IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 1, JANUARY 2005 197 Dynamic Thermal Modelling of Power Transformers Dejan Susa, Matti Lehtonen, and Hasse Nordman Abstract—The aim of this paper is to introduce hot-spot and top-oil temperature thermal models for more accurate temperature calculations during transient states based on data received in a normal heat run test (i.e., the top oil in the tank of the transformer and the average winding-to-average oil gradient). Oil viscosity changes and loss variation with temperature are taken into account. The new thermal models will be validated using experimental (fiber-optic test) results obtained at varying load current on a 250-MVA-ONAF-cooled unit, a 400-MVA-ONAF-cooled unit and a 605-MVA-OFAF-cooled unit. The results are also compared with the IEEE—Loading guide (1995) Annex G method. R Index Terms—Hot-spot temperature, power transformer, top-oil temperature. NOMENCLATURE A C g h H I i K k L n Area. Overshoot factor (maximum of the function ). Electrical capacitance. A constant. A constant. Specific heat of fluid. Thermal capacitance. Oil thermal capacitance. Winding thermal capacitance. Grashof number. Gravitational constant. Rated average winding-to-average oil temperature gradient. Heat transfer coefficient. Hot-spot factor. Load current. Subscript indicates initial. Load factor. Oil thermal conductivity. Characteristic dimension length, width, or diameter. A constant. Nusselt number. Prandtle number. DC losses per unit value. Eddy losses per-unit value. DC losses (in watts). Manuscript received June 19, 2003; revised August 21, 2003. Paper no. TPWRD-00301-2003. D. Susa and M. Lehtonen are with the Power Systems Laboratory, Helsinki University of Technology, Espoo FIN-02015 HUT, Finland (e-mail: Dejan.Susa@hut.fi; Matti.Lehtonen@hut.fi). H. Nordman is with ABB Oy, Vaasa 65101, Finland (e-mail: hasse. firstname.lastname@example.org). Digital Object Identifier 10.1109/TPWRD.2004.835255 rated p.u. Eddy losses (in watts). Stray losses (in watts). Heat generated by total losses. Heat generated by load losses. Heat generated by no-load losses. Ratio load losses at rated current to no-load losses. Nonlinear thermal resistance of the oil. Nonlinear winding-to-oil thermal resistance. Winding thermal resistance. Insulation thermal resistance. Oil density. Coefficient of thermal cubic expansion of the oil. Cinematic viscosity of the oil. Oil viscosity. Ambient temperature. Top-oil temperature. Hot-spot temperature. Rated top-oil temperature rise over ambient temperature. Rated hot-spot temperature rise over top-oil temperature. Oil time constant. Winding time constant. Subscript indicates rated value. Subscript indicates per-unit value. I. INTRODUCTION A CCORDING to the IEC 354 loading guide for oil-immersed power transformers , the hot-spot temperature in a transformer winding consists of three components: the ambient temperature rise, the top-oil temperature rise, and the hotspot temperature rise over the top-oil temperature. It is assumed that during a transient period, the hot-spot temperature rise over the top-oil temperature varies instantaneously with transformer loading, independently of time. The variation of the top-oil temperature is described by an exponential equation based on a time constant (oil time constant). The top-oil time constant suggested by the IEC loading guide is 150 min for the oil natural-air natural cooling mode (ONAN) and for the oil natural-air forced cooling mode (ONAF). In the case of the oil forced-air forced cooling mode (OFAF), it is suggested to use the oil temperature of the oil leaving the winding. This has not been practiced in the industry due to difficulties recording the temperature of the oil leaving the winding. The suggested oil-time constant for OFAF-cooled units is 90 min according to IEC, whereas the IEEE Loading Guide  suggests a design-specific calculation method for the top-oil time constant. In this case, the time constant is calculated for each transformer unit separately depending on both the cooling mode (ONAN, ONAF, OFAF) and the masses of the 0885-8977/$20.00 © 2005 IEEE 198 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 1, JANUARY 2005 various components comprising a transformer (such as the oil, tank, radiators, insulation, clamps, and unenergized windings). During the last ten years, fiber-optic probes have been used by many authors – in order to obtain as accurate values for transformer temperatures as possible. It has been recorded that there is a time delay between the top-oil temperature rise in the tank and the hot-spot temperature rise. The hot-spot temperature increases rapidly at a time constant equal to the time constant of the windings. During transient states, this results in winding hottest spot temperatures higher than those predicted by the present IEC loading guide for oil-immersed power transformers . By analyzing