energies Article A Simplified Top-Oil Temperature Model for Transformers Based on the Pathway of Energy Transfer Concept and the Thermal-Electrical Analogy Muhammad Hakirin Roslan 1,2 , Norhafiz Azis 2,3, *, Mohd Zainal Abidin Ab Kadir 2 , Jasronita Jasni 2 , Zulkifli Ibrahim 2,4 and Azalan Ahmad 5 1 2 3 4 5 * Faculty of Engineering, Universiti Pertahanan Nasional Malaysia, Kem Sg Besi 57000, Malaysia; hakirinroslan@gmail.com Centre for Electromagnetic and Lightning Protection Research, Universiti Putra Malaysia, Serdang 43400, Malaysia; mzk@upm.edu.my (M.Z.A.A.K.); jas@upm.edu.my (J.J.); zulib78@gmail.com (Z.I.) Institute of Advanced Technology (ITMA), Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia Faculty of Engineering Technology, Universiti Teknikal Malaysia Melaka, Durian Tunggal 76100, Malaysia Malaysia Transformer Manufacturing Sdn. Bhd, Ampang 54200, Selangor, Malaysia; azalan@mtm.com.my Correspondence: norhafiz@upm.edu.my; Tel.: +60-3-8946-4373 Received: 28 September 2017; Accepted: 24 October 2017; Published: 11 November 2017 Abstract: This paper presents an alternative approach to determine the simplified top-oil temperature (TOT) based on the pathway of energy transfer and thermal-electrical analogy concepts. The main contribution of this study is the redefinition of the nonlinear thermal resistance based on these concepts. An alternative approximation of convection coefficient, h, based on heat transfer theory was proposed which eliminated the requirement of viscosity. In addition, the lumped capacitance method was applied to the thermal-electrical analogy to derive the TOT thermal equivalent equation in differential form. The TOT thermal model was evaluated based on the measured TOT of seven transformers with either oil natural air natural (ONAN) or oil natural air forced (ONAF) cooling modes obtained from temperature rise tests. In addition, the performance of the TOT thermal model was tested on step-loading of a transformer with an ONAF cooling mode obtained from previous studies. A comparison between the TOT thermal model and the existing TOT Thermal-Electrical, Exponential (IEC 60076-7), and Clause 7 (IEEE C57.91-1995) models was also carried out. It was found that the measured TOT of seven transformers are well represented by the TOT thermal model where the highest maximum and root mean square (RMS) errors are 6.66 β¦ C and 2.76 β¦ C, respectively. Based on the maximum and RMS errors, the TOT thermal model performs better than Exponential and Clause 7 models and it is comparable with the Thermal-Electrical 1 (TE1) and Thermal-Electrical 2 (TE2) models. The same pattern is found for the TOT thermal model under step-loading where the maximum and RMS errors are 5.77 β¦ C and 2.02 β¦ C. Keywords: top-oil temperature thermal model; pathway of energy transfer; thermal-electrical analogy; nonlinear oil thermal resistance; transformers 1. Introduction Thermal modeling is one of the important studies for the estimation of the top-oil temperature (TOT) and hot-spot temperature (HST) in transformers. TOT is defined as the average of the tank outlet oil temperature and the oil pocket temperature [1]. Meanwhile, HST consists of ambient temperature (AT), TOT rise, and HST rise [1]. One of the common approaches to carry out thermal modeling of transformers is through a numerical network thermal model. Currently, there are two standards that utilize this approach to determine HST and TOT, known as IEC 60076-7 and IEEE C57.91-1995 [1,2]. Energies 2017, 10, 1843; doi:10.3390/en10111843 www.mdpi.com/journal/energies Energies 2017, 10, 1843 2 of 15 These standards use the exponential function for constant loading and the differential function for dynamic loading for determination of HST and TOT [1,2]. A number of studies were also carried out to develop alternative thermal models for HST and TOT [3–18]. The majority of these studies were carried out based on the thermal-electrical analogy that represents the heat transfer equivalent circuit in the form of a resistor capacitor (RC) circuit [3]. Based on this concept, the current source analogy represented the heat input due to losses, whilst the nonlinear resistance analogy represented the effect of air or oil cooling convection currents [3]. TOT is one of the important components to determine HST in transformers. A model based on the thermal-electrical analogy was developed [5,6] to determine the TOT thermal model. In these studies, the nonlinear thermal resistance was expanded through inclusion of the Nusselt, Prandtle, and Grashof numbers. The loss variation, viscosity changes with temperature, equivalent thermal capacitances, oil time constants for different designs, and oil circulation were also considered in [5–7]. The TOT obtained by TOT thermal model in [5] was quite close to the measured TOT, as compared to Annex G model in IEEE C57.91-1995 for transformers with oil natural air forced (ONAF) and oil forced air forced (OFAF) cooling modes. The same finding was found for transformers with either oil natural air natural (ONAN) or ONAF, and OFAF cooling modes in [6]. The TOT thermal models in [5–7] utilize a number of input parameters which increase the accuracy of the simulated TOT. A Levenberg–Marquardt model was proposed in [8] to determine the input parameters, such as TOT rise at rated, oil time constant, and constant n, based on in-service TOT measured data. Based on the case study of transformers with ONAF and OFAF cooling modes, it was found that the TOT thermal model based on the Levenberg–Marquardt approach could represent the measured TOT better than the Clause 7 model in IEEE C57.91-1995 [8]. This model requires the in-service measured TOT information, which may not be available for some transformers in order to determine the input parameters. A TOT thermal model was introduced in [9,10] based on the concept of outdoor and indoor heat transfer flows of transformers. The indoor TOT thermal model equivalent circuit required a considerable number of thermal resistances input parameters, such as windings, oil, core, tank, cooling air, ventilation holes, walls, and ceiling. The model also considered the heat generated in the distribution cabinet, qcabin , such as heat from busbars, fuses, disconnectors, and switches. Based on the case study of a transformer with an ONAN cooling mode, the TOT thermal model could represent quite well the measured TOT for indoor and outdoor conditions. The TOT thermal models in [9,10] are beneficial for comprehensive thermal analysis for individual outdoor or indoor transformers. Nonlinear thermal resistance is one of the important parameters in the thermal-electrical analogy thermal models. This parameter was commonly determined based on the Nusselt, Prandtle, and Grashof numbers according to heat transfer theory [5,8–11]. Through this approach, the oil viscosity was included in the final form of the nonlinear thermal resistance equation due to its sensitivity toward temperature. The viscosity coefficient was also calibrated in [12] in order to determine the nonlinear thermal resistance for the transformer with a rating over 6300 kVA. Another approach to determine the nonlinear thermal resistance was based on measured values outside the tank and