IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009 1257 A Simple Model for Calculating Transformer Hot-Spot Temperature Dejan Susa, Member, IEEE, and Hasse Nordman Abstract—A simple model for calculating the hot-spot temperature is introduced. The model is based on the hot-spot to ambient gradient. The model considers the changes of the oil viscosity and winding losses with temperature. The results are compared with temperatures calculated by IEEE Annex G method and measured results at varying load for the following transformer units: 250-MVA ONAF, 400-MVA ONAF, and 605-MVA OFAF. Index Terms—Hot-spot temperature, oil viscosity, top-oil temperature, winding losses. Weight of core and coil assembly (in kilograms). Weight of the oil (in kilograms). Weight of the tank and fittings (in kilograms). Correction factor of oil. No-load losses. DC losses per unit value. Relative winding eddy losses, per unit of dc loss. NOMENCLATURE Specific heat capacity of winding material. Eddy losses (per unit value). Stray losses (in watts). Specific heat capacity of core. Specific heat capacity of the tank and fittings. DC losses (in watts). Heat generation. Specific heat capacity of oil. Heat generated by total losses. A constant. Heat generated by winding losses. Oil thermal capacitance. Nonlinear winding to ambient thermal resistance. Winding thermal capacitance. Thermal capacitance of the core. Current density at rated load. Thermal capacitance of the tank and other metal parts. Subscript indicates the ultimate value. Constant. Load current. Portion of the core losses in the total transformer losses. Subscript indicates initial. Load factor. Portion of the stray losses in the total transformer losses. Weight of core (in kilograms). Minute. Weight of the tank and fittings (in kilograms). Portion of the winding losses in the total transformer losses. Weight of oil (in kilograms). Oil viscosity. Weight of winding material (in kilograms). Ambient temperature. Top-oil temperature. Top-oil temperature rise over ambient. Manuscript received January 25, 2008; revised January 16, 2009. Current version published June 24, 2009. This work was supported in part by the SINTEF Energy Research Department, Trondheim, Norway. Paper no. TPWRD-000362008. D. Susa is with the SINTEF Energy Research Department, Trondheim NO-7465 , Norway (e-mail: [email protected] sintef.no). H. Nordman is with the ABB, Power Transformers, Vaasa 65101, Finland (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRD.2009.2022670 0885-8977/$25.00 © 2009 IEEE Hot-spot temperature. Rated hot-spot to ambient temperature gradient.W Winding time constant. Winding time constant. Oil time constant. 1258 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009 Winding time constant. R Subscript indicates rated value. pu Subscript indicates per unit value. I. INTRODUCTION P OWER transformers represent the largest portion of capital investment in transmission and distribution substations. In addition, power transformer outages have a considerable economic impact on the operation of an electrical network. One of the most important parameters governing a transformer’s life expectancy is the hot-spot temperature value. The classical approach has been to consider the hot-spot temperature as the sum of the ambient temperature, the top-oil temperature rise in tank, and the hot-spot-to-top-oil (in tank) gradient , . During the last 20 years, fiber-optic probes have been used by many authors – in order to obtain as accurate values for transformer temperatures as possible. When the load is increased, it takes some time before the corresponding oil circulation adapts its speed  due to lower temperature and higher oil viscosity at the preceding loads. Consequently, the hot-spot temperature rises rapidly during the first 10–20 min with a time constant that is equivalent approximately to the winding time constant  (Fig. 1). Nevertheless, this time period is different for each transformer and it is very dependent on the transformer design. Conveniently, it has been further observed that 50% of the temperature change occurs during the rapid rise. When the temperature threshold level is reached, the oil circulation is established at a rate, which is defined as critical, preventing further rapid temperature rise. Therefore, the hot-spot temperature will continue to rise slower with a time constant that is equivalent to the top-oil time constant (Fig. 1). On the other hand, at the transformer cold start, 75% of the temperature change occurs during the rapid rise period due to more harsh initial oil conditions (i.e., initial oil speed is zero). In contrast, the initial oil circulation prior to the load decrease is faster than it will be under the load considered. Therefore, the temperature will now decrease rapidly with a time constant that is equivalent to approximately the winding time constant. Once 50% of the final temperature drop is reached, the oil velocity will be much lower than initially. As a result, the temperature starts decreasing slowly with a time constant equivalent to the top-oil time constant. This paper presents a simpler