Improved thermal network model for hot spot location in traction transformer windings

Obtaining accurate location information of the hot spot in advance is a prerequisite for locating the hot spot. To tackle this problem, we propose an improved thermal network model, which provides a quantitative method for placing fiber optic sensors at hot-spot locations. The initial step is to construct an enhanced thermal network model that takes into account the thermal resistance of the boundary layer using heat transfer theory. Then, we compare this model with Computational Fluid Dynamics (CFD) in example calculations to validate its effectiveness and precision. The results show that our proposed model can accurately and rapidly calculate temperature distribution within winding areas, enabling the identification of hot spots. This information is reliable for installing monitoring equipment to measure transformer temperatures.


Introduction
The traction transformer is an essential energy conversion device that significantly affects the train's running state.And its cooling method is usually directed forced oil.As of the end of 2022, China has a total operating mileage of 155,000 kilometers for railways, with high-speed rail accounting for 42,000 kilometers.Chinese transportation system heavily relies on railway transportation, which serves as its backbone.Due to its significance, any negligence in monitoring the status of transformers can cause serious damage to life and property.For instance, the fire accident caused by the traction transformer failure on train G281 on January 25, 2018 has attracted great attention from industry personnel.It is now clear that further research into the thermal characteristics of transformers is necessary.Therefore, this paper will focus on conducting thermal analysis of transformers, comparing various methods' advantages and disadvantages, and selecting the most effective research approach to ensure both speed and reliability in the research process.
There are two main methods for obtaining advanced hot-spot location information of transformer windings: numerical calculation and thermal network modeling.In order to study the hot spot temperature of transformers, Xu Jing et al. used CFD software to establish a multi-physics field heating model of fluid-solid-thermal coupling oil-immersed transformers and analyzed the thermal characteristics of transformers.However, despite their accuracy and precision, these methods require significant computing resources, which can reduce time efficiency and limit their practical applications [1]   .
The thermal network model method is another approach for analyzing thermal characteristics.It offers both computational efficiency and accuracy, making it a favorable alternative to CFD.This method relies on the principles of heat transfer and thermal-electric analogy.Yifan Chende et al. proposed a heat network model considering electromagnetic loss; the distribution characteristics of the heat source inside the dry transformer along the cooling pipe were also taken into consideration. [2].However, the modeling process may not be comprehensive enough to include all necessary parameters such as thermal resistance, which can result in decreased accuracy during calculations [3] .It is precisely this characteristic of the thermal network that researchers are constantly improving it.
In this paper, we propose an enhanced thermal network model that incorporates the thermal resistance of the boundary layer to measure and calculate the hot spot temperature in the transformer winding region.This model combines the advantages of CFD and the thermal network model.We also compare it with CFD to verify its effectiveness.

Transformer cooling oil unit thermal network model with winding insulation paper
In traction transformer windings, the cooling oil flow is laminar due to the structure of the oil pump and oil baffle.

Thermal Resistance of Boundary layer
The cooling oil in the transformer winding area has high viscosity, resulting in a thin boundary layer near the surface of the insulation paper.Although small, this boundary layer significantly affects heat transfer through oil flow in this region.In our thermal network model for the transformer winding area, we assume that the velocity of oil flow in the channel is much greater than that of the boundary layer.The thickness of the boundary layer can be determined by using the local similarity solution method on the Blasius equation [4][5]  In parallel plate oil flow, its value is twice the distance between parallel plates [6] .Lastly, 0  represents the dynamic viscosity of oil flow.

Thermal Resistance of Convection
The convective thermal resistance on both sides of the cooling oil flow channel can be denoted as The equation group [7][8] has been selected to calculate the required local Nusselt number value based on the oil flow characteristics in the winding area of oil-immersed transformers: where z + represents non-dimensional distance:

Thermal network model of winding insulation paper-conductor unit
The copper loss in the winding conductor generates heat, which is then transferred to the surface of the insulation paper through thermal conduction.Figure 2 shows the thermal network model for the insulation paper-conductor unit in a winding.In the insulation paper-conductor unit of winding, thermodynamics laws dictate that the conduction thermal resistance can be represented by where pi d − is the thickness of insulating paper on the surface of conductor unit i; i A represents the area perpendicular to the direction of heat flow on the surface of conductor unit i; and p G denote the thermal conductivity coefficient of insulating paper material.

Transformer winding area thermal network topology model
Figure 3 shows a thermal network topology model of the transformer winding area with r blocks of wire pancakes and Q turns per block, where there are P×Q blocks of conductors in total (P, Q=1, 2, 3, ...).The process of solving both thermal and electrical circuits is similar, using the nodal voltage method concept.To illustrate this approach, we consider the first coil as an example where we list down the temperature-solving equation for each node in the winding thermal network: In the given equation, n=11 denotes the total number of nodes present in a set of winding conductors within a linear coil.According to sections 1.3, 1.4, and 1.5, the thermal resistance values are calculated, and equations are solved.

CFD Modeling Example
The research in this section focuses on a 6.3 MVA oil-immersed traction transformer with a rated voltage of 25 kV/1.5 kV and a low-voltage winding.Specifically, we will examine one of the cooling channels as an example and conduct research and verification on hot spot positioning in the winding area using the specific structures and dimensions shown in Figure 4.The physical parameters for each part of the cooling channel are provided in Table 1.