the measured results of tested power transformers, it was noticed that the hot-spot temperature rise over top-oil temperature at load changes is a function depending on time as well as the transformer loading (overshoot time-dependent function) (Fig. 1) , . The maximum values and ) for different shapes of this function (=overshoot factor transformers, different loadings, different oil circulation modes in the windings (zig-zag and axial), and different cooling modes are given in . It has also been noticed that the top-oil temperature time constant is shorter than the time constant suggested by the present loading guide, especially for the ONAN and ONAF cooling modes . Based on those results, a new exponential calculation method was also proposed for the hot-spot temperature variation at the varying load . The thermal overshoot phenomenon (Fig. 1) has also been strongly investigated by Linden Pierce. The obtained results are published in scientific papers ,  and as a parallel dynamic hot-spot temperature calculation procedure in the IEEE Loading Guide (Annex G in ). This paper is an attempt to develop a generalized model, which is able to explain the principal background of the previous findings. The paper presents a new temperature calculation method based on heat transfer theory, application of the lumped capacitance method, the thermal-electrical analogy, and a definition of nonlinear thermal resistance. The basic approach has already been proposed by Swift . As a new idea, the present work focuses specifically on the nonlinear thermal resistance of the transformer oil. The method presented in this paper takes into account oil viscosity changes and loss variation with temperature; it is validated using experimental results, and it is compared with the IEEE-Annex G . II. TRANSFORMER THERMAL MODELS In trying to analyze the transformer temperature problem, the basic analogy between thermal and electrical processes is given below (Table I) , . At this point, it may be useful to define thermal resistance and thermal capacitance as the material’s ability to resist heat flow and to store heat, respectively. The foregoing analogy assumes that the thermal characteristics of a material are constant, that is, they are not changing with temperature. In order to use this electrical–thermal analogy for the transformer temperature calculation, it will be necessary to modify further the lumped capacitance method by introducing a Fig. 1. Normalized time variation of hot-spot temperature rise above top-oil temperature f (t) (in tank) for a step increase in load current . TABLE I THERMAL–ELECTRICAL ANALOGY nonlinear thermal resistance, which takes into account changes in the transformer oil thermal characteristics with temperature. The validity of the lumped capacitance method with respect to power transformers is given in ,  and the general validity of the method shown in . The transformer oil has thermal characteristics strongly dependent on temperature as presented in Table II, , where oil viscosity dependency on temperature is most pronounced. The nonlinear thermal resistance will be clearly defined in following sections for both the top-oil temperature model and the hot-spot temperature model. A. Top-Oil Temperature Model The top-oil temperature model is given as a thermal circuit (Fig. 2), based on the thermal–electrical analogy and heat transfer theory , . where total losses; heat generated by no-load losses; heat generated by load losses; thermal capacitance of the oil; top-oil temperature; nonlinear thermal resistance; ambient temperature. The heat generated by both no-load and load transformer losses is represented by two ideal heat sources . The ambient temperature is represented as an ideal temperature source . according to heat The nonlinear oil thermal resistance transfer theory  is given by the following equation: (1) where is the heat transfer coefficient, and is the area SUSA et al.: DYNAMIC THERMAL MODELLING OF POWER TRANSFORMERS 199 TABLE II THERMAL CHARACTERISTICS OF TRANSFORMER OIL TABLE III EMPIRICAL VALUES FOR CONSTANTS C AND n top-oil to ambient temperature gradient . By substituting (3), (4), and (5) in (2), the following expression is obtained: (6) The variation of viscosity with temperature is much higher than the variation of other oil physical parameters, Table II, , . Therefore, all of the oil physical parameters except the viscosity in (6) will be replaced by a constant and (6) will be solved for the heat transfer coefficient as follows: (7) where is assumed to be a constant, expressed as (8) Fig. 2. Top-oil temperature model. Hence, the nonlinear thermal resistance is inversely proportional to the heat transfer coefficient, whose dependence on temperature is explained in the text to follow. Based on heat transfer theory, the natural convection oil flow around vertical, inclined, and horizontal plates and cylinders can be described