but still accurate temperature calculation method, taking into account the findings mentioned before. With alternate switching between two different time constants, (the short one being equal to the winding time constant and the long one being equal to the top-oil time constant), the model is based only on the hot-spot-to-ambient air gradient. The thermal model is based on heat-transfer theory –, numerous transformer thermal tests and reports – and –, application of the lumped capacitance method, the thermal-electrical analogy, and the concept of thermal resistance between winding insulation surface and ambient (i.e., air). Fig. 1. Hot-spot and top-oil temperatures of the 120-kV winding in the 400-MVA ONAF-cooled transformer at 1.29-p.u. constant load. The model presented in this paper takes oil viscosity changes and loss variation with temperature into account. The changes in transformer time constants due to changes in the oil viscosity are also accounted for. The model requires an iterative calculation procedure. The models are validated by using experimental results, which have been obtained from a series of thermal tests performed on three different power transformers: (250-MVA ONAF, 400-MVA ONAF, and a 605-MVA OFAF-cooled unit), , . The model is tied to measured parameters that are readily available (i.e., data obtained from a normal heat-run test performed by the transformer manufacturer at commissioning). II. THERMAL MODEL A. Thermal Circuit The thermal circuit for the hot-spot temperature rise over the ambient based on heat-transfer theory, –,  and thermal-electrical analogy – is given in Fig. 2 with the following elements: heat generated by winding losses; thermal capacitance of the winding; hot-spot temperature rise over ambient; winding to ambient nonlinear thermal resistance. The steady-state temperature rise equation for the natural convection and heat-exchange phenomena between the winding insulation surface, (where the sensors are located), and the ambient is given as follows: (1) SUSA AND NORDMAN: SIMPLE MODEL FOR CALCULATING TRANSFORMER HOT-SPOT TEMPERATURE 1259 TABLE I THRESHOLDS FOR ALL COOLING MODES Fig. 2. Hot-spot temperature rise thermal circuit. valid only for ONAN and ONAF cooling mode where is the hot-spot-to-ambient temperature gradient. is a function of the fluid properties and winding characteristic dimensions and is considered to be a constant . is constant that is partly based on experimental results obtained from thermal tests , . The sensors locations have been discussed in , , and . is the viscosity variation with temperature (in kilograms per millisecond), given by the following equation : (2) where the viscosity is evaluated at the value of where the load current is given as and the rated current is given as ; the rated hot-spot to ambient gradient (9) ; the heat generated by the winding losses (10) where is the loss per unit value given as given by (11) (3) is the hot-spot temperature and is the ambient temperature. The nonlinear thermal resistance between the winding insulation surface and the ambient of the transformer is characterized by (4), which is derived from (1) (4) The differential equation for the thermal circuit in Fig. 2 is given as follows: is the winding loss’s dependence on the hotwhere spot temperature (12) and describe the behavior where of the dc and eddy losses as a function of temperature . The dc losses vary directly with temperature, whereas the is the temeddy losses vary inversely with temperature. perature factor for the loss correction, equal to 225 for aluis the hot-spot temperaminium and 235 for copper. ture. Finally, (5) becomes (5) (13) Now, if the following parameters are defined: the thermal resistance as (6) stand for the rated and relative where subscripts and values, respectively, the winding time constant (7) stands for the rated winding time constant where given in Section II-C and it is assumed that , ; the load factor (8) The final solution of (13) for the load increase and decrease is given in the following section. The corresponding values for the temperature threshold level and oil viscosity exponent are given in Tables I and II. B. Complete Model 1) Load Increase: The hot-spot temperature increases to a level corresponding to a load factor of K (14) The initial hot-spot rise over ambient is . The hot-spot rise calculated for the end of previous load step is used as the is initial hot-spot rise for the next load step calculation. the ultimate hot-spot rise given by the following equation: (15) 1260 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009 is the time delay correction for the following condition: TABLE II LOAD STEPS FOR THE 250-MVA TRANSFORMER (25) Equation (21) is applicable here as well. Equations require an iterative calculation procedure. 