1. Introduction
Liquid immersed power transformers are devices transferring a large amount of electric power with high efficiency. Due to high currents through the winding and the magnetic field through the core, there are heat losses that can reach several hundred thousand kW for large power transformer transformers (>100 MVA). The losses are small compared to the transferred power, meaning the power efficiency of the transformer is high, but the amount of the generated heat is considerable and leads to an increase of temperature inside the transformer. The temperature of the liquid and solid insulation materials should be limited to avoid a sudden failure due to bubbling phenomena or accelerated insulation ageing as a result of the cumulative effect of the increased temperature over a longer time period.
A majority of LIPTs operate in conditions of variable load and ambient temperature. Due to the high thermal inertia of the transformer, there is a time delay between the change of the load and the change of the LIPT components’ temperature. This implies that the temperature will change in time, but more slowly, enabling loading over the rated load during short time periods.
This application is of practical interest and can be used, for example, to overload the transformer during planned daily peak loads, instead of replacing the transformer with a larger one or adding another transformer to the substation. An overload can be useful in emergency cases where the transformer should take over the load due to the outage of some other grid elements. The grid operator needs to take decisions regarding the safe overloading of the transformer and, for this purpose, it is necessary to perform calculations of the critical temperatures inside the transformer, using dynamic thermal models (DTM). The necessity of thermal monitoring, based on the DTMs, in order to optimize the usage and increase the reliability of LIPTs in electrical power systems is presented by Tenbohlen et al. [
1].
In order of complexity, the physics-based DTMs can be classified in the following way: (1) ordinary differential equation (ODE), with standard or experimentally determined parameters, (2) detailed dynamic THNM and (3) computational fluid dynamics (CFD) calculations.
Simple ODEs are the preferred approach used in international loading guides (IEC 60076-7 and IEEE C57.91) [
2,
3]. More precisely, such models consist of a small number of ODEs being solved analytically, when possible, or numerically using finite difference or other solvers. Their main advantage is fast execution, low computational resources, and easy programming. In the loading guides [
2,
3] ODEs are established as lumped models, where transformer parts are represented with one temperature—the winding with insulation hottest-spot (in further text “hot-spot”) and the top insulation liquid (in further text “top-oil”). The easiest way, but with lower accuracy, is to apply such simple models using the recommended parameters given in standards for different transformer types. In such an approach, complex dependences of the hot-spot and top-oil temperatures on the characteristics of the materials used, the cooling equipment, and the construction of the active part are not considered. An improvement in the accuracy is made after the parameters of the models are determined from the results of the temperature-rise test on a transformer in short-circuit [
4], experiments with both winding and core losses [
5,
6], or field data. Although the individual influences of the above stated parameters are not considered, their cumulative effect on the temperature values is roughly modeled. The model from [
3], enhanced by continuously updating parameters based on the transformers’ real-time thermal response [
7], is a possibility to overcome some disadvantages of ODEs.
The main problem with the lumped model is to adequately consider the vertical temperature gradient. Since the oil temperature increases along the height of the winding, and the winding hot-spot appears around its top, the majority of the models are based on the mixed top pocket oil temperature (top-oil). As shown in [
8], the time variation of the top winding oil consists of one slowly changing component (linked with the bottom-oil temperature) and one small delay with the winding temperature. The thermal model in Annex G [
3] is based on the bottom-oil temperature. Although the application of models based on the bottom-oil can better capture the phenomena, none of the models accurately calculates the dynamic change of the vertical winding oil gradient. It depends on the heat transferred from the windings to the oil and the oil flow, which is determined by the equilibrium of the pressure in closed oil circulation loops. Another key issue with existing lumped models is that the heat accumulation is determined according to the change of some local oil or winding temperature. In reality, the temperature over the volume of the parts (winding, core, oil) is changing. Therefore, to determine the complete accumulated heat, the temperature distribution has to be known. Heat accumulation might be roughly modeled by introducing correction coefficients for the mass and specific heat (for example, in [
3], for ONAN and ONAF cooling, two thirds of the weight of the tank and 86% of the specific heat of the oil were used) while calculating the thermal capacity in a lumped model. Another possibility is to determine the parameters characterizing heat accumulation using laboratory or field measurements (examples can be found in [
9,
10]). Some researchers are trying to create hybrid models by combining the lumped models of some transformer parts with the distributed parameter models for the other. This possibility is explored and experimentally validated for the special type of traction transformer [
11].