average winding temperatures from the short-circuit heating experiment in [19]. The nonlinear thermal resistance was represented by an exponential form in [20,21] where the element of the heat conductance matrix and constants were determined based on a genetic algorithm (GA) [20,21]. Recently, a moisture-dependent nonlinear thermal resistance was proposed in [22] where it considered the cellulose thermal conductivity and area/thickness. This paper presents a simplified approach to estimate the TOT based on a proposed concept of the pathway of energy flow that is adopted from heat transfer theory to construct the nonlinear thermal resistance and lumped capacitance circuit. An alternative approximation of the convection coefficient, h, is proposed, which eliminates the requirement of the viscosity for application in the thermal-electrical analogy. The main contribution of this study is the application of the pathway of energy based on heat transfer principles and the redefinition of nonlinear thermal resistance. The derivation of the TOT thermal model is presented and tested based on the measured TOT of seven transformers with either Energies 2017, 10, 1843 3 of 15 ONAN or ONAF cooling modes. The TOT thermal model is also tested with the step-loading of a Energies 2017,with 10, 1843 3 of 15 transformer an ONAF cooling mode from previous studies. Energies 2017, 10, 1843 3 of 15 2. Heat Transfer Theory 2. Transfer Theory Application in Transformers 2. Heat Heat Transfer TheoryApplication Applicationin inTransformers Transformers 2.1.2.1. Concept of of Heat Transfer in Transformers 2.1. Concept Concept of Heat Heat Transfer Transfer in in Transformers Transformers The concept ofof approachesto evaluatethe the heat transfer The concept energy transfer one of the simplest heat transfer in The concept ofenergy energytransfer transferisis isone oneof of the the simplest simplest approaches approaches totoevaluate evaluate the heat transfer in in transformers. In general, heat transfer occurs there is a temperature difference in a medium or transformers. In general, heat transfer occurs when there is a temperature difference in a medium transformers. In general, heat transfer occurs when there is a temperature difference in a medium or or between media [23]. transformers, the heat sources losses which between media [23]. originatefrom fromload loadand andno-load no-load losses which between media [23].InIn Intransformers, transformers,the the heat heat sources sources originate originate from load and no-load losses which distribute through oils and transfer to the tank/radiator of the transformer [24]. HST is the hottest distribute through oils and transfer to the tank/radiator of the transformer [24]. HST is the hottest distribute through oils and transfer to the tank/radiator of the transformer [24]. HST is the hottest temperature inside aa transformer and normally located windings, and the coolest temperature temperature inside a transformer temperature inside transformerand andisis isnormally normallylocated locatedatat atwindings, windings,and and the the coolest coolest temperature temperature is is AT, which is located outside the transformer. Based on the heat transfer theory, heat or will AT,iswhich is located outside thethe transformer. theory,heat heatororenergy energy will AT, which is located outside transformer.Based Basedon onthe the heat heat transfer transfer theory, energy will transfer from the hottest temperature to the coolest temperature; in transformers, from HST to the transfer from the hottest intransformers, transformers,from from HST transfer from the hottesttemperature temperatureto to the the coolest coolest temperature; temperature; in HST to to thethe conduction and convection through all media, papers, oils, AT via conduction andconvection convectionthrough through all all media, media, including including windings, insulation papers, oils, ATAT viavia conduction and includingwindings, windings,insulation insulation papers, oils, and tanks/radiators. This pathway of energy transfer concept is applied in this study to determine and tanks/radiators.This Thispathway pathwayofofenergy energytransfer transferconcept conceptisisapplied appliedininthis thisstudy studytotodetermine determinethe and tanks/radiators. the resistance equivalent circuit, which can be in 1. the thermal thermal resistance equivalent circuit, which canseen be seen seen in Figure Figure thermal resistance equivalent circuit, which can be in Figure 1. 1. Figure 1. Pathway of energy in transformers. Figure Figure 1. 1. Pathway Pathway of of energy energy in in transformers. transformers. Where conductionthrough through copper copper of the thethe Where q1 ππis thethe conduction the winding, winding,πq22 isisthe theconduction conductionthrough through 1 is Where 1 is the conduction through copper of the winding, π2 is the conduction through the insulation paper, π is the convection from insulation paper to the oil, π is the convection from the insulation paper, q is the convection from insulation paper to the oil, q is the convection from 3 4 3 4 insulation paper, π3 is the convection from insulation paper to the oil, π4 is the convection from thethe tank, is conduction through the tank/radiator is oiloil to to thethe tank, q5 ππis55 the conduction through of transformers, transformers,ππθβπ is the theHST, HST, ππθπππ hs is oil is oil to the tank, is the the conduction throughthe thetank/radiator tank/radiator of of transformers, the HST, βπ πππ is the TOT, and π is the AT. Comprehensive analysis of the thermal conductivity through insulation theisTOT, andand θ a isππthe AT. Comprehensive analysis of the thermal conductivity through insulation the TOT, π is the AT. Comprehensive analysis of the thermal conductivity through insulation paper can characterized based on consideration on the geometry is paper can bebe characterized porouscharacteristics. characteristics.Fractal Fractal geometry paper can be characterizedbased basedon onconsideration consideration on the porous porous characteristics. Fractal geometry is is among the common approaches that can be used to model the pore structure for modelling the among thethe common approaches thatthat can can be used to model the pore for modelling the thermal among common approaches be used to model the structure pore structure for modelling the thermal through paper In the thermal of thermal conductivity conductivity through insulation insulation paperIn[25,26]. [25,26]. In this this study, theconductivity thermal conductivity conductivity of conductivity through insulation paper [25,26]. this study, thestudy, thermal of insulation insulation paper was represented by a single thermal resistance for the simplification of the thermal insulation paper was represented by a single thermal for resistance for the simplification of the resistance thermal paper was represented by a single thermal resistance the simplification of the thermal resistance equivalent resistancecircuit. equivalent circuit. circuit. equivalent Nonlinear Thermal Resistances 2.2. Nonlinear ThermalResistances Resistances 2.2.2.2. Nonlinear Thermal All media in transformers have their own thermal which act like dissipation elements. media transformershave havetheir theirown own thermal thermal resistances resistances elements. AllAll media inin transformers resistanceswhich whichact actlike likedissipation dissipation elements. The thermal resistance is nonlinear due to the presence of the convection coefficient, h. Detailed The thermal resistance is nonlinear due to the presence of the convection coefficient, h. Detailed The thermal resistance is nonlinear due to the presence of the convection coefficient, h. Detailed descriptions of the nonlinear thermal resistance are presented Section 3.1. Based on the pathway descriptions nonlinear thermalresistance resistanceare arepresented presentedinin inSection Section3.1. 