3) Winding Exponent and Change Levels: The viscosity ex0.5). In ponent for all cooling modes is equal to 0.5 (i.e., addition, the temperature change thresholds are given in Table I. C. Time Constants 1) Winding Time Constant: The winding time constant  is as follows: The function describes the relative increase of the hot-spot temperature rise until the corresponding threshold is reached (16) for (26) for (27) where is the winding time constant prior to the change winding time constant in minutes at the rated load; (17) hot-spot to ambient temperature gradient at the rated load; is the rated winding time constant given in Section II-C. will be replaced by function once The function the threshold level for a given load is reached as suggested in Table I relative winding eddy losses, per unit of dc loss, corrected for the hot-spot temperature; current density in A/mm2 at the rated load. (18) is the winding time constant after the change 2) Top-Oil Time Constant: The top-oil time constant at the rated load is given as follows: (28) (19) is the rated oil time constant given in Section II-C. is the time delay correction for the following condition: where rated top-oil time constant (in minutes); (20) rated top-oil temperature rise over ambient temperature (in Kelvin (K)); (21) is the time when the function reaches the corresponding threshold level. Equations require an iterative calculation procedure. 2) Load Decrease: The hot-spot temperature decreases to a level corresponding to a load factor of K (22) The function describes the relative decrease of the hot-spot temperature rise until the corresponding threshold is reached (23) The function will be replaced by function once the change level for a given load is reached as suggested in Table I (24) total supplied losses (total losses) (in watts (W)) a the rated load; equivalent thermal capacitance of the transformer oil . The equivalent thermal capacitance of the transformer oil for transformers with external cooling and a zigzag oil flow through the windings is given by (29) where weight of the winding material (use only the excited parts) (in kilograms); weight of the core (in kilograms); weight of the tank and fittings (in kilograms); SUSA AND NORDMAN: SIMPLE MODEL FOR CALCULATING TRANSFORMER HOT-SPOT TEMPERATURE 1261 TABLE III MAXIMUM AND AVERAGE ERROR FOR THE 250-MVA TRANSFORMER SM: simple model; IEEE: IEEE-Annex G weight of the oil (in kilograms); specific heat capacity of the winding material and ) in , ; ( specific heat capacity of the core , ; 0.13) in specific heat capacity of the tank and fittings 0.13) in , ; specific heat capacity of the oil , ; Fig. 3. Hot-spot temperature of the 118-kV winding in the 250-MVA ONAFcooled transformer. 0.51) in correction factor for the oil in the ONAF, ONAN, and OFAF cooling modes; correction factor for the oil in the ODAF cooling mode; portion of the stray losses in the total losses; portion of the core losses in the total losses; portion of the winding losses in the total losses. Equation (29) is an empirical formula based on observations from different thermal tests and the modeling that has already been performed and validated in the author’s previous work , . The equivalent thermal capacitance of the transformer oil for transformers without either external cooling or guided horizontal oil flow through the windings (where the lack of radiators and the lack of the horizontal oil flow through the winding directly affects the oil flow inside the transformer tank, thus slowing down the cooling process) is calculated according to the IEEE Loading Guide-Annex G  and . III. COMPARISON The measured temperature results, which are recorded for three different transformer units during different varying load tests, are compared by the new calculation method presented in this paper and the IEEE Annex G method. The input data for both methods are given in the Appendix. The maximum and average errors obtained for both models are given in Tables III, V, and VII. The maximum error is obtained as the maximum difference between the measured and calculated curve. The average error is obtained as the sum of the data values divided by the number of data values. The error plots are also shown in Figs. 3–5. A. The 250-MVA ONAF The rated voltages of the 250-MVA transformer were 1.5%/118/21 kV. The windings were seen from the limb side, the 118-kV and 230-kV main windings, the regulating winding, and the 21-kV tertiary winding. The connection was YNyn0d11, and the short-circuit impedance in the 250/250-MVA main direction was 12%. The oil flow through the windings was guided by oil guiding rings in a zigzag pattern. The transformer was equipped with a total of 16 fiber-optic sensors, eight in the 118-kV winding and eight in the 230-kV winding, according to the principles explained in . In total, 14 thermocouples were located in the tie plates and outer core packets at the top level of the main windings of phase B. In addition to the normal delivery tests, including the ONAN and ONAF heat-run tests, the following load tests were performed on the unit operating in the ONAF cooling mode: • constant load current; 1.28 p.u.; duration 13.5 h; • constant load current; 1.49 p.u.; duration 15 h; • varying load current Table II. The measured hot-spot temperature results of the hottest winding and sensor, recorded during the varying load current test, are compared with the results obtained from the thermal models in Fig. 3. The maximum