by the following empirical correlation : (2) where and are constants dependent on whether the oil circulation is laminar or turbulent (Table III) . , Prandtle number , and The Nusselt number are described by the following equations, Grashof number respectively : (3) (4) and is the viscosity variation with temperature, kg/(ms), given by the following equation : (9) where the viscosity is evaluated at the top-oil temperature. The and its parameters are depicted as a function of constant temperature in Fig. 3. It is seen that at the normal operation temperatures of power transformers, especially over 40 C, the factor is practically constant. An example of oil viscosity variation with temperature (compared with other physical properties of transformer oil from Fig. 3) is shown in Fig. 4. The differential equation for the thermal circuit shown in Fig. 2 is (10) If we substitute the equation for nonlinear thermal resistance (1), into (10), the following equation is obtained: (5) (11) where characteristic dimension, length, width, or diameter; gravitational constant; oil thermal conductivity (Table II); oil density (Table II); oil thermal expansion coefficient (Table II); specific heat of oil (Table II); oil viscosity (Table II); Then, by substituting (7) for the heat transfer coefficient , the differential equation is changed to (12) 200 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 1, JANUARY 2005 Fig. 3. Physical properties of transformer oil. Next, if we define the variable Fig. 4. Oil viscosity variation with temperature. oil viscosity, as TABLE IV CONSTANT n FOR THE TOP-OIL THERMAL MODEL (13) and the following constants: the rated nonlinear thermal resistance (14) the rated top-oil temperature rise over ambient temperature (15) the rated top-oil time constant (Appendix A) (16) the ratio of load losses at rated current to no-load losses (17) and the load factor (18) is the rated current. where is the load current and Equation (12) is then reduced to its final form It is assumed that oil circulation inside the transformer tank is laminar and so the constant for that particular flow type is equal to 0.25 (Table III). For transformer cold start or when the oil velocity inside the transformer tank is equal to zero, the constant will take different values for different cooling modes (Table IV). The values for the constant are based on our own tests and on , , and . In order to calculate the top-oil temperature from the differential equation (19), a numerical analysis method such as Runge–Kutta should be used and the equation elements should be classified as follows: — — — — , constants: , , input variables: , , ; output variables: independent variable: . ; ; The application of the top-oil temperature model for the varying load calculation on three different power transformers will be given in Section III. B. Hot-Spot Temperature Model (19) which forms the basic model for top-oil temperature calculation. The importance of the oil viscosity temperature variation is that it affects both the oil thermal resistance and top-oil time constant. Similar to the theory given for the top-oil temperature model, the hot-spot temperature model is also represented as a thermal circuit (Fig. 5). where heat generated by load losses; winding thermal capacitance; SUSA et al.: DYNAMIC THERMAL MODELLING OF POWER TRANSFORMERS 201 If we substitute the equation for the nonlinear thermal resistance (23) into (25), the following equation is obtained: (26) Fig. 5. Then, by substituting the equation for the heat transfer coefficient (24) into (26), the differential equation is changed to Hot-spot temperature model. hot-spot temperature; nonlinear winding to oil thermal resistance; oil temperature. The heat generated by load losses is again represented as an ideal heat source and the oil temperature forms an ideal temperature source . The nonlinear thermal resistance is defined by the heat transfer theory, which has already been applied to the top-oil thermal model, as explained below. The nonlinear winding to oil thermal resistance is given by the following equation: If we again define the oil viscosity as a variable that (27) (Fig. 4) such (28) and the following constants: the rated nonlinear hot-spot to top-oil thermal resistance (29) (20) is the winding thermal resistance, is where is is oil thermal the insulation thermal resistance, and resistance. By comparing the resistances given in (20), the following thermal correlations are obtained (21) (22) the rated hot-spot temperature rise over top-oil temperature as (30) where is the hot-spot factor and is the rated average winding to average oil temperature gradient.  