3.1.Based Based on on the the pathway pathway of descriptions of of thethe nonlinear thermal of energy transfer concept shown in Figure 1, the heat transfer thermal resistance equivalent circuit of energy transfer concept shown in Figure 1, the heat transfer thermal resistance equivalent circuit is energy transfer concept shown in Figure 1, the heat transfer thermal resistance equivalent circuit is represented by Figure 2. is represented by Figure 2. represented by Figure 2. Figure 2. Thermal resistance equivalent equivalent circuit. Figure Figure 2. 2. Thermal Thermal resistance resistance equivalent circuit. circuit. According Figure three locations of which are , ππππθ,, and The According 2, there thereare are three locations of temperatures, are , θoilππ,ππ..and Accordingtoto toFigure Figure 2, 2, there are three locations of temperatures, temperatures, whichwhich are π πβπ and The θ a . βπ , ππππ hs heat sources in transformers originate from load losses, π , and no-load losses, π , which are added ππ’ ππ The heat sources in transformers originate from load losses, q , and no-load losses, q , which heat sources in transformers originate from load losses, πππ’ , andcu no-load losses, πππ , whichf eare addedare in the thermal resistance equivalent circuit, as shown in Figure 3. added inthermal the thermal resistance equivalent as shown in 3. Figure 3. in the resistance equivalent circuit,circuit, as shown in Figure Energies 2017, 10,10, 1843 Energies 2017, 1843 Energies 2017, 10, 1843 of 15 44 of of415 15 Energies 2017, 10, 1843 4 of 15 Figure 3. Thermal resistance equivalent equivalent circuitwith with heatsource. source. Figure Figure 3. 3. Thermal Thermal resistance resistance equivalent circuit circuit with heat heat source. Figure 3. Thermal resistance equivalent circuit with heat source. Previous study showed that thermal resistance tank/radiator could be discarded due Previous study showed that thethe thermal resistance ofof tank/radiator Previous study showed that the thermal resistance of tank/radiatorcould couldbe bediscarded discardeddue duetoto tothe Previous study characteristic showed that the thermal resistance of tank/radiator could be close discarded due[3]. to In the perfect conductor where the thermal resistance was practically to zero the perfect conductor characteristic where the thermal resistance practically close to zero perfect conductor characteristic where the thermal resistance waswas practically close to zero [3].[3]. InIn this thestudy, perfectthe conductor characteristic where the thermal resistance was practically close to zero [3]. In this focus is on the TOT thermal resistance that relates with the convection thermal this study, the focus is on the TOT thermal resistance that relates with the convection thermal study,this thestudy, focus the is on the TOT that relates the with convection thermal thermal resistance of focus is tank. on thermal the TOTresistance thermal resistance thatwith relates thelaw convection resistance of the to the Due this reason, application of of the oil oil to to thethis tank. Due to to this reason, the theof application of Fourier’s Fourier’s law of of heat heat conduction conduction theresistance oil to the tank. Due reason, the application Fourier’s law of heat conduction and the dual resistance of the oil to the tank. Due to this reason, the application of Fourier’s law of heat conduction and the dual phase lag method are not considered in the computation. The final form of the heat and the dual phase lag method are not considered in the computation. The final form of the heat phase lag method are not considered in the computation. The final form of the heat transfer thermal and the dual phase lag method are not considered in the computation. The final form of the heat transfer thermal resistance equivalent circuit can be re-drawn, as shown in Figure 4. transfer thermal resistance equivalent circuit can be re-drawn, as shown in Figure 4. resistance equivalent circuit can be re-drawn, as shown in Figure 4. transfer thermal resistance equivalent circuit can be re-drawn, as shown in Figure 4. Figure 4. Thermal resistance equivalent circuit with a heat source and without the thermal resistance Figure 4. Thermal resistance heat source and without the thermal resistance Figure Thermal resistanceequivalent equivalentcircuit circuit with the thermal resistance Figure 4. 4. Thermal resistance equivalent circuit with aaheat heatsource sourceand andwithout without the thermal resistance of the tank/radiator. of the tank/radiator. of the tank/radiator. of the tank/radiator. 2.3.2.3. Lumped Capacitance Lumped Capacitance Lumped Capacitance 2.3.2.3. Lumped Capacitance The current form of the thermal circuit Figure 4 can be used The current form of the thermalresistance resistance equivalent equivalent circuit inin Figure 4 can onlyonly be used underunder The The current current form form of of the the thermal thermal resistance resistance equivalent equivalent circuit circuit in in Figure Figure 44 can can only only be be used used under under steady state condition. In order to determine HST and TOT under transient, the lumped capacitance steady state condition. In order to determine HST and TOT under transient, the lumped capacitance steady state condition. In order to determine and TOT under transient, the lumped capacitance steady state condition. In order to determine HST and TOT under transient, the lumped capacitance approach was applied.The TheBiot Biotnumber number of the the thermal resistance must bebebe determined before thisthis approach was applied. thermal resistance must determined before this approach was applied. must before approach was applied.The TheBiot Biotnumber number of of the thermal thermal resistance resistance must be determined determined before this approach can be implemented. The Biot number is the ratio of the thermal resistance of conduction approach can be implemented. The Biot number is the ratio of the thermal resistance of conduction approach can be implemented. The Biot number is the ratio of the thermal resistance of conduction approach can be implemented. The Biot number is the ratio of the thermal resistance of conduction over the convection [3,23]. In transformers, the Biot number is less than 1 since the thermal resistance over convection [3,23]. In transformers, number is thermal resistance over thethe convection [3,23]. transformers, the number isless lessthan than111since sincethe the thermal resistance over the convection [3,23].In In transformers, the Biot Biot number is less