and average errors are given in Table III. B. The 400-MVA ONAF The rated voltages of the transformer were 410 6 1.33%/ 120/21 kV. The windings were, seen from the limb side: 120-kV and 410-kV main windings, a regulating winding, and a 21-kV 1262 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009 TABLE IV LOAD STEPS FOR THE 400-MVA TRANSFORMER TABLE V MAXIMUM AND AVERAGE ERROR FOR THE 400-MVA TRANSFORMER SM: simple model; IEEE: IEEE-Annex G Fig. 4. Hot-spot temperature of the 410-kV winding in the 400-MVA ONAFcooled transformer. probes (eight in each winding), and the tie plates, outer core packets, and yoke clamps had a total of 37 thermocouples. Additional load tests with ONAF cooling were the following: • constant load current: 1.0 p.u.; duration 12 h: • constant load current: 1.29 p.u.; duration 10 h; • constant load current: 1.60 p.u.; duration 15 h; • varying load current (Table IV). The measured hot-spot temperature results of the hottest winding and sensor, which were recorded during the varying load current test, are compared with results obtained from the thermal models in Fig. 4. The maximum and average errors are given in Table V. C. The 605-MVA OFAF Fig. 5. Hot-spot temperature of the 362-kV winding in the 605-MVA OFAFcooled transformer. tertiary winding. The connection was YNynd, and the short-circuit impedance in the 400/400-MVA main direction was 20%. The oil flow through the windings was guided by the oil guiding rings in a zigzag pattern. The main windings in this transformer are representative of two basic cases: 1) “restricted oil flow” (2-mm radial spacers in the 120-kV winding) and 2) “unrestricted oil flow” (3-mm radial spacers in the 410-kV winding). The main windings were equipped with a total of 16 fiber-optic The 605-MVA transformer was a generator stepup (GSU) unit with the windings seen from the limb side: part of the HV winding (i.e., 362 kV-winding), the double shell LV winding (i.e., 22 kV-winding), and the main part of the HV winding. The oil circulation through the windings was guided by the oil guiding rings in a zigzag pattern in such a way that the oil flow through the LV winding was restricted (2-mm radial spacer) and through the HV winding unrestricted (3-mm radial spacer). The transformer was not a sealed OD (i.e., the oil circulation was not forced through the winding block). In total, 24 fiber-optic sensors were installed in the top disc/turns of the outer shell of the LV winding and the outer part of the HV winding. In addition to the normal heat-run tests, the following load tests were made with OFAF cooling: • constant load current: 1.00 p.u.; duration 12 h; • constant load current: 1.30 p.u.; duration 1.2 h; • varying load current (Table VI). The measured hot-spot temperature results of the hottest winding and sensor, which were recorded during the varying load current test, are compared with results obtained from the thermal models and are shown in Fig 5. SUSA AND NORDMAN: SIMPLE MODEL FOR CALCULATING TRANSFORMER HOT-SPOT TEMPERATURE 1263 TABLE VI LOAD STEPS FOR THE 605-MVA TRANSFORMER TABLE VII MAXIMUM AND AVERAGE ERROR FOR 605-MVA TRANSFORMER SM: simple model; IEEE: IEEE-Annex G. The maximum and average errors are given in Table VII. IV. CONCLUSION The athors have already developed a few transformer thermal models –. All models take into account the oil viscosity change with temperature as one of the parameters defining the temperature curve. Also, the hot-spot to top-oil temperature gradient and the top-oil temperature rise are defined as two separate systems but cascadely interconnected. Thus, any change of the top-oil temperature will affect the hot-spot to top-oil gradient and further on the hot-spot temperature. Similarly, by switching from the winding time constant to the top-oil time constant, the model presented in this paper takes into account this additional top-oil effect on the hot-spot temperature rise. A new feature in the thermal model developed in this paper is that it is based directly on the hot-spot -to-ambient air gradient without splitting up this gradient into the two gradients hot-spot-to-top oil and top oil-to-ambient air. The oil viscosity effect and loss change with temperature are also taken into account. The model is based on an exponential iterative calculation procedure. Nevertheless, more rigid and more precise mathematical procedures could be applied as well. The authors have decided to use an exponential approach to follow well-known temperature calculation procedures given in  and . Comparably, the results obtained by the IEEE Annex G method and the results plotted by the proposed model are in good agreement with the measured values for the load increase and load decrease. However, one could conclude that models yield values either on the conservative side or with reasonable accuracy. The main advantage of the proposed model is a reduced number of the input data and its simplicity. Nevertheless, both models develop higher error at the load increase. This can be straightforwardly observed in Figs. 3–5. The models simply predict a much faster initial temperature rise Fig. 6. Computing