The level of details in THNM enables the calculation of the oil flow distribution needed to determine the vertical oil gradients. The detailed static THNMs [
12], implemented in software design tools such as [
13], are spreading in transformer engineering practice and can be applied to transformers built with conventional or emerging insulating materials.
Contrary to the static THNM, dynamic THNM is still at a low-level technology readiness level (TRL). Although Seitlinger [
14] published in 2000 the idea of application of dynamic THNM and its postulates, being patented in 1999 [
15], detailed THNM was not widely used for temperature calculation during dynamic grid operating conditions, although there is a strong technical need for this. The possibility of developing the dynamic detailed THNM of LIPT winding by rearranging the existing static version is proposed in [
16], and the further research of winding hydraulic behavior during transients is presented in [
17].
One of the discouraging factors for applying detailed dynamic THNM is that detailed construction and the material characteristics are to be known, and that is not often the case, especially for the older transformers. Practically, it is realistic to envisage application to new transformer units, considering that a proper agreement between the transformer producer and the user is met.
Of significant importance in the context of digitalization, the detailed dynamic THNM can be associated with the development of the concept of the transformer digital twin. A digital twin can be defined as a scalable virtual replica of a physical asset that, through automatically updated data and simulation tools, can continuously monitor and predict the condition and behavior of its real-life counterpart, with the goal of optimizing the latter’s performance. The dynamic THNM provides a high-fidelity virtual replica of the thermal behavior of the LIPT.
The modern tendency in different problem-solving areas is the application of artificial intelligence (AI) and deep learning. It can be used in the process of modeling the behavior of technical systems in different operating conditions. Machine learning (ML) and AI are referred to as “black-box” models or data-driven models, as opposed to physics-based models. There are “gray-box” models, or hybrid models, which include more physics and start from a better description of the model structure [
18]. Another modern research direction is the application of reduced order modeling (ROM). This is the technique of reducing, for example using AI or other mathematical representations, the complexity of the model (in terms of computational time and used memory) while preserving the model’s accuracy. One of the possibilities is to run off line the CFD calculations and to use their results as the input data for the ROM training, which generates a “black-box” model that mimics the reality. The ROM is valid only within the area of scenarios covered by the set of physics-based simulations (steady and transient states).
More than a hundred years of developments on transformer heat transfer and loading [
19] indicate that it is impossible to grasp the complexity of the problem using a small number of ODEs tuned with experimental data. Some of those models are useful in some specific applications, but none of them are general enough to cover all possible scenarios. The key problem is that it is not possible to describe all relevant physical influences with a few differential equations. Although there is much less experience in practical application of ML and AI, due to the same reasons stated for simple lumped models, it is not expected that AI will become a panacea for dynamic thermal model of LIPTs. The detailed dynamic THNM is deeply based on the hydraulic and heat transfer theory and the preliminary findings indicate that it is applicable in real-time operation with variable load, cooling stage, tap position, and cooling medium temperature.
This paper explains the fundamental equations of detailed dynamic THNM. It was found that the dynamic THNM requires a completely different approach to the one used for static THNM [
12]. Static THNM is built on the concept that oil ducts and conductors are the elements of the networks, while the solver concept in dynamic THNM is moving towards a 1D finite-element model (FEM).
An important practical issue is the execution time of the developed software that should be shorter than real time. The experimental set-up from McGill University used for the model validation [
20] is of moderate complexity but it can be used for the demonstration of the execution time of detailed dynamic THNM applied to a real transformer. The real-time applicability disqualifies the use of CFD. For example, in [
21] it has been reported that it would take about 30 days of computational time using 464 processors to simulate 30 min of flow and the thermal behavior in a transformer-scale model. To overcome these limitations of CFD and get results faster, there is a possibility of neglecting a considerable number of physical phenomena [
22]. The consequence of this is lower accuracy and a reduced set of results. With such simulations, only the global distribution of the flows and the temperature are obtained.
This paper presents a numerical solver applicable for the transient temperature calculation of a heated solid immersed in liquid circulating in a closed loop. The method is implemented and tested for the case of natural convection fluid flow, which is the most difficult process of the dynamic THNM. The method is tested on a simple experimental set-up with water circulating in a single loop. In the following stage of the research, the model will be expanded using the detailed THNM developed before ([
12,
23]), to achieve practical application for real transformer operation.
Section 2 briefly revisits the detailed THNM used for steady-state simulation.