than since thermal resistance of conduction is normally smaller than convection [3]. Through application of the the lumped capacitance normally smaller thanconvection convection [3]. Through lumped capacitance of of conduction is isnormally than [3]. Throughapplication applicationof the lumped capacitance of conduction conduction normallysmaller smaller convection [3].thermal Through application ofofthe the lumped capacitance method, the is capacitance is addedthan to the heat transfer resistance equivalent circuit which can method, the capacitance is added to the heat transfer thermal resistance equivalent circuit which can method, the capacitance is added to the heat transfer thermal resistance equivalent circuit which can method, the capacitance is 5. added the heat transfer thermal circuit which can be represented by Figure Theretoare two capacitances that areresistance included, equivalent known as oil and winding be represented by Figure 5. There are two capacitances that are included, known as oil and winding be be represented byEach Figure 5.5.There capacitances that are are included, included, known and winding represented by Figure Thereare aretwo two capacitances that known asas oiloil and winding capacitances. of the capacitances represent the storage element of either the oil or winding in capacitances. Each the capacitances represent the storage element either the oil winding capacitances. Each ofof the capacitances represent the storage capacitances. Each of the capacitances represent the storageelement elementofof ofeither eitherthe theoil oiloror orwinding windinginin inthe the thermal resistance equivalent circuit. the resistance equivalent circuit. the thermal thermal resistance equivalent circuit. thermal resistance equivalent circuit. Figure 5. Thermal resistance equivalent circuit with lumped capacitance. Figure 5. Thermal resistance equivalent circuit with lumped capacitance. Figure 5. Figure 5. Thermal Thermal resistance resistance equivalent equivalent circuit circuit with with lumped lumped capacitance. capacitance. 2.4. Thermal-Electrical Analogy The thermal-electrical analogy parameters are through variable, across variable, dissipation element, and storage element [3]. The through variable, heat source, q, in the thermal approach is analogous to current, I in the electrical approach. Meanwhile, the across variable, temperature, θ, in the thermal approach is analogous to voltage, v, in the electrical approach. The dissipation element Energies 2017, 10, 1843 5 of 15 2.4. Thermal-Electrical Analogy The thermal-electrical analogy parameters are through variable, across variable, dissipation element, Energies 2017, 10,and 1843storage element [3]. The through variable, heat source, q, in the thermal approach is5 of 15 analogous to current, I in the electrical approach. Meanwhile, the across variable, temperature, π, in the thermal approach is analogous to voltage, v, in the electrical approach. The dissipation element variable, thermal resistance, , ,ininthe thermalapproach approach is analogous to electrical resistance, variable, thermal resistance,Rπ the thermal is analogous to electrical resistance, π ππ , in Rel , thπ‘β electricalapproach, approach,while while the storage thermal capacitance, πΆπ‘β , C inththe in thethe electrical storageelement elementvariable, variable, thermal capacitance, , inthermal the thermal approach is analogous electrical capacitance, capacitance, πΆππ thethe electrical approach. Based on Figure the approach is analogous totoelectrical C,elin , in electrical approach. Based on5Figure 5 thermal-electrical circuit can be divided into two models, known as the HST and TOT models, as the thermal-electrical circuit can be divided into two models, known as the HST and TOT models, shown in Figure 6a,b, respectively. In this study, only the TOT model was developed and examined. as shown in Figure 6a,b, respectively. In this study, only the TOT model was developed and examined. (a) (b) Figure 6. Thermal-electrical analogy circuit: (a) hot-spot temperature; and (b) top-oil temperature. Figure 6. Thermal-electrical analogy circuit: (a) hot-spot temperature; and (b) top-oil temperature. 3. Top-Oil Temperature Thermal Model 3. Top-Oil Temperature Thermal Model 3.1. Definition of Nonlinear OilOil Thermal 3.1. Definition of Nonlinear ThermalResistance Resistance According to the transfer theory in in [5,23], resistanceisisdefined definedasas the According to heat the heat transfer theory [5,23],the thenonlinear nonlinearoil oil thermal thermal resistance theproduct inverse product of convection coefficient, h andAarea, A which is by given by Equation (1).nonlinear The inverse of convection coefficient, h and area, which is given Equation (1). The nonlinear oil thermal resistance, π πππ (1) in Equation refers to oil-to-tank nonlinear oil-to-tank thermal resistance, oil thermal resistance, Roil in Equation refers to (1) nonlinear thermal resistance, Roil −tank in π πππ−π‘πππ in Figure 6b: Figure 6b: 1 (1) Rπ oilπππ== (1) hA βπ΄ The convection coefficient, h hininEquation beaffected affectedbyby multiple factors, including The convection coefficient, Equation (1) (1) may may be multiple factors, including the the medium thermal characteristics. According to the heat transfer theory, the convection coefficient, h medium thermal characteristics. According to the heat transfer theory, the convection coefficient, varies with temperaturedifference difference between between the tank/radiator) and the h varies with thethe temperature theobject object(transformers (transformers tank/radiator) andfluid the fluid (transformers In this case,h hisisapproximated approximated by in in Equation (2) [23]: (transformers oil) oil) [23].[23]. In this case, bythe theexpression expression Equation (2) [23]: β = πΆ(ππ − π∞ )π h = C (θs − θ∞ )2n (2) (2) where n and C are constants and the unit of C is W/m ·K(1+n). The constant, n, contributes to the nonlinearity of the convection coefficient, h. In Equation (2), π represents the TOT, ππππ , and π∞ where n and C are constants and the unit of C is W/m2 ·K(1+n)π . The constant, n, contributes to the represents the AT, ππ . The TOT rise, βππππ , is determined by subtracting ππ from the ππππ where it nonlinearity of the convection coefficient, h. In Equation (2), θ represents the TOT, θ , and θ∞ s oil can be simplified to Equation (3): represents the AT, θ a . The TOT rise, βθoil , is determined by subtracting θ a from the θoil where it can be simplified to Equation (3): (3) β = πΆ(βππππ )π n h = C βθ ( ) oil to the final proposed equation of nonlinear (3) Substitution of Equation (3) into Equation (1) leads oil thermal resistance as shown in Equation (4): Substitution of Equation (3) into Equation (1) leads to the final proposed equation of nonlinear oil thermal resistance as shown in Equation (4): Roil = 1 CA(βθoil )n (4) The thermal resistance is nonlinear due to the presence of the constant n, which represents the heat transfer convection modes in transformers, as shown in Equation (4). The redefinition of the nonlinear oil thermal resistance based on approximation of convection coefficient, h, in Equation (4) is among the key contributions of this study. A comparison with similar