is The rated winding time constant (31) for hot-spot temperatures measured on the surface of the insulated conductors , . Thus, the final equation for the nonlinear winding to oil thermal resistance is (23) Equation (23) is similar to (1) for the top-oil temperature model; therefore, the equation for the heat transfer coefficient is completely analogous to the heat transfer coefficient in (7) and the load loss’s dependence on temperature is (32) and describe the dc and where eddy losses variation with temperature, respectively. The dc losses vary directly with temperature, whereas the eddy losses vary inversely with temperature. It follows that the final equation is (24) where the viscosity is again evaluated at the top-oil temperature and is now the hot-spot to top-oil temperature gradient. The differential equation for the thermal circuit shown in Fig. 5 is (25) (33) which is the basic model for the hot-spot temperature. In analogy to (19), the equation takes into account the change of thermal resistance and winding time constant due to the oil viscosity temperature variation. The loss variation with temperature is also included in (33). 202 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 1, JANUARY 2005 Similar to the same assumption made in the top-oil thermal model that oil flow inside the transformer tank is laminar, the value for constant , where oil circulation is already formed, will be equal to 0.25. The constant will have different values in the case of a transformer cold start, that is, when the oil velocity is equal to zero. Values for constant for different cooling modes and different oil circulation conditions are given in Table V. By using the same calculation method as used in the top-oil temperature model (the Runge–Kutta method), the hot-spot temperature can be calculated from the differential equation (33), where the equation elements are classified in the following ways. , , — Constants , , . , (note that —represents — Input variables , the “output variable” in the top-oil differential equation, so that (19) and (33) form a cascaded connection). . — Output variables — Independent variable . The hot-spot temperature model will also be applied on three power transformer units and the results will be discussed in Section III. III. COMPARISON WITH MEASURED RESULTS The measured temperature results, which are recorded for three different transformer units during different varying load tests–Table VI, Table VII, , are compared with results obtained by both the calculation method presented in this paper referred to as the thermal model in graphics below, and the IEEE-Annex G method  (Figs. 6–11). The results plotted by the proposed thermal model agree with measured values with good accuracy especially in the case of the top-oil temperature, at both load increase and load decrease. The results obtained by IEEE-Annex G method also follow generally the measured results for the hot-spot temperature well, whereas the top-oil temperature results show less accuracy. The input data for the IEEE-Annex G calculation method are given in  and Table VIII below. In order to make Table VIII reasonably short, the symbols used in Annex G of  are used directly without a verbal explanation (the readers are referred to Section G.6 of ). The input data for the suggested thermal model are given in Table IX. IV. CONCLUSION Both top-oil temperature and hot-spot temperature thermal models are proposed for power transformers. The models are based on heat transfer theory, application of the lumped capacitance method, the thermal-electrical analogy, and definition of nonlinear thermal resistance. The key factor is that models take account of variations in oil viscosity and winding resistance. As a reference temperature for the oil viscosity evaluation, the top-oil temperature in the tank is used for both models. The constant , which defines the shape of the thermal curve, is given for both different cooling modes and different oil flow condi- TABLE V CONSTANT n FOR THE HOT-SPOT THERMAL MODEL TABLE VI TESTED TRANSFORMER UNITS TABLE VII PERFORMED LOAD TESTS Fig. 6. Hot-spot temperature of the 118-kV voltage winding of the 250-MVA ONAF-cooled transformer. tions. The constant’s values are based on our own test and on , , and . The thermal models are applied on three transformer units at varying load and the results are compared to measured results, and to results obtained by the IEEE-Annex G method. It is shown that the thermal models yield results that compare very well with measured results, especially for the top-oil temperature. The results obtained by the IEEE-Annex G model are also very good for the hot-spot temperature calculation but less accurate for the top-oil temperature. SUSA et al.: DYNAMIC THERMAL MODELLING OF POWER TRANSFORMERS Fig. 7. Top-oil temperature of the 250-MVA ONAF-cooled transformer. 