence of the oil velocities in the different ducts. V9
IV. THE MODEL From the cooling point of view, a transformer presents a complex structure. A simplification is therefore helpful to better understand the physical phenomena inside, but the simplified model has to bring out the elements involved in the calculation of the transformer winding’s temperatures. As a first application, we considered a disc winding transformer. Fig. 3 shows the shape of a winding part between two barriers. Winding conductors produce calorific energy that is collected by the moving fluid. This structure allows studying the fluid behavior and the heat transfer in an easy manner.
V8
5
6
3
V7
V6 V5
3 V4
4
2 1
V3 2
V2
V1 1 Fig. 4. Model of oil velocities in a winding part.
Three laws allow putting into equations the interdependence of the fluid velocities: the continuity law, the laws of Darcy and Weisbach. A. Continuity Law The continuity law expresses the balance of the flow rates at each fluid separation point.
Fig. 3. Disc winding cooling principle. Case of four discs consisting of six conductors each.
B. Pressure Losses The law of Darcy takes into account the friction of the fluid against the ducts surface, while the Weisbach law deals with the turbulences created either through direction changes of the fluid or by crossing between two fluids coming from different ducts. These losses are expressed in (1).
3
∆P ( v ) =
ρ Réf 2 v η ⋅φ ⋅ L ⋅v +ξ ⋅ 2 2 Dh 2
VI. HEAT TRANSFER CALCULATION MODEL (1)
C. Flows in Piping Systems In order to solve the module equations, the piping systems theory has been used [1]. It can be compared to the electrical circuit theory. The following conditions have to be verified: 1. 2. 3.
The algebraic sum of the pressure losses is zero. Continuity is verified at each node: input flow rates are equal to output flow rates. For each duct, the pressure losses verify the laws of Darcy and Weisbach.
Every conductor can represent a different heating source because of the eddy current distribution [3]. The heat transfer from the winding to the coolant occurs in two stages. At first, heat is transferred by conduction from the heating surface to the insulation between conductor and oil; the temperature of the oil particles touching the insulation will therefore increase. Then, through their movement created thermically (natural convection) or by a general fluid displacement (forced convection), they will be mixed to other particles situated in colder regions and transmit so their thermal energy. This can be illustrated in Fig. 5 in case of one disc having six turns.
V3
D. Meshes Law The application of this law on the example of Fig. 4 gives:
ΔPV4 +ΔPV7 +ΔPV8 -ΔPV6 -ΔPV3 -ΔPV2 = 0
(2)
ΔPV4 +ΔPV5 -ΔPV3 -ΔPV2 = 0
(3)
ΔPV7 +ΔPV8 -ΔPV6 -ΔPV5 = 0
(4)
In general, setting ‘j’ as the disc number, this law becomes for the internal meshes (i.e. 2 and 3 in Fig. 4):
ΔPV3j+1 +ΔPV3j+2 -ΔPV3j -ΔPV3j-1 = 0
(5)
and for the external meshes (i.e. mesh 1in Fig. 4), with ‘jmax’ corresponding to the module’s total number of discs: j max
∑ ΔP j =1
V3j+1
-ΔPV3j -ΔPV2 +ΔPV3jmax+2 = 0
(6)
T4;P4
T3;P3 ∆T3,1 ∆T3,2 ∆T3,3 ∆T3,4 ∆T3,5 ∆T3,6 α3,1 α3,2 α3,3 α3,4 α3,5 α3,6 ∆T2,1 V2
α2,1
1
2
3
4
5
6
α4,1
∆T4,1 V4
α1,1 α1,2 α1,3 α1,4 α1,5 α1,6 ∆T1,1 ∆T1.2 ∆T1,3 ∆T1,4 ∆T1,5 ∆T1,6 T1;P1
V1
T2;P2
Fig. 5. Thermal model of a winding’s disc composed of six turns.
A. Heat Transfer Equations The steady state of the system is expressed through two equations [2]. The fluid temperature increase is deduced from (13).
Q o −o = c p ⋅ ρ ⋅ Ac ⋅ v ⋅ ∆T = c p ⋅ m ⋅ ∆T = q ⋅ Ac (13)
E. Nodes Law For the example of Fig. 4, the nodes law gives:
The transmitted energy per second from the conductor to the fluid is expressed in (14)
4+ m 2= m input m
(7)
7+ m 5= m 4 m 5+ m 3= m 6 m 8+ m 6= m output m
(8) (9) (10)
In general, for the nodes on the output side of the disc ‘j’, one obtains: 3j-2 = m 3j+1 + m 3j-1 m (11) and for the input side
3j + m 3j+2 = m 3 j+3 m
Q c −o = α ⋅ Ac ⋅ (Tcond − Thuile )
According to Fig. 6, the energetic balance between two points of the fluid is:
Q ( o−o )T1 = Q ( o−o )T0 + Q c −o
(15)
This gives (16) after development.
c p1 ⋅ ρ1 ⋅ Ac ⋅ v ⋅ T1 = c p0 ⋅ ρ 0 ⋅ Ac ⋅ v ⋅ T0 + (12)
(14)
+ α 0 ⋅ Ac ⋅ (Tcond − Toil )
(16)
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B. Equations of the conductors temperatures For a conductor not situated at a disc end, the energetic balance between the turn losses and the fluid in both surrounding ducts is:
Qc-o 1
0
T0
T1
Qo-o
Fig. 6. Thermal energy increase from point 1 to point 2.