forms of nonlinear thermal resistances such as Susa and Tang models was also carried out. Energies 2017, 10, 1843 6 of 15 The nonlinear thermal resistance for Susa Model can be obtained based on the following approach. The relationship of Nusselt, Prandtle, and Grashof numbers is given by Equation (5) [5]: Nu = C [ Gr · Pr ]n (5) The Nusselt, Prandtle, and Grashof numbers can be determined based on Equations (6)–(8): h· L k (6) coil ·µ k (7) L3 ·ρoil 2 · g · β·(βθoil ) µ2 (8) Nu = Pr = Gr = Substitutions of Equations (6)–(8) into Equation (5) lead to the Equation (9): 3 n h· L L ·ρoil 2 · g· β·(βθoil ) coil ·µ · = C· k k µ2 (9) Since only viscosity is sensitive toward temperature, Equation (9) can be reduced to Equation (10) known as the convection coefficient, h [5]: h = C1 ·( βθoil n ) µ (10) where C1 is obtained from Equation (11): in h 1− n 3n−1 C1 = C · ρoil 2 · g· β·k( n ) · L( n ) ·coil (11) Substitution of Equation (10) into (1) leads to Equation (12) which is the final form of the nonlinear thermal resistance for Susa model [5]: Roil = µn CA(βθoil )n (12) The nonlinear thermal resistance for Tang model can be derived based on the following approach. According to [20,21], the thermal conductance can be represented by Equation (13): G = a(βθoil )b (13) where G is the element of the thermal conductance matrix, βθoil is the TOT rise, a and b are the constants. The reciprocal of Equation (13) leads to the final form of nonlinear thermal resistance for Tang model, as shown in Equation (14) [20,21]: Roil = 1 a(βθoil )b (14) The proposed nonlinear thermal resistance does not consider viscosity as in the Susa model. The nonlinear thermal resistance in the Susa model considers multiple oil parameters, such as characteristic length, L, oil thermal conductivity, k, specific heat of oil, coil , oil density, ρoil , gravitational constant, g, and oil thermal expansion coefficient, β, while the proposed nonlinear thermal resistance only considers the constant C that is derived from the heat transfer theory. The derivation of nonlinear thermal resistance in the Susa model is different from the proposed nonlinear thermal resistance where it is based on Nusselt, Prandtle, and Grashof numbers. Meanwhile, the nonlinear thermal Energies 2017, 10, 1843 7 of 15 resistance in the Tang model is derived based on the exponential form of heat conductance matrix element. The constants, a and b in the Tang model are obtained based on GA through the in-service measured data. 3.2. Derivation of Top-Oil Temperature Thermal Model Based on the TOT thermal-electrical analogy equivalent circuit in Figure 6b, the differential equation representation can be determined based on Equation (15) [5]: q f e + qcu = Coil dθoil (θ − θ a ) + oil dt Roil (15) Substitution of Equation (4) into Equation (15) leads to Equation (16): q f e + qcu = Coil dθoil (θ − θ a ) + oil 1 dt n (16) CA(βθoil ) The nonlinear oil thermal resistance at rated, Roil, rated , TOT rise at rated, βθoil,rated , and oil time constant, τoil , can be determined according to Equations (17)–(19): Roil, rated = 1 CA(βθoil,rated )n βθoil,rated = q f e + qcu rated · Roil,rated τoil = Roil,rated ·Coil (17) (18) (19) The ratio of load losses at the rated to the no load losses, R can be computed according to Equation (20): qcu R= (20) qfe The ratio of the load current to the load current at rated, K can be calculated according to Equation (21): I K= (21) Irated Substitutions of Equations (17)–(21) into Equation (16) lead to Equation (22), which is the final form of the TOT thermal model: 1 + R·K2 dθ ( θ − θ a )1+ n ·βθoil,rated = τoil oil + oil 1+R dt βθoil,rated n (22) The TOT thermal model in Equation (22) is in the same form as in [8], however the approach in terms of derivation of the TOT thermal model is different. The nonlinear oil thermal resistance, Roil , in [8] was obtained through Nusselt, Prandtle, and Grashof numbers. The TOT model in [8] also took into consideration on the influence of oil viscosity in the TOT thermal model. In addition, the input parameters such as the TOT rise at rated, βθoil,rated , oil time constant, τoil and constant, n, in [8] were estimated based on the Levenberg-Marquardt method. The TOT thermal model in Equation (22) does not require the information on the viscosity for determination of nonlinear oil thermal resistance, Roil where simplification is carried out through approximation of convection coefficient, h, of the heat transfer convection between the oil and the transformer tank. 3.3. Input Data The TOT thermal model was tested on seven transformers with ONAN and ONAF cooling modes. The seven transformers were named as TX1, TX2, TX3, TX4, TX5, TX6, and TX7. In addition, the TOT Energies 2017, 10, 1843 8 of 15 thermal model was also tested on the step loading of a 250 MVA transformer with ONAF cooling mode obtained from [1,5] and named as TX8. Both TX3 and TX4 have identical design except the cooling mode for TX3 is ONAN and TX4 is ONAF. The input parameter for these transformers ratings are shown in Table 1 for constant loading and Table 2 for step loading. The oil time constant, τoil , was calculated based on [2] using the input parameters obtained from the nameplate rating such as the weight of core and coil in kg, the weight of the tank and fittings in kg, and transformer oil in L. Based on the convection heat transfer, the value of constant, n is 0.25 for laminar oil circulation [27], which indicates free convection (ONAN cooling modes). It is used to control the sensitivity of the TOT model curve which depends on the cooling modes and oil circulation of transformers. The constant, n, has different values for forced convection (ONAF and OFAF cooling modes) [5,6]. In this study, the constant n for TE1 and TE2 were calibrated based on the measured TOT in order to increase the accuracy of the simulated TOT. The same procedure was carried out for the TOT thermal model, which can be seen in Table 3. Table 1. Input parameters for the TOT thermal model. Transformer/Winding Parameters TX1 TX2 TX3 TX4 TX5 TX6 TX7 11/0.433 kV 33/11 kV 33/11 kV 33/11 kV 132/11 kV 132/33 kV 132/33 kV 300 ONAN 1.00 4.52 30.3 129 31.9 0.8 0.5 0.8 15,000 ONAF 1.00 7.56 51.23 152 44.9 0.8 0.5 0.9 30,000 ONAN 1.00 10.47 59.2 200 29.7 0.8 0.5 0.8 30,000 ONAF 1.00 11 58.87 207 29.9 0.8 0.5 0.9 30,000 ONAN 1.00 5.79 38.9 239 31.8 0.8 0.5 0.8 60,000 ONAN 1.00 4.66 44.07 295 32.3 0.8 0.5 0.8 90,000 ONAF 1.00 10 47.87 165 41.7 0.8 0.5 0.9 Rating (kVA) Cooling Modes Load (p.u.) R βθoil,rated (K) τoil (min) θoil,i (K) (IEC) Oil exponent, x (IEC) Constant, k11 (IEEE) Exponent, n Table 2. Input parameter for TOT thermal model for step loading [1,5]. Transformer/Winding Parameters TX8 250/118 kV Rating (kVA) Cooling Modes 250,000 ONAF 1.00 pu/3 h + 0.60 pu/3 h + 1.50 pu/2 h + 0.30 pu/3 h + 2.10 pu/0.33 h 10.17 38.3 168 38.3 Load (p.u.) R βθoil,rated (K) τoil (min) θoil,i (K) Table 3. Constant n used in this study. Transformers TX1 TX2 TX3 TX4 TX5 TX6 TX7 TX8 Constant n TE1 TE2 Thermal Model 0.1 0.75 0.75 0.75 0.4 0.4 0.25 0.1 1 0.9 0.9 0.9 0.95 0.95 0.95 0.95 0.1 1 1 1 0.65 0.65 0.5 0.25 Energies 2017, 10, 1843 9 of 15 3.4. Comparison with Previous Thermal-Electrical and Standard Models In this study, two thermal-electrical