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1 Study of a Transformer Thermal System Pedro de Carvalho Peixoto de Sousa Barros Department of Electrical and Computer Engineering (DEEC) - Instituto Superior Técnico (IST) Abstract—This project has as goal the study and modulation of a dry single-phase transformer thermal system, with 1kVA of electrical power. During this study is identified and characterized the different zones of the transformer on which the temperature is considered homogeneous. It is presented a lumped model as well as the methodology to determinate its parameters. Tests are made for different load situations, quantifying the model error and giving particular attention to overload and short-circuit situations, being outstanding the importance of using models in order to foresee with precision the maximum temperature of the transformer as well as its overload capacity. I also presented a introductory study of the thermal behavior of the transformer and its modulation when it finds a regimen of forced cooling convection. Index Terms—Transformer, parameters, lumped parameters. thermal model, distributed I. INTRODUCTION circuit situations in order to analyse its thermal behaviour and to calculate its overload capacity. Finally, I made an introductory study with the objective to verify the thermal behaviour of the transformer when it finds a cooling forced convection and how this regimen can increase the transformer efficiency. II. IDENTIFICATION OF HOMOGENEOUS ZONES A transformer has in its constitution some kind of materials, namely copper, iron and some types of isolation. Each one of these materials have different thermal characteristics, as such, normally it’s expected that the temperature in the transformer is not the same on its different points. To become possible to read the temperature in the different regions of the transformer, I used type J thermocouples in a single-phase transformer of 1kVA and nominal voltage of 240/120V as it shows in the Fig.1. T HE project and the overload capacity of electric machines, namely transformers, are strongly conditioned by its thermal performance. Due to difficulties of modulating the thermal system, exists the conscience that the advantages of materials are not explored integrally in the construction of transformers, so for precaution reasons, are fixed exaggerated safety margins for its operation. Nowadays there are numerical calculation methods capable to deal with complicated geometries and represent on the same time several phenomenas (heat and mass transportation) that if are used carefully and systematically can allow the creation of more numerical models, namely the thermal component of the transformer. In this work, I have done a thermal study of the transformer, in order to be possible to know its dynamics as well as its hottest zone. The knowledge in identifying and studying the hottest zone of the transformer is directly related with the lifetime reduction of electric machines as the material aging depends on the temperature they are subjected to. It is intended to identify several temperature rises in different zones of the transformer, once the simplest models estimate a homogeneous temperature over all the transformer area. In this study I considered a thermal lumped model, as well as the justification and characterization of its parameters, becoming important the realization of several tests to validate the model and determinate the error. A study is made for overload situations as well as short Fig. 1. Thermocouples. Nu-Core, A-A winding, B-B winding, BC-BC winding. The transformer has three windings, being the internal winding (the one next to the core) named as winding A. In the 2 external windings, named as winding B and C, the nominal tension is 120V and when connected in series allows a transformation ratio of 1:1. Please note that the thermocouples from T1 to T6 are assembled under the iron core and the T7 and T8 are out of the core window. The thermocouple T9 has been assembled over the upper part of the core in order to be possible to read its temperature. When a test is carried out in the transformer, for instance a short circuit test, it shows that the transformer temperature is not homogeneous, existing zones where the temperature is higher and others where it is lower. In Fig. 2 is showed the temperature rise for a short circuit test where it’s possible to verify the non homogeneity in transformer temperature, and conclude that it is lower in the iron core (T9). 55 T1 T2 T3 T4 T5 T6 T7 T8 T9 Temperature Rise (ºC) 50 45 40 35 30 25 Fig. 4. Temperature variation as the readings move away to the external layers of the copper. variation occurs in the isolation material, whereas in the copper and in the iron it stays practically constant. The temperature differences in the previous figures can be explained appealing to Fig. 5, where it is represented the heat flux on the zone in analysis. 