The thermal model of a duct situated between two discs with six turns is represented in Fig. 7. The thermal balance in the whole duct is obtained in applying (16) for each of the twelve turns surrounding it.
2,1
2,2
α5,1
α5,2
T3,1
T3 P3
2,3
2,4 α5,4
α5,3
T3,2
2,5
T3,3
2,6 α5,6
α5.5
T3,4
T3,5
T4 P4
V5T5, α3,2
α3,1 1,1
1,2
α3,5
α3,4
α3,3 1,3
1,4
1,5
α3,6
+T T R j ,i ⋅ I 2j ,i = α 4 j −3,i ⋅ Ac 0 ⋅ Tcond j ,i − 2 j −1,i −1 2 j −1,i + 2 (20) +T T + α 4 j −5,i ⋅ Ac 0 ⋅ Tcond j ,i − 2 j +1,i −1 2 j +1,i 2 VII. RESULTS AND ANALYSIS At first, a Matlab algorithm has been used, which is based on an iterative method. We have considered a winding with forty-six discs. Three oil guiding rings are situated at discs number twelve, twenty-three and thirty five. In order to test the model described above in a simple case, we supposed that every disc has only one turn. The different parameters are shown in Fig. 9. Table I shows the comparison between maximum temperatures calculated and measured.
1,6
Fig. 7. Example where the thermal model has two discs. The fluid is heated by twelve turns.
The balance for any pair of turns belonging to different discs and situated in front of each other (Fig. 7) is given respectively in (17) and (18).
Q ( o − o )T3 + Q C 2,1− o + Q C1,1− o = Q ( o − o )T3,1
∆ T 3i+1 v 3i+1
T 2i-3 p 2i-3
c p3,1 ⋅ ρ3,1 ⋅ Ac ⋅ v ⋅ T3,1 = c p3 ⋅ ρ3 ⋅ Ac ⋅ v ⋅ T3 +
( ⋅ A ⋅ (T
) )
α cond2,1
c
cond2,1
− Toil
cp
2 j −1 , i + 1
2 j −1 , i
⋅ ρ 2 j −1,i ⋅ Ac ⋅ v 3 j −1 ⋅ T2 j −1,i +
T2 j −1,i +1 + T2 j −1,i + 2
D 3i+2 A c2
T cond A c0
∆ T 3i-2 v 3i-2
D 3i
D 3i-1
∆ T 3i-1 v 3i-1
∆ T 3i v 3i T 2i p 2i
∆ T 3i-3 v 3i-3
T 2i-2 p 2i-2
Fig. 9. Simplified model with one turn per disc.
⋅ ρ 2 j −1,i +1 ⋅ Ac ⋅ v 3 j − 1 ⋅ T2 j −1,i +1 =
α 4 j − 5 ,i +1 ⋅ Ac ⋅ Tcond j −1,i +1 −
D 3i+1
T 2i+2 p 2i+2
(18)
Generally, in case of two discs, where ‘j’ (≥2) is the disc number and ‘i’ the turn number, (18) is transformed in (19).
cp
A c0 A c1
T 2i-1 p 2i-1
(17)
α cond1,1 ⋅ Ac ⋅ Tcond1,1 − Toil +
∆ T 3i+2 v 3i+2
T 2i+1 p 2i+1
(19)
T2 j −1,i +1 + T2 j −1,i α 4 j − 3,i ⋅ Ac ⋅ Tcond j ,i +1 − 2 For the vertical ducts, other corresponding equations are used.
TABLE I COMPARISON BETWEEN CALCULATION AND MEASUREMENTS.
Load Case
Optical fiber measurement
Calculation without pressure losses
Calculation with pressure losses
Nominal
97.5°
69°
98°
50% Overload
104.2°
86.8°
114.6°
The complete program results are given in form of temperature evolution versus the number of the turn. Fig. 10 represents four different curves. Starting from the top left, they show respectively the conductor temperature, the fluid temperature in the vertical right duct, the one of the horizontal duct, and finally in the vertical left duct.
5
• •
Concerning the running time, another method dealing with matrix manipulations will replace the actual iterative one. Implementation of the eddy current effect in the various conductor turns.
V9
T5,2 V8
T5,1
T5 P5
α7,3
α7,2
α7,1
T5,5
T5,3
T5,6
T6 P6
α7,4
α7,5
α7,6
2,4
2,5
2,6
α8,1
α6.1 2,1
V7
2,2
α5,1
A. Results Analysis The simplified approach demonstrates that the assumptions and formulae used, give acceptable results in term of temperature figures. The examination of the curves points out temperature steps. These correspond to the fluid deviation through the oil guiding rings. Actually, the oil arriving from point B in Fig. 11 has been heated through the precedent disc conductors. In addition, the guiding rings store thermal energy that is distributed further to the fluid; this phenomenon has not been taken into account in the actual model.