models in [3,5] were used to evaluate the performance of the TOT thermal model. Thermal-Electrical 1 (TE1) model was proposed by [5] as seen in Equation (23). The model in Equation (24) was introduced by [3] and termed as Thermal-Electrical 2 (TE2) model: 1 + R·K2 dθ ( θ − θ a )1+ n ·µ pu n ·βθoil,rated = µ pu n ·τoil oil + oil 1+R dt βθoil,rated n (23) 1 1 1 + R·K2 dθ · (βθoil,rated ) n = τoil oil + (θoil − θ a ) n 1+R dt (24) The TOT thermal model was also compared with the existing models in standards known as the Exponential model in IEC 60076-7 and the Clause 7 model in IEEE C57.91-1995, as seen in Equations (25) and (26) [1,2]. The constants, x, k11 , and n used for both the Exponential and Clause 7 thermal models were chosen based on Table 1. ! x 1 + R·K2 θoil = θ a + βθoil,i + βθoil,rated · − βθoil,i · f 1 (25) 1+R −t θoil = θ a + (βθ TO,U − βθ TO,i )· 1 − e τoil + βθ TO,i (26) Maximum error and root mean square (RMS) error were used to analyze the simulated TOT, as can be seen in Equations (27) and (28): maximum error = θoil, calculated − θoil, measured s RMS error = 2 ∑in=1 (θoil, calculated − θoil, measured ) n (27) (28) where θoil, measured is the measured TOT, θoil, simulated is the simulated TOT, and n is the number of samples. 4. Results and Discussions 4.1. Constant Loading The comparison between the TOT thermal model and the existing TOT Thermal-Electrical, Exponential (IEC 60076-7) and Clause 7 (IEEE C57.91-1995) models can be seen in Figures 7–13. For TX1, the simulated TOT for all thermal models overshoot during the first 300 min of the temperature rise test as seen in Figure 7. The maximum errors for the TOT thermal, TE1, TE2, Exponential, and Clause 7 models are 5.65 β¦ C, 5.98 β¦ C, 5.03 β¦ C, 4.79 β¦ C, and 7.92 β¦ C. Meanwhile, the RMS errors are 2.72 β¦ C, 2.89 β¦ C, 2.41 β¦ C, 2.23 β¦ C, and 4.02 β¦ C. The measured TOT for TX2 is well represented by the TOT thermal, TE1, TE2, and Exponential models, as shown in Figure 8. However, the simulated TOT based on the Clause 7 model is lower than the measured TOT during the first 300 min of the temperature rise test. The maximum errors of the TOT thermal, TE1, TE2, Exponential, and Clause 7 models are 3.72 β¦ C, 2.77 β¦ C, 2.09 β¦ C, 2.15 β¦ C, and 12.92 β¦ C, while the RMS errors are 1.34 β¦ C, 1.33 β¦ C, 0.97 β¦ C, 1.03 β¦ C, and 5.33 β¦ C. The simulated TOT based on TE2 and Exponential models are much closer to the measured TOT than the TOT thermal, TE1, and Clause 7 models for TX3. The same pattern is observed for TX4 and it is expected since both transformers have the same design, except for its cooling modes. For TX3, the maximum errors for TOT thermal, TE1, TE2, Exponential, and Clause 7 models are 6.53 β¦ C, 6.32 β¦ C, 0.91 β¦ C, 3.30 β¦ C, and 8.25 β¦ C, while for TX4, the values are 6.66 β¦ C, 5.71 β¦ C, 3.35 β¦ C, 8.37 β¦ C, and 8.24 β¦ C. On the other hand, the RMS errors of TX3 for the TOT thermal, TE1, TE2, Exponential, and Clause 7 Energies 2017, 10, 1843 10 of 15 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 0 0 Clause 7 Clause TE1 7 TE1 Model Thermal Thermal TE2 Model Exponential TE2 Exponential 100 100 200 200 300 300 400 500 600 (min) 600 400 Time500 Time (min) 24 24 20 20 16 16 12 12 8 8 4 4 0 0 -4 1000 -4 1000 Measured Thermal MeasuredModel TE1 Thermal Model TE2 TE1 IEC TE2Exponential IEEE Clause 7 IEC Exponential Error IEEE Clause 7 Error 700 700 800 800 900 900 Error Temperature Error Temperature (°C)(°C) Top-oil Temperature Top-oil Temperature (°C)(°C) models are 2.76 β¦ C, 2.49 β¦ C, 0.56 β¦ C, 2 β¦ C, and 5.17 β¦ C, while for TX4, the values are 2.36 β¦ C, 2.28 β¦ C, 1.38 β¦ C, 3.61 β¦ C, and 5.20 β¦ C, respectively. The TOT thermal, TE1, and TE2 models can represent the measured TOT for TX5 quite well as compared with the Exponential and Clause 7 models. The maximum and RMS errors for TOT thermal Energies 2017, 10, 1843 10 of 15 β¦ β¦ model are10, 1.43 Energies 2017, 1843C and 0.67 C. For TE1, TE2, Exponential and Clause 7 models, the maximum 10errors of 15 β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ are 1.42 C, 1.64 C, 7.44 C, and 5.30 C, while the RMS errors are 0.70 C, 1.04 C, 3.72 C, and 2.65 Clause 7 model for TX7 is lower than the measured TOT during the initial stage of the temperatureC, respectively. same pattern is observed for TX6 and except the simulated based the Clause model for TX7 is lower than the measured TOT during the initial stage of the temperature rise test.7 On theThe other hand, the simulated TOT based onTX7, the Exponential model is TOT higher thanon the Clause model for TX7 lower than measured during stage the temperature rise test. 7On theduring other hand, the500 simulated TOT basedTOT onerrors the Exponential isofhigher thanTE2, the measured TOT theisfirst min.the The maximum for the TX6initial for model TOT thermal, TE1, rise test. On the other hand, the simulated TOT based on the Exponential model is higher than the measured TOT during the first 500 min. The maximum errors for TX6 for TOT thermal, TE1, TE2, Exponential, and Clause 7 models are 2.01 °C, 1.98 °C, 2.85 °C, 8.04 °C, and 9.04 °C, while for TX7, measured TOT during the first 500 min. The maximum errors for TX6 for TOT thermal, TE1, TE2, Exponential, 7 models 2.01 1.98 °C, °C, 8.04 °C, 9.04 °C, while the values areand 2.14Clause °C, 2.03 °C, 2 °C,are 6.67 °C,°C, and 6.84 °C.2.85 Meanwhile, theand RMS errors of TX6for forTX7, the β¦ C, 1.98 β¦ C, 2.85 β¦ C, 8.04 β¦ C, and 9.04 β¦ C, while for TX7, Exponential, and TE2, Clause 7°C, models are the values areTE1, 2.14 °C, 2.03 2 °C, 6.67 °C, and °C. Meanwhile, errors TX6°C, forand the TOT thermal, Exponential, and2.01 Clause 76.84 models are 0.81 °C,the 0.72RMS °C, 1.54 °C,of4.54 β¦ C, 2.03 β¦ C, 2 β¦ C, 6.67 β¦ C, and 6.84 β¦ C. Meanwhile, the RMS errors of TX6 for the values are 2.14 TOT thermal, TE1, TE2, Exponential, and Clause 7 models are 0.81 °C, 0.72 °C, 1.54 °C, 4.54 °C, and 4.24 °C, while for TX7, the values are 0.95 °C, 0.88 °C, 1 °C, 3.39 °C, and 4.33 respectively. β¦ C, 0.72 β¦ C, 1.54 β¦ C, 4.54 β¦ C, the°C, TOT thermal, TE1, TE2, Exponential, and Clause models areand 0.814.33 4.24 while for TX7, the values are 0.95 °C, 0.88 °C, 1 7°C, 3.39 °C, °C, respectively. and 4.24 β¦ C, while for TX7, the values are 0.95 β¦ C, 0.88 β¦ C, 1 β¦ C, 3.39 β¦ C, and 4.33 β¦ C, respectively. 90 90 80 80 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 0 0 Exponential Thermal Model Exponential TE2 Thermal Model TE2 TE1 Clause 7 TE1 Clause 7 100 100 200 200 300 300 400 500 600 (min) 600 400 Time500 Time (min) 700 700 48 48 40 Measured 40 Thermal MeasuredModel 32 32 TE1 Thermal Model 24 TE2 TE1 24 IEC TE2Exponential 16 IEEE Clause 7 IEC Exponential 16 Error IEEE Clause 7 8 8 Error 0 0 -8 -8 -16 -16 -24 800 900 1000 -24 800 900 1000 Figure TOT based different thermal models TX2. Figure 8. 8. TOT based onon different thermal models forfor TX2. Figure 8. TOT based on different thermal models for TX2. Error Temperature Error Temperature (°C)(°C) Top-oil Temperature Top-oil Temperature (°C)(°C) Figure 7. TOT based on different thermal models for TX1. Figure TOT based different thermal modelsfor forTX1. TX1. Figure 7. 