CFD Numerical Analysis
The model's boundary conditions are as follows: the oil velocity at the entrance transformer is 0.2 m/s, and the temperature is 300 K.Each turn of wire has a heating power of 99.79 W. The outlet is set as a pressure outlet with a defined value of 0. Insulation sleeves and oil guide plates are set as adiabatic boundaries based on thermal conductivity material characteristics.The entire simulation process took approximately 93 minutes.The results indicate that the hottest spot was found in the eighth turn conductor of the second line cake on the oil inlet side, reaching a temperature of 348.500K. Figure 5 displays the distribution of winding conductor temperatures, while Figure 6 shows the oil flow velocity distribution.As shown in Figure 6, there is a layer of almost zero velocity oil flow near the surface of the insulating paper on both sides of the oil flow, whether it is in the vertical or horizontal oil ducts.This proves the existence of the boundary layer in the oil flow.

Validation of the Enhanced Thermal Network Model
In this article, we have developed an enhanced thermal network model that allows us to accurately determine the cooling channel on a single device within a mere 4.88 seconds.Our findings have verified the hotspot location identified through CFD analysis, which is situated at the second disk and eighth turn conductor on the oil inlet side.Figure 7 illustrates a comparison of temperature distribution calculation results for each disk, while Table 2 presents error analysis outcomes.It should be noted that disks and conductors are numbered starting from 1, with the oil inlet serving as the reference point.. From Figure 7, it is evident that the temperature values and trends obtained through the improved thermal network model presented in this paper are largely consistent with those from CFD simulations.The program-calculated hotspot temperature of 350.953K differs from the maximum error in CFD simulation results by only 2.404 K. Further details on calculation errors at other positions can be found in Table 2.The table data indicates that the average temperature's MAE in each disk is not more than 1.83 K. Additionally, the AE_Hot-Spot falls within 2.82 K.The temperature's MArE is below 0.54%, and its MSE is 4.0687, while the eMSE of temperature is 2.0171.The e2 values range from 0.9893 to 0.9988, indicating that this distributed thermal network model is highly reliable and accurate in calculating temperature distribution.

Conclusion
This article presents a thermodynamic analysis of the thermal resistance in the traction transformer winding area.In addition to considering the general conductor and oil flow thermal resistances, we establish a boundary layer thermal resistance that is often overlooked in research.Our findings reveal a MArE of winding temperature ranging from 0.35% to 0.54% and an e2 ranging from 0.9893 to 0.9988.The calculation results of the thermal network model for traction transformers have been validated against CFD results.This model not only calculates the temperature distribution in the winding area but also accurately locates hot spots.Compared to CFD, it reduces computation time from 93 minutes to 4.88 seconds on the same computer, improving computational efficiency by over 99%.The research findings in this paper will greatly impact the practical application of engineering.For example, thermal simulation is faster than CFD and can precisely locate hot spots in transformers.This information can be used to analyze the thermal aging of transformers and optimize their heat dissipation structure.Additionally, the speed and reliability of thermal network calculations are highly valuable for research purposes compared to resource-intensive CFD simulations.This further encourages us to delve deeper into studying the thermal network model.

Figure 1
shows the thermal network model for an insulation paper-winding-cooling-oil unit.In this figure, To represents the intermediate temperature of the oil flow between winding conductors T and T' in the oil flow channel.
As a result, we use the concept of boundary layer thermal resistance Oj R − to represent heat transfer in this layer between the oil flow channel and left/right insulation paper.The calculation equation for viscous oil thermal resistance Oj R thermal conductivity coefficient of transformer oil.j A refers to the surface area perpendicular to the direction of heat flow on surface j of the oil passage.Finally, OS  − denotes the thickness of the boundary layer in the oil passage.

z 0 
Re reflects the flow characteristics of fluid at a distance z from the cooling oil flow leaving the inlet of the oil passage.It can be calculated using this equation: 2023 5th International Conference on Energy, Power and Grid (ICEPG 2023) Journal of Physics: Conference Series 2703 for the density of the cooling oil, () 0 vz for the flow velocity of the cooling oil at point z, and h D for hydraulic diameter.

hL
3).It can be calculated based on convective heat transfer theory.is the convective heat transfer coefficient between the oil passage j surface and the insulation paper surface.Its calculated values are shown in the following two equations: Ns represents the average Nusselt number of the respective conductor; I is the geometric length of convective heat transfer surface; a z and b z indicate the positions of the upper and lower surfaces of insulating paper in the coordinate system, while s z N signifies the local Nusselt number at point z.


Re represents the eeynolds number and Pr represents the rrandtl number.The calculation equations for Re and Pr are as follows: stands for the velocity of oil flow; 0 is the density of oil flow; 0  signifies the dynamic viscosity of oil flow; and 0 c indicates the specific heat capacity of oil.

Figure 3 .
Figure 3. Topological model of thermal network in transformer winding region.

Figure 4 .
Figure 4. rhysical structure of cooling channel for oil-immersed traction transformer.

Figure 7 .
Figure 7.The calculation results of thermal network model and CFD simulation.

Table 2 .
Error analysis of improved thermal network model and CFD simulation results