algorithm for the simple model. compared to the measured one. In addition, the higher overload of the error is more pronounced as the oil viscosity effect is underestimated. In other words, the established oil circulation is 1264 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009 TABLE VIII INPUT DATA FOR THE THERMAL MODELS Fig. 7. Load profile during an extended heat-run test of a transformer. the model could be used for the real-time online hot-spot temperature computation as an integrated part of a monitoring system, (i.e., indirect hot-spot measurement). Furthermore, the model application will allow both transformer manufactures and users to run different loading and ambient scenarios and, by analyzing the results, improve the transformer design (costs, size, and load carrying capacity). Input data necessary for the suggested thermal model. at a much higher rate than it is accounted for by the presented models. The way to overcome the problem is in the further improvement of the winding time constant equation assuming that the oil viscosity equation is correct. Therefore, future work should consider additional thermal tests and investigations to derive general and simpler winding and top-oil time constants calculation procedures. The concept for the model application in transformer monitoring, the model computation algorithm, Fig. 6, and the model validation are given in the Appendix, respectively. APPENDIX A. Transformer Monitoring A transformer online monitoring system, which collects information from several measurable variables, should also include real-time application of the thermal models to provide an accurate picture of the operating condition of the transformer, allowing the operator to detect the early signs of faults and correct them. In general, the monitoring system identifies faults by comparing the results of measurements with prediction of the models. Consequently, the complete application of the suggested model is only possible in the systems used for transformers equipped with the fiber-optic sensors as is the case with all developed hot-spot thermal models. However, B. Model Algorithm The algorithm that describes the steps to follow in order to calculate the hot-spot temperature is given in Fig. 6. C. Model Validation The model can be validated in an extended heat-run test (Fig. 7) made on a transformer with installed fiber-optic sensors by using fitting and extrapolation techniques. Note that these techniques should be applied in a manner consistent with the modeling presented in this paper. An application example as well as a corresponding mathematical procedure are given in . The extended heat-run test consists of a regular heat-run test with added an overload test (Fig. 7). The overload should be applied three hours after the cooling curve is recorded in order to obtain a prolonged cooling curve as well. 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Power Del., vol. 16, no. 2, pp. 171–175, Apr. 2001.  W. H. Tang, Q. H. Wu, and Z. J. Richardson, “Equivalent heat circuit based power transformer thermal model,” Proc. Inst. Elect. Eng., Elect. Power Appl., vol. 149, no. 2, pp. 87–92, Mar. 2002.  W. H. Tang, Q. H. Wu, and Z. J. Richardson, “A simplified transformer thermal model based on thermal-electric analogy,” IEEE Trans. Power Del., vol. 19, no. 3, pp. 1112–1119, Jul. 2004. Dejan Susa (S’05–M’06) was born in Split, Croatia, on May 22, 1972. He received the D.Eng. degree in electrical engineering from the University of Nis, Nis, Serbia, in 2000, and the M.Sc. and D.Sc. degrees from the Helsinki University of Technology, Espoo, Finland, in 2002 and 2005, respectively. He was with the Power Systems Laboratory, Helsinki University of Technology, from 2001 to 2006. He has been with the Center for Power Transformer Monitoring, Diagnostic and Life Management, Monash University, Clayton, Australia, since 2006. Currently, he is with SINTEF Energy Research Department, Trondheim, Norway. He is working on different power transformer research topics (losses, temperatures, moisture, gasses, online monitoring). Dr. Susa is a member of Norwegian IEC National Committee and of IEC MT1 (loading guide for oil-immersed power transformers), IEC MT2 (ability to withstand short circuit), and IEC MT6 (temperature rise). Hasse Nordman (M’08) was born in Overmark, Finland, in 1945. He received the Ph.D. degree in mathematics from the Abo Akademi University, Turku, Finland, in 1977. From 1970 to 1982, he was with ABB Corporate Research (formerly Stromberg Research Centre), Vaasa, Finland, working on current-related phenomena (losses, temperatures, short-circuit forces) in electric power equipment. Since 1982, he has been with the Development Engineering Department in the Power Transformer Division of ABB, Vaasa. He is also the leader of the global ABB R&D activity “Load Losses and Thermal Performance.” Dr. Nordman is a member of CIGRE, Chairman of the Finnish National Committee in the IEC Power Transformer Technical Committee (TC 14), and Convenor of the Maintenance Team MT1: Revision of IEC 354: Loading guide for oil-immersed power transformers.