20 15 10 5 5.2 5.4 5.6 5.8 6 Time (h) Fig. 2. Temperature rise. Short-Circuit test. In the copper windings, the temperature is lower in the winding A, and it gradually rises as we read the temperature in the external windings, reaching its higher reading (for the short circuit test) in the winding B (T4 and T5) and coming back to lower reading in the winding C. As an example, is presented in the Figs. 3 and 4 the temperature inside windings, as well as its variation as the readings move away to the external layers of the copper. Fig. 3. Temperature inside the winding in the central zone of the transformer. Please note that the Fig. 3 and 4 were made with simulation programs that solve the equations for a distributed parameters model. To simplify the simulation, it was only presented the central zone of the transformer, where were assembled the thermocouples from T1 to T6. In Fig. 4, it is possible to verify that the temperature Fig. 5. Total heat flux [W/m2] Due to the large contact surface with the air temperature, the iron core has a very small increase of temperature when compared with the windings, acting as a heat exchanger. This way it explains the fact the temperature is lower near the core. As the reads move away from the core, the temperature of the winding A rises, because the heat as more difficulty to flow due to the various isolation layers. As was already explained, the hottest temperature is verified in winding B, what it would be expected to happen as it is an internal winding having several isolations layers, and therefore making the heat flux difficult through it. In addition, the winding B connected in series with winding C, has more 18 meters of length than winding A. With more copper, for a giving current it will appear more losses, therefore the temperature will be higher. In the external part of winding C, the temperature comes back to lower values, as it is a outside zone of the winding and the heat is easily dissipated to the air around. In thermocouples T7 and T8, that are not assembled underneath the iron core, when analyzing Fig. 2, we verify that T7 as the same temperature of T5, as both are assembled between winding B and C and near the same wire, however, T8 has a substantially lower reading as it is assembled near the exterior, revealing that the temperature is not constant along the wires on the same layer, being higher on the central zone of the winding and getting lower as it moves for the wires near 3 the exterior. Of everything that was observed it is evidenced that the transformer as several zones with different temperatures, indicating clearly that the temperature is not homogeneous. However, it becomes difficult to make the study of the transformer considering all the different temperatures. In order to be possible to establish a lumped model, it will be considered zones with material homogeneity, as well as zones where it can be found heat sources and using the Biot number [14] is possible to establish a criterion where we can say that a specific region of the transformer is assumed as having a homogeneous temperature. According with (1), it can be assumed that the temperature is uniform if the Biot number (NB) is lower than 0,1. This means that the thermal resistance for internal conduction is much less than the thermal resistance for convection at the surface of the transformer. NB = h× L k (1) Fig. 6. Lumped model. Nu-Core, A-A winding, BC-BC winding. each region and Pi represents the heat generation sources. The Gi-j conductances represent the resistance to the heat exchange between regions i and j. Once the winding A is an internal winding, it was verified that most of the heat produced in this winding is lost for the core and winding BC and not directly to the air. Due to this fact, the existence of a conductance between winding A and the air is not considered. A. Determination of the model parameters The thermal equations of the presented model are, in the matricial form, represented by (3) where C× V L= A (2) In (1), h is the heat transmission coefficient for convection, L is the ratio between the volume of the body (V) and the area of its surface (A) and k is the thermal conductivity. Knowing the transformer geometry and the materials properties is possible to calculate the Biot number for the different zones of the transformer. Therefore, for the iron core, I got a Biot Number of 6x10-3, being this value of 1,4x10-3, 1,5x10-3 and 60x10-6 for the windings A, B and C respectively. As the Biot number is lower than 0,1 we can assume a temperature homogeneity in each one of the considered zones. With experimental tests, it was verified that the windings B and C can be considered as one only zone of homogeneous temperature. Please note that in all zones it was considered the hottest point. III. LUMPED MODEL One of the goals of this work is the determination of a lumped model on which will be possible its use in a simple way with the lowest error. For that, it was considered each of the transformer regions with a current source (that corresponds to the heat generation on that material), a capacitor (to represent the heat accumulation) and a resistor (to represent the temperature variation in the transformer due to the heat flux). Fig. 6 shows the different transformer zones, being ∆θi the variation of temperature of zone i, Ci the thermal capacity of d∆θ + G × ∆θ = P dt (3) In which C represents the matrix of the thermal capacities, ∆θ the vector of temperature rise for each region of the transformer, G the matrix of thermal conductances and P the vector of the heat source for each zone where it is verified the heat generation. From (3), it is possible to verify that in a steady state regimen we get (4), G × ∆θ = P ⇔ ∆θ = G −1 × P ⇔ ∆θ = R × P (4) being R the matrix of the thermal capacities. To solve (3), it is necessary to determine the model parameters. Vector P is an entrance value, being possible to represent it depending on the transformer load, whereas the ∆θ vector represents the temperature variation for each specific part of the transformer when related to a specific load. This way, there are two variables to be calculated: the matrix of thermal capacities C, and the matrix of thermal resistances R (the matrix G is obtained by inverting the matrix R). 