7. TOT based onon different thermal models Energies 2017, 10, 1843 Energies 2017, 10, 1843 Energies Energies2017, 2017,10, 10,1843 1843 Exponential Exponential Exponential Clause 7 Clause 7 TE1 Clause TE2 7TE1 TE2 TE1 Thermal Model Thermal Model TE2 Thermal Model 100 100 100 200 200 200 300 300 300 400 500 600 400 Time 500 (min) 600 400 Time 500(min) 600 Time (min) 64 64 Measured 64 Measured Thermal Model 56 Measured Thermal Model 56 TE1 Model 5648 Thermal TE1 48 TE2 TE1 4840 TE2 IEC Exponential TE2 40 IEC Exponential IEEE Clause 7 4032 IEC Exponential IEEE Clause 7 32 Error IEEE ErrorClause 7 3224 Error 24 2416 16 168 8 80 0 0 -8 -8 -8-16 800 900 1000 -16 800 900 1000 -16 800 900 1000 Error Temperature (°C) Error Temperature (°C) Error Temperature (°C) Top-oil Temperature (°C) Top-oil Temperature (°C) Top-oil Temperature (°C) 100 100 10090 90 9080 80 8070 70 7060 60 6050 50 5040 40 4030 30 3020 20 2010 10 10 0 00 0 0 0 11 of 15 11 of 15 1111ofof1515 700 700 700 Figure 9. TOT based on different thermal models for TX3. Figure 9. TOT based on different thermal models for TX3. Figure9.9.TOT TOTbased basedonondifferent differentthermal thermalmodels modelsfor forTX3. TX3. Figure Exponential Exponential TE2 Exponential TE2 TE2 TE1 TE1 TE1 100 100 100 200 200 200 Thermal Model Thermal Model Thermal Model Clause 7 Clause 7 Clause 7 300 300 300 400 500 600 400 Time 500 (min) 600 400 Time 500(min) 600 Time (min) 80 80 Measured 80 Measured 70 Thermal Model 70 Measured Thermal Model 70 TE1 Thermal Model 60 TE1 60 TE2 TE1 60 TE2 IEC Exponential 50 TE2 50 IEC Exponential IEEE Clause 7 5040 IEC Exponential IEEE Clause 7 40 Error IEEE ErrorClause 7 4030 Error 30 3020 20 2010 10 100 0 0 -10 -10 -10 -20 800 900 1000 -20 800 900 1000 -20 800 900 1000 Error Temperature (°C) Error Temperature (°C) Error Temperature (°C) Top-oil Temperature (°C) Top-oil Temperature (°C) Top-oil Temperature (°C) 100 100 10090 90 9080 80 8070 70 7060 60 6050 50 5040 40 4030 30 3020 20 2010 10 10 0 00 0 0 0 700 700 700 Figure 10. TOT based on different thermal models for TX4. Figure 10. 10. TOT TOT based based on on different different thermal thermal models Figure models for for TX4. TX4. Figure 10. TOT based on different thermal models for TX4. Clause 7 Clause 7 Exponential Clause 7 Exponential Thermal Model Exponential TE2 Thermal Model TE2 Thermal Model TE1 TE2 TE1 TE1 100 100 100 200 200 200 300 300 300 400 500 600 400 Time 500 (min) 600 400 Time 500(min) 600 Time (min) 28 28 Measured 28 Measured Thermal Model 24 Measured Thermal Model 24 TE1 Model Thermal 24 TE1 TE2 20 TE1 TE2 20 IEC Exponential TE2 20 IEC Exponential IEEE Clause 7 16 IEC Exponential 16 IEEE Clause 7 Error 16 IEEE ErrorClause 7 12 Error 12 12 8 8 8 4 4 4 0 0 0 -4 800 900 1000 -4 800 900 1000 -4 800 900 1000 Error Temperature (°C) Error Temperature (°C) Error Temperature (°C) Top-oil Temperature (°C) Top-oil Temperature (°C) Top-oil Temperature (°C) 80 80 80 70 70 70 60 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 10 10 0 00 0 0 0 700 700 700 Figure 11. TOT based based on on different different thermal Figure 11. TOT thermal models models for for TX5. TX5. Figure 11. TOT based on different thermal models for TX5. Figure 11. TOT based on different thermal models for TX5. Energies2017, 2017,10, 10,1843 1843 Energies Energies 2017, 10, 1843 12of of15 15 12 12 of 15 56 56 Measured Measured Thermal Model 48 Thermal Model 48 TE1 TE1 TE2 40 TE2 40 IEC Exponential IEC Exponential 32 IEEE Clause 7 IEEE Clause 7 32 Error Error 24 24 Top-oil C)C) Top-oilTemperature Temperature(°(° 70 70 60 60 50 50 Clause 7 Clause 7 Exponential Exponential 40 40 TE1 Thermal Model TE1 Thermal Model TE2 TE2 30 30 20 20 16 16 8 8 10 10 0 0 0 0 Error C)C) ErrorTemperature Temperature(°(° 80 80 0 0 100 100 200 200 300 300 400 500 600 400 Time 500(min) 600 Time (min) 700 700 800 800 900 900 -8 1000 -8 1000 Figure12. 12. TOT TOTbased basedon ondifferent differentthermal thermalmodels modelsfor forTX6. TX6. Figure Figure 12. TOT based on different thermal models for TX6. Exponential Exponential TE1 TE1 Thermal Model Thermal Model TE2 TE2 Clause 7 Clause 7 100 100 200 200 300 300 400 500 600 400 Time 500(min) 600 Time (min) 28 28 Measured Measured Thermal Model 24 Thermal Model 24 TE1 TE1 20 TE2 20 TE2 IEC Exponential 16 IEC Exponential 16 IEEE Clause 7 IEEE Clause 7 12 Error 12 Error 8 8 4 4 0 0 -4 -4 -8 800 900 1000 -8 800 900 1000 Error C)C) ErrorTemperature Temperature(°(° Top-oil C)C) Top-oilTemperature Temperature(°(° 90 90 80 80 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 0 0 700 700 Figure Figure13. 13. TOT TOTbased basedon ondifferent differentthermal thermalmodels modelsfor forTX7. TX7. Figure 13. TOT based on different thermal models for TX7. 4.2. Step Step Loading Loading 4.2. 4.2. Step Loading The performance performance of of the the TOT TOT thermal thermal model model is is also also tested tested on on step step loading loading of of TX8 TX8 as as shown shown in in The The performance of the TOT thermal model is also tested on step loading of TX8 as shown in Figure14. 14.The Thesimulated simulatedTOT TOT based TOT thermal models are quite to measured Figure based onon TOT thermal andand TE1TE1 models are quite closeclose to measured TOT. Figure 14. The simulated TOT based on TOT thermal and TE1 models are quite close to measured TOT. The simulated TOT on based both and Exponential models overshoot slightly overshoot the The simulated TOT based bothon TE2 andTE2 Exponential models slightly once the once loading TOT. The simulated TOT based on both TE2 and Exponential models slightly overshoot once the loading increases 0 to and 1.0 p.u. and to On 1.5 p.u. On the otherthe hand, the simulated TOTon based increases from 0 tofrom 1.0 p.u. 0.6 to 1.50.6 p.u. the other hand, simulated TOT based the loading increases from 0 to 1.0 p.u. and 0.6 to 1.5 p.u. On the other hand, the simulated TOT based on the 7Clause slightly lower than measured TOT the same loading period. A Clause model 7ismodel slightlyislower than measured TOT during theduring same loading period. A significant on the Clause 7 model is slightly lower than measured TOT during the same loading period. A significantbetween deviation between simulated based onmodel Exponential model TOT and measured TOT deviation simulated TOT based onTOT Exponential and measured is found once theis significant deviation between simulated TOT based on Exponential model and measured TOT is found once the loading decreases from to 0.3 p.u. The for maximum for TE1, the TOT TE1, loading decreases from 1.5 to 0.3 p.u. The1.5 maximum errors the TOTerrors thermal, TE2,thermal, Exponential, found once the loading decreases from 1.5 to 0.3 p.u. The maximum errors for the TOT thermal, TE1, β¦ C, while TE2,Clause Exponential, andare Clause models are4.22 5.77β¦ °C, 5.70β¦°C, 4.22 7.76 °C, 7.70 °C, and °C,errors while are the and 7 models 5.77 β¦7C, 5.70 β¦ C, C, 7.70 C, and the7.76 RMS TE2, β¦Exponential, and Clause 7 models are 5.77 °C, 5.70 °C, 4.22 °C, 7.70 °C, and 7.76 °C, while the β¦ C,°C, β¦ C,1.91 β¦ C,°C, RMS C, errors 2.02 2 °C, and 2.84 °C, respectively. 