C
B A
V4
α5,2
α5,4
α5.5
T3,5
T3,3
V6
α5,6
T3,6
T4 P4
V5 α3,1
α3,2
α3,3
1,1
1,2
1,3
α3,4
α3,5
α3,6
α2,1
α1,2
α1.1 T1 P1
α5,3
T3,2
T3,1
T3 P3
Fig. 10. Temperature curves in the transformer winding and oil ducts.
2,3
T1,1
α1,3
T1,2 V2
1,4 α1,4
T1,3
1,5
1,6
α1,5
α1,6
T1,5
T1,6
α4,1
V3
T2 P2
Fig. 12. Complete model, with two discs and six turns each.
VIII. CONCLUSION This paper presented a calculation philosophy to determine the thermal behavior of a power transformer. A first Matlab program has been written, where some assumptions have been made to verify the temperature calculation method. Comparison with measurements showed the validity of the approach. In the next program version many improvements listed and discussed in this paper, will be implemented. The simulation time will also be shortened through an appropriate modification of the programming method.
Fig. 11. Detail explaining the temperature step.
IX. REFERENCES B. Model Improvement As mentioned before, there are several improvements to be done in the next version of this program: • The variation of certain parameters in function of the temperature; this concerns the conductor resistance, the dynamic viscosity and the heat transfer coefficient. • The pressure losses due to turbulences, according to Weisbach’s law, will also be taken into account. • The heat flux across the oil guiding ring will be added to the existing material features. • Obviously, the real number of turns per disc will be introduced, which is already possible in the actual program (Fig. 12).
[1]
J.-P Corbat & Ph. De Werra, "Mécanique des fluides," Ecole d’Ingénieurs et d’architectes de Fribourg, Fribourg, Switzerland, September 1998. [2] Frank P. Incorpera & David P. DeWitt, "Fundamentals of Heat and Mass Transfer,"’ ISBN 0-471-38650-2, 5th edition, 2002, pp. 52-326. [3] J. El Hayek, "Parameterized Eddy Current Model of a Power Transformer," presented at the 7th Int. Conf. on Modeling and Simulation of Electric Machines, Converters and Systems, Montreal, Canada, 2002. [4] M. Caprotti et D. Ansermot, "Modèle thermique d’un transformateur refroidi à l’huile," rapport de projet de diplôme, Ecole d’ingénieurs et d’architectes de Fribourg, Fribourg, Switzerland, Dec. 1999. [5] J. Declercq & W. Van Der Veken, "Accurate Hot Spot Modeling in A Power Transformer Leading to Improved Design and Performance," Pauwels Trafo Belgium, Technology Development, 1999 [6] R. M. Del Vecchio & P. Feghali, "Thermal Model of a Disk Coil with Directed Oil Flow," North American Transformer, Milpitas USA, 1999.
6 [7] [8]
H. Schad, "Fiberoptical Distributed Temperature Sensing of Temperature Profiles of a Transformer Model Coil," presented at the int. SPIE-conf. Fiber Optic and Laser Sensors VI, Boston, September 1988. P. Boss, J.F. Ravot, P. Mathis & H. Brändle, "Mesure du profil de température dans les transformateurs à l’aide d’un capteur distribué à fibre optique," Cigré Symposium, Berlin, 1993.
Joseph El Hayek (M’1997) was born in Lebanon, on November 11, 1963. He graduated in Electrical Engineering from the Swiss Federal Institute of Technology, Lausanne (EPFL), Switzerland. After two years as assistant in the Electromechanics and Electrical Machines Laboratory of the EPFL, he worked for ABB Sécheron Ltd (Geneva) in the field of Power and Traction Transformers where he led the Engineering and tests department, and later the worldwide ABB Center of Excellence for Traction Transformers until end 1996. Since 1997, he is professor of Electrical Machines at the University of Applied Sciences of Western Switzerland (HES-SO) in Fribourg, Switzerland. He is technical committee’s member of the center of Excellence ‘Energies’ of the HES-SO. His employment experience includes also the ABB Transportation Systems (today Bombardier Transportation) in Zürich. His special field of interest is the multi-windings transformers design and modeling.
Philippe De Werra was born in 1942, he received his diploma in Mechanical Engineering from the Federal Institute of Technology, Zurich (Switzerland) in 1966. After a short period at the Institute of Control and Steam Installations of the same School, he worked in the department of steam and nuclear installations at Sulzer in Winterthur, sector thermal calculations and control of high pressure boiler and special heater, then, from 1975, in the department of development of the Swiss Federal Armament Works, sector high velocity solicitations and testing of metals. Since 1988, he teaches fluid mechanics and elementary material knowledge at the University of Applied Sciences in Fribourg (Switzerland).
Samuel Marguet was born in Fribourg, on November 14, 1976. In 2000, he obtained a BSc degree in Mechanical Engineering from the University of Applied Sciences of Western Switzerland, Fribourg, Switzerland. He worked as project engineer in the department of chemical Engineering in 2001, and is working in the department of Electrical Engineering of the same university since November 2001. His special field of interest is the mechanics of fluids and heat and mass transfer.