1) Thermal capacities calculation Knowing the transformer geometry and its materials, it is possible to characterize with some accuracy the thermal capacity. Cth = ρVC p = mC p So it is possible to write the matrix C as follows: (5) 4 0 CA 0 0 6253 ,63 0 = 0 C BC 0 0 485 ,70 0 609 ,18 0 0 (6) 60 2) Thermal resistance calculation The calculation of the thermal resistances values is slightly more complicated than the thermal capacities. Once the transformer geometry is complex, it becomes easier to calculate the thermal resistances values with experimental tests. Observing the steady state equation (4) it is shown that heating each different parts of the transformer and measuring the temperature variation in all the other parts, it is possible to calculate the thermal resistances values. To heat each zone of the transformer, it was made a test with direct current, it means that knowing the current value, it is possible to calculate the heat generated in each zone (power losses). This type of test is valid when is intended to heat the windings. Once is not possible to heat only the iron core (it is impossible to heat it without heating also the windings), to verify the iron core heating, it was made an open circuit test where was monitored the total electric power on the transformer and the current in the winding A. Therefore, with the mentioned tests, it becomes possible to calculate the matrix R. Then, the thermal resistances matrix is as follows: R Nu R = R Nu −A R Nu −BC R Nu −A RA R A −BC R Nu −BC 0,65 0,66 0,54 R A−BC = 0,66 2,33 1,42 R BC 0,54 1,42 2,19 (7) where the main diagonal line represents all the resistances connected to a specific zone (Nu, A the BC) and outside the diagonal are represented the thermal resistances between each two zones of the transformer. Inverting the matrix R it is possible to obtain the thermal conductances matrix G. R G Nu − Ar + G Nu − A + G Nu −BC − G Nu − A − G Nu − BC −1 =G = − G Nu − A G Nu − A + G A − BC the steady state. Afterwards, was made a simulation with the previously presented model in order to validate it. − G Nu − BC − G A −BC + G Nu − BC Temperature Rise (ºC) C Nu C = 0 0 TNu-Simul. TA-Simul. TBC-Simul. TNu TA TBC 50 40 30 20 10 0 0 2 4 6 8 10 Time (h) Fig. 7. Steady-State test. From Fig. 7, it is verified that the obtained results for the simulation approaches the experimentally values. It is also observed that the transformer shows a maximum variation of temperature, for the operation at nominal power rate, about 58ºC for the copper windings and 28ºC in the iron core. As it would be expected, it is evidenced that the thermal behaviour of the windings is different from the thermal behaviour of the iron core, being its time constants about 3600 (1 hour) and 8200 seconds (2 hours and 18 minutes), respectively. For this test, it is evidenced that the winding A losses are about 10,4W being 13,9W for the winding BC, noticing that this difference is due to the longer length of the winding BC. Through a test open circuit test (carried out for the model parameters calculation) it was verified that the iron core losses are about 20W. TABLE I MAXIMUM ERROR 8,0ºC |Absolute Error| 1,5ºC Relative Error % 18,8 41ºC 3ºC 7,3 41ºC 3ºC 7,3 Temperature Simulated Experimental TNu 9,5ºC TA 44ºC TBC 44ºC IV. EXPERIMENTAL RESULTS AND SIMULATIONS On table I is presented the maximum error for this test, verifying that it occurs during the transient state. It’s noticed that the relative error in the core is high during this test, however, it is verified that it occurs at extremely low temperature with a small absolute error. On the other hand, the relative error in windings is lesser, but occurring an increase of the absolute error. After reaching the temperature steady state the error decreases to values lower than 3,5% in the core and 3,8% in the windings. A. Steady State operation For this test, the transformer was placed supplying energy to a resistive load at nominal power rated, reading the temperature in various parts of the transformer until it reaches B. Overloads and Short circuits In this chapter it is intended to study which is the maximum overload capacity of the transformer, doing tests and simulations, in order to assure the normal operation of the − G A − BC G BC−Ar + G A − BC 2,23 − 0,49 − 0,23 = − 0,49 0,82 − 0,41 − 0,23 − 0,41 0,78 (8) 5 1) Overloads In Fig. 8 it is presented the experimental and computational result obtained for a test where the transformer operates at nominal power rate, afterwards operating at a temporary overload, increasing the current to a value 10% higher than its nominal current. Afterwards it decreases the current until it reaches 90% of its nominal value. 