2.02 2 β¦ C,are 1.91 4.78 and°C, 2.844.78 respectively. RMS errors are 2.02 °C, 2 °C, 1.91 °C, 4.78 °C, and 2.84 °C, respectively. Energies Energies 2017, 2017, 10, 10, 1843 1843 13 of of 15 15 13 40 Measured Thermal Model TE1 TE2 IEC Exponential IEEE Clause 7 Error Top-oil Temperature (°C) 90 80 70 60 50 40 35 30 25 20 15 Exponential TE2 Thermal Model TE1 10 Clause 7 30 5 20 0 10 -5 0 0 100 200 300 400 500 600 Time (min) 700 800 900 Error Temperature (°C) 100 -10 1000 Figure 14. TOT based on different thermal models for TX8. 4.3. Discussions Under constant loadings of seven transformers, the highest maximum maximum and RMS errors for the TOT thermal model are 6.66 β¦°C C and 2.76 β¦°C, C, which are lower than Exponential and Clause 7 models where the highest maximum errors are 8.37 β¦°C C and 12.92 β¦°C C while the highest RMS error are 4.54 β¦°C C β¦ and 5.33 5.33 °C, respectively. The performance of TE1 TE2 models are slightly better than TOT C, respectively. The performance of TE1 and and TE2 models are slightly better than TOT thermal β¦ C while thermal modelthe where themaximum highest maximum 6.325.03 °C and 5.03 °C theRMS highest RMS model where highest errors areerrors 6.32 β¦are C and thewhile highest error are β¦ β¦ error 2.892.41 °C and °C. Meanwhile, under step loading, thethermal TOT thermal performs 2.89 are C and C. 2.41 Meanwhile, under step loading, the TOT modelmodel performs betterbetter than than Exponential and Clause 7 models, and comparable TE1TE2 and models. TE2 models. The model TE1 model is Exponential and Clause 7 models, and comparable with with TE1 and The TE1 is able able to represent the TOT better than the TOT thermal model owing to its comprehensive to represent the TOT better than the TOT thermal model owing to its comprehensive consideration consideration onHowever, the viscosity. due to the minimal information by the TOT on the viscosity. due However, to the minimal information requirement byrequirement the TOT thermal model, thermal model, the performance could be considered as reasonable as alternative approach the performance could be considered as reasonable as alternative approach to determine the TOT to of determine theFurthermore, TOT of transformers. Furthermore, the TOT thermal transformers model is able to different evaluate transformers. the TOT thermal model is able to evaluate with transformers different cooling modesτoil since the oil time constant,toπthe is calculated according to cooling modeswith since the oil time constant, is calculated according respective cooling modes. πππ the respective cooling modes. 5. Conclusions 5. Conclusions Based on the proposed pathway of the energy transfer concept and thermal-electrical analogy through utilization of the nonlinear thermal and lumped capacitance method, an alternative Based on the proposed pathway of theresistance energy transfer concept and thermal-electrical analogy approachutilization to determine is developed. key component of the TOT thermal model is the through of the theTOT nonlinear thermalThe resistance and lumped capacitance method, an of the convection h, proposed on heatoftransfer theory that alternative approximation approach to determine the TOTcoefficient, is developed. The keybased component the TOT thermal eliminates the alternative requirementapproximation of the viscosity.ofThe thermal model was tested on seven transformers model is the theTOT convection coefficient, h, proposed based on heat with either ONAN ONAF cooling modes. Based the caseThe study, thethermal TOT thermal model can transfer theory that or eliminates the requirement of theon viscosity. TOT model was tested β¦C represent the measured TOT quite well where the cooling highestmodes. maximum and errors arethe 6.66 on seven transformers with either ONAN or ONAF Based on RMS the case study, TOT β¦ C, respectively. Under step loading, the performance of TOT thermal model is comparable and 2.76 model thermal can represent the measured TOT quite well where the highest maximum and RMS with types°C ofand thermal where the Under simulated is quite to measured TOT thermal and the errorsother are 6.66 2.76 models °C, respectively. stepTOT loading, theclose performance of TOT β¦ C and 2.02 β¦ C. Overall, the performance of the TOT thermal model maximum and RMS errors are 5.77 model is comparable with other types of thermal models where the simulated TOT is quite close to is comparable and TE2 models better than°C theand Exponential and Clause 7 models. measured TOTwith and the the TE1 maximum and RMS and errors are 5.77 2.02 °C. Overall, the performance of the TOT thermal model is comparable with the TE1 and TE2 models and better than the Acknowledgments: The authors would like to thank Ministry of Education, Universiti Pertahanan Nasional Exponential Clause Putra 7 models. Malaysia andand Universiti Malaysia for funding under Putra IPB schemes, (GP-IPB/2014/9440801). Special thanks to Malaysia Transformer Manufacturing Sdn. Bhd. for the technical support. Acknowledgments: The authors would like to thank Ministry of Education, Universiti Pertahanan Nasional Author Contributions: The research study was carried out successfully with contribution from all authors. Malaysia Universiti Putra Malaysia under Putra IPB were schemes, (GP-IPB/2014/9440801). Special The main and research idea, simulation worksfor andfunding manuscript preparation contributed by Muhammad Hakirin thanks Malaysia Transformer Sdn.preparation Bhd. for the technical Roslan.to Norhafiz Azis contributedManufacturing on the manuscript and researchsupport. idea. Mohd Zainal Abidin Ab Kadir, Jasronita Jasni and Zulkifli Ibrahim assisted on finalizing the research work and manuscript. Azalan Ahmad Author Contributions: The research study was carried out successfully with contribution from all authors. The main research idea, simulation works and manuscript preparation were contributed by Muhammad Hakirin Roslan. Norhafiz Azis contributed on the manuscript preparation and research idea. Mohd Zainal Abidin Ab Energies 2017, 10, 1843 14 of 15 gave several suggestions from the industrial perspectives. All authors revised and approved the publication of the paper. Conflicts of Interest: The authors declare no conflict of interest. Nomenclature A C Coil Cwdg f1 h I K n qcu qfe R Rwdg R paper R paper−oil Roil −tank Rtank/radiator Roil Roil,rated Rth−hs θa θoil θhs θoil,i θoil, calculated θoil, measured βθoil,rated βθ TO,U βθ TO,i τoil µ pu Area Parameter with unit W/m2 ·K(1+n) Oil thermal capacitance Winding thermal capacitance Increase exponent function of Top-oil temperature Convection coefficient Load current Load factor A constant Heat generated by load losses Heat generated by no load losses Ratio of load losses at rated current to no load losses Winding thermal resistance Paper thermal resistance Nonlinear paper to oil thermal resistance Nonlinear oil to tank thermal resistance Tank/radiator thermal resistance Nonlinear oil thermal resistance Nonlinear oil thermal resistance at rated Hot-spot thermal resistance Ambient temperature Top-oil temperature Hot-spot temperature Initial value of Top-oil temperature Simulated value of Top-oil temperature Measured value of Top-oil temperature Top-oil temperature rise at rated Ultimate Top-oil temperature rise Initial Top-oil temperature rise Oil time constant Per unit value of oil viscosity References 1. 2. 3. 4. 5. 6. 7. 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