203 Fig. 11. Top-oil temperature of the 605-MVA OFAF-cooled transformer. TABLE VIII INPUT DATA FOR COMPUTER PROGRAM IN THE LOADING GUIDE (1995) Fig. 8. Hot-spot temperature of the 410-kV voltage winding of the 400-MVA ONAF-cooled transformer. Fig. 9. Top-oil temperature of the 400-MVA ONAF-cooled transformer. APPENDIX A. Top-Oil Time Constant The top-oil time constant calculation method is based on the calculation method suggested by the IEEE Guide for Loading Mineral-Oil-Immersed Transformers. The top-oil time constant at rated load is given by the following equation : Fig. 10. Hot-spot temperature of the high-voltage winding of the 605-MVA OFAF-cooled transformer. (A.1) 204 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 1, JANUARY 2005 TABLE IX INPUT DATA FOR THE SUGGESTED MODEL  L. W. Pierce, “An investigation of the thermal performance of an oil filled transformer winding,” IEEE Trans. Power Del., vol. 7, no. 3, pp. 1347–1358, Jul. 1992.  , “Predicting liquid filled transformer loading capability,” IEEE Trans. Ind. Applicat., vol. 30, no. 1, pp. 170–178, Jan./Feb. 1994.  G. Swift, T. S. Molinski, and W. Lehn, “A fundamental approach to transformer thermal modeling, part I—theory and equivalent circuit,” IEEE Trans. Power Del., vol. 16, no. 2, pp. 171–175, Apr. 2001.  G. Swift, T. S. Molinski, and R. Bray, “A fundamental approach to transformer thermal modeling, part II—field verification,” IEEE Trans. Power Delivery, vol. 16, no. 2, pp. 176–180, Apr. 2001.  W. Lampe, L. Pettersson, C. Ovren, and B. Wahlström, “Hot-spot measurements in power transformers,” in Proc. Cigre, Rep. 12-02, Int. Conf. Large High Voltage Electric Systems, Aug. 29–Sep. 6 1984. where rated top-oil time constant (in min); rated top-oil temperature rise over ambient temperature [in kelvins (K)]; supplied losses (total losses) [in watts ], at rated load; transformer thermal capacity ( C). The thermal capacity is given by the following equation for the ONAN, ONAF, and OFAF cooling modes: (A.2) is the weight of the oil in kilograms (kg). where In the equation for the thermal capacity, 86% of the specific heat of the oil was used . Equation (A.2) is an empirical formula based on the modeling performed in this paper. REFERENCES  Loading Guide for Oil-Immersed Power Transformers.  IEEE Guide for Loading Mineral-Oil-Immersed Transformers, IEEE Std. C57.91-1995.  F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed. New York: Wiley, 1996.  K. Karsai, D. Kerenyi, and L. Kiss, Large Power Transformers. New York: Elsevier, 1987.  H. Nordman and M. Lahtinen, “Thermal overload tests on a 400 MVA power transformer with a special 2.5 pu short time loading capability,” IEEE Trans. Power Del., vol. 18, no. 1, pp. 107–112, Jan. 2003.  H. Nordman, N. Räfsbäck, and D. Susa, “Temperature responses to step changes in the load current of power transformers,” IEEE Trans. Power Del., vol. 18, no. 4, pp. 1110–1117, Oct. 2003. Dejan Susa (S’05) was born in Split, Croatia, on May 22, 1972. He received his Diploma Engineer degree in electrical engineering from the University of Nis Electrical Engineering faculty, Yugoslavia, in 2000. He received the M.Sc. degree from the Helsinki University of Technology, Espoo, in 2002. He is currently pursuing the Ph.D. degree in the Department of Electrical Engineering, Helsinki University of Technology. Currently, he is doing research work (transformer short time overloading capability) at the Helsinki University of Technology, Power Systems Laboratory. He is a member of the Finnish National Committee in the IEC Power Transformer Technical Committee (TC 14), and a member of the Maintenance Team MT1: Revision of IEC 354: Loading guide for oil-immersed power transformers. Matti Lehtonen was born in 1959. He received the Master’s and Licentiate degrees in electrical engineering from Helsinki University of Technology, Espoo, Finland, in 1984 and 1989, respectively, and the Ph.D. degree in technology from Tampere University of Technology, Tampere, Finland, in 1992. Currently, he is a Professor of IT applications in power systems at Helsinki University of Technology. He was with VTT Energy, Espoo, Finland. His research interests include earth fault problems, harmonic-related issues, and applications of information technology in distribution automation and distribution energy management. Hasse Nordman was born in Overmark, Finland, in 1945. He received the Ph.D. degree in Mathematics from the Abo Akademi University, Turku, Finland, in 1977. Since 1994, he is the leader of the global ABB R&D activity “Load Losses and Thermal Performance” at Business Area Power Transformers. From 1970 to 1982, he was with ABB Corporate Research, Vaasa, Finland (formerly Stromberg Research Centre), working on current related phenomena (losses, temperatures, short-circuit forces) in electric power equipment. From 1982 to 1994, he was with the Development Engineering Department in the Power Transformer Division of ABB, Vaasa. Dr. Nordman is a member of CIGRE. He is the chairman of the Finnish National Committee in the IEC Power Transformer Technical Committee (TC 14), and the convenor of the Maintenance Team MT1: Revision of IEC 354: Loading guide for oil-immersed power transformers.