70 TNu-Simul. TA-Simul. TBC-Simul. TNu TA TBC Temperature Rise (ºC) 60 50 40 As an example, the laboratory transformer, as it was already referred, at rated power it produces power losses of 20W in the iron core, being 10,4W and 13,9W in the windings A and BC, respectively. Once this transformer has class H isolation, it’s possible a maximum temperature variation of 125ºC with an ambient temperature of 40ºC and 15ºC error margin for the hottest point, it means that this point can reach 180ºC. Fig. 9 represents a simulation where initially the transformer operates at rated power, later on increasing 2,5 times the winding losses it produces losses of 26W in the winding A and 34,8W in winding BC. The losses in iron core can be considered constant once these only depend on the supplied voltage value. 120 Temperature Rise (ºC) transformer and avoiding failures. Regarding the short circuits, this study becomes important mainly for the project of transformer circuit breakers and its time actuation reaction, which has to be fast enough to avoid major failures. TNu-Simul. TA-Simul. TBC-Simul. 100 80 60 40 30 20 20 1 2 3 4 Time (h) 5 6 7 Of course that when it reaches an overload state increasing the 10% the current, the windings verify an increase on losses of 20% and a temperature rise. As it is possible to see in table II, the error of the simulation values, for the hottest point, is about 2% for the windings and 3% for the iron core. TNu 4 6 Time (h) 8 10 12 Fig. 9. Maximum Overload. Fig. 8. Overload test. Temperature 2 TABLE II MAXIMUM ERROR: HOT-SPOT |Absolute Simulated Experimental Error| 30ºC 31ºC 1ºC Relative Error % 3,1 TA 66,1ºC 67,3ºC 1,2ºC 1,8 TBC 65,1ºC 64,2ºC 0,9ºC 1,4 To study temporary overloads situations, a model that calculates which is the temperature variation with a small error shows particularly useful, it means, for a given operation state of the transformer, it is possible to calculate with some accuracy the maximum temperature of the hottest point, if it operates under overload, as well as which is the maximum time that the transformer can support overload situations without isolation damages. This model can also be useful for the thermal design of the transformer as knowing the maximum temperature reached it becomes possible to manufacture smaller transformers and this way avoiding the usual over sizing of electric machines. As it is presented, the model foresees that the winding temperature variation reaches the limit value, which is the 125ºC. Assuming that the windings resistance does not change with the temperature, it is possible to calculate the theoretical current value that flows trough them for the considered situation, and it is verified that the values reaches 6,6 amperes. Once the nominal current of the transformer is 4,16 amperes, a current of 6,6 amperes corresponds to an increase of 58% in the nominal current, reveals that the laboratory transformer is clearly oversized. In the previous simulation, it was presented an extreme situation of overload, on which the transformer finds an operating state during a period of time long enough to reach a steady state. However, it is possible to put the transformer operating with higher values than the ones mentioned previously if the operating time doesn’t exceed the period of time necessary to reach its maximum value of 125ºC. The Fig. 10 presents some simulations for different current values. In this simulation, it is assumed as beginning point the temperature variation in steady state, making afterwards a current change multiplying it by 1.58, 1.7, 2 and 2.5. The situation where the nominal current is multiplied by 1.58 was already presented, being the maximum overload situation. As we increase the current value, it is verified that the temperature reaches quickly the maximum value. 6 Temperature Rise (ºC) 100 80 60 40 20 0 t1 t2 1 t3 2 Time (h) 3 4 5 Fig. 10. Overload Time. For the situations considered in this simulation, it is also verified that it is possible to operate the transformer with a current 70% higher than the nominal value during a period of time about 80 minutes (t3), decreasing this period to 25 minutes if the current is double of its nominal value (t2). If the current reaches 2.5 times the nominal value, the period of time that the transformer can operate overloaded without any isolation damages decreases to 12 minutes (t1). With this simulation, it is also intended to enhance the importance of the thermal behaviour of the iron core, it means, the iron core working as a heat exchanger how it contributes for the windings cooling operating under overload conditions. Observing again Fig. 10, it is verified that for operating states of soft overloads ( near rated values) the iron core constant thermal time has an important role on the global transformer cooling, it means, the heat generated by the transformer is slowly dissipated by the iron core making the temperature of the hottest point to raise also gradually. As the overload becomes more violent (extremely higher than the rated conditions) the heat generated by the windings reaches extremely high values, producing a quickly temperature increase. The heat generated in the windings needs time to spread into the iron core, being then dissipated by this one. As the windings heat increase time is sufficiently lower than the period of time necessary to spread and dissipate the heat trough the core, the cooper temperature will increase quickly that the core temperature. Take for instance the situation where the current is 2.5 times the nominal current. The temperature in the winding BC takes approximately 12 minutes (t1) to reach its maximum value. At that exactly time, it is verified that the iron core temperature practically doesn’t change. 2) Short circuits The use of a proper thermal model allow us to calculate efficiently the period of time that the transformer can operate with short circuit, knowing which is the maximum overload protection actuation time. In Fig. 11 is presented a short circuit situation in the secondary winding of the transformer, where the current is 30 times higher the nominal current. 120 Temperature Rise (ºC) TNu - 1.58*In TBC - 1.58*In TNu - 1.7*In TBC - 1.7*In TNu - 2*In TBC - 2*In TNu - 2.5*In TBC - 2.5*In 120 TNu TA TBC 100 80 60 40 20 -1 -0.5 0 0.5 1 1.5 Time (s) 2 2.5 3 3.5 Fig. 11. Short-Circuit. In the figure above it is possible to observe that the iron core does not have time to heat, reason why its behaviour can be disgard, considering just the windings behaviour. To project the overload protections, it is verified that the transformer can remain in operation under short circuit during 2.75 seconds from which can occur failure. Therefore, the protection actuation time must be lower than the previously referred time. C. Forced Convection In the previous chapters, it was presented a study for the transformer thermal behaviour during a natural convection situation. In this chapter it is intended to present an introductory study for a transformer thermal behaviour when it operates under forced cooling convection. As it is presented, the hottest point of the transformer, under nominal conditions, is located inside the copper windings. Cooling under forced convection has as goal the temperature reduction in that point. As already referred, it was observed that the iron core acts as a heat exchanger. Having verified this, a fan was used pointing directly to iron core and this was increasing, efficiently, the heat exchange. To calculate the model parameters, it was proceeded as previously referred, doing tests with direct current in order to heat each winding separately and a open circuit test in order to heat only the core. This way, it was possible to calculate the model parameters when the transformer operates under forced cooling convection, being the thermal resistance matrix given by (9): 0,23 0,25 0,21 R = 0,25 1,82 1,05 0,21 1,05 1,84 (9) Inverting the matrix is then possible to obtain the thermal conductances matrix. 7 5,20 − 0,55 − 0,28 R = G = − 0,55 0,88 − 0,44 − 0,28 − 0,44 0,83 Comparing the values obtained for matrix G with forced convection (10) and for the matrix G with natural convection (8), it is observed that almost all the values are similar to the exception to the value obtained for the core conductance. This fact was expected, once only cooling the core with forced convection, increases the thermal conductance between the core and the air, GNu-ar. As the iron core is the only part of the transformer where the forced convection acts, all the other conductance values don’t have significant variations. With the intention of comparing the behaviour of the lumped parameters model under forced convection with the real temperature evolution, a test has been made where the transformer, that was initially at ambient temperature, placing it later on operating under nominal conditions until the temperature rises up to the steady state. Fig. 12 presents the experimental results and the results obtained with the simulation. Comparing the results obtained with forced convection with the ones obtained with natural convection, it is verified a global temperature reduction in the transformer giving the indication that the position chosen for the fan has been efficient. 45 Temperature Rise (ºC) 40 35 TNu-Simul. TA-Simul. TBC-Simul. TNu TA TBC 30 25 TABLE III MAXIMUM ERROR: FORCED COOLING CONVECTION |Absolute Relative Temperature Simulated Experimental Error| Error % TNu 5,8ºC 4,5ºC 1,3ºC 28,8 (10) −1 20 28,5ºC 25,8ºC 2,7ºC 10,5 29,1ºC 26,5ºC 2,6ºC 9,8 From table III, where it is possible to read the model error, we can verify an increase of it when comparing with the same test but with natural convection. This error increase is due to the fact that we are not modelling correctly the forced convection, it means, representing the complex phenomena as the convection, which involves mass transportation, just trough a resistance, what could not be sufficient and therefore increasing the error value. V. CONCLUSION This report presented a lumped model for a dry single-phase transformer, considering it with several zones with homogeneous temperature. The model revealed accuracy, with small error when calculating the transformer temperatures. With the considered model, it was possible to foresee the maximum temperature of the hottest point for situations of overload and short circuits. With the presented tests, it was intended to point out the importance of using models of temperature forecast for a transformer in order to be possible to calculate the overload capacities and the protections time actuation. Placing the transformer in a situation of forced cooling convection, an increase of the model error is verified, this way concluding that modelling complex phenomena as convection requires another kind of study. 15 REFERENCES 10 [1] 5 0 0 TA TBC [2] 1 2 3 Time (h) 4 5 6 Fig. 12. Forced cooling convection. During this test it is observed that the maximum temperature variation in the windings is about 43ºC and without forced convection the temperature variation increases to 58ºC. Regarding the core, it is verified a reduction to 12ºC compared with the 28ºC with natural convection. 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