Section 3 describes the experimental set-up that was used for the validation of the new dynamic THNM. The details of the heat exchanger are reported in
Section 4.
Section 5 considers the application of the static THNM to the experimental set-up. In
Section 6, the basics of dynamic THNM are presented, with emphasized reasons as to why it is not possible to follow the approach of building a model analog to the steady state. The comparison of the static and dynamic THNM simulations with the experimental results is presented in
Section 7.
2. About Detailed Thermal-Hydraulic Network Models
In a LIPT, the oil flows up inside the active part of the transformer, increasing its temperature due to the heat transfer from the windings and the core, and then it flows down through the cooler, decreasing its temperature due to the heat transfer to the outer cooling medium (air or water). These heat transfers depend on the temperature differences between the active part and the oil, and between the oil and air (water). The relation between heat transfer and the temperature differences depends on the construction inside the tank and the outer cooling arrangement. Each temperature gradient, especially the vertical oil temperature gradient (∆θ
ovw), depends on the oil flow.
Figure 1 shows the simplified temperature diagram of the transformer, for the case of two windings (no core and no oil by-pass in the tank). θ
bo represents the temperature rise of the bottom oil–oil exiting the cooler and entering the windings, θ
to the temperature rise of oil entering the cooler, θ
hs the hot-spot winding temperature rise,
g the average winding temperature minus average temperature of oil in the winding, and the hot-spot factor
HS. The critical temperature in a transformer with a well guided stray flux transformer appears in one the windings, whereby it depends on the winding constructions and might appear in either HV or LV windings.
By using detailed THNM, the flow rate of circulating oil and the distribution of the oil flow and its temperatures at the inlet and outlet of the transformer parts (winding, core, and cooling equipment) can be determined. The detailed THNM is built as a “bottom to top” structure [
12], meaning that the networks are first established and solved for each of the elements (windings, core, cooler, etc.), and then they are connected to the global hydraulic network. Solving the global hydraulic network gives the overall distribution of oil. Solving the thermal and hydraulic networks of the transformer elements provides the distribution of the flows inside them, the temperature of the oil in each duct, and for a winding, the temperature of each conductor. The detailed THNM is based on the equations describing: (a) the conservation of heat, (b) the conservation of mass, and (c) the pressure equilibrium in closed loops. Solving the corresponding non-linear equations is a challenging but achievable task.
The most useful result is the location of the winding hot-spot and its temperature under the specified loss distribution. From this result, the hot-spot factors from
Figure 1 (
HS) can be determined for each winding [
23]. The model provides the values of temperatures measured in the standard temperature-rise test, meaning it satisfies the requirements in the transformer design phase. Some examples of its application are presented in [
24,
25]. Up to now, the static THNM could be applied in transformer thermal design, but not to simulating dynamic grid operating conditions. The fundamental differences between the concepts of building static and dynamic detailed THNMs and the first implementation of dynamic THNM and its testing on a small-scale model [
20] are described in the next sections.
4. Discussion about the Heat Exchanger
The log-mean temperature difference (LMTD) method has been applied. It is based on heat exchanger equations [
26]:
In the literature, such as [
26], the equations for convection heat transfer coefficients (CHTC) for different geometries can be found. In the experimental set-up, the closed-loop water circulating through the heat exchanger (CHTC α
w,hex,ts) flows through a pipe [
27], and the outer water circuit (CHTC α
w,o,hex) flows through an annulus [
28]. In LIPTs, the heat exchanger is more complex (a two-pass shell and tube is often used, where the water flows through the circular tubes and the oil flows across the bank of circular tubes). In the literature [
26,
29,
30], the equations for such cases can be found. The CHTC depends on the flow rate and the fluid temperature and thus these values differ for different operating points. Calculating the CHTC at each transformer operating point is theoretically the ideal approach, but for applying it, the detailed construction of the heat exchanger has to be known. Such data are generally not available for commercial heat exchangers. Another practical issue is the validity and accuracy of the equations for CHTC from the literature. For this reason, a different approach is used herein. After neglecting small heat resistance to heat conduction through the pipe (term
in Equation (3)) it can be written:
(Equation (5) is written for the case of a heat exchanger on a transformer, where CHTC on the insulation liquid side is α
f and on the water side α
w). In general, for optimized design of the heat exchanger at the rated operating point, the α
f,r Sf is close to α
w,r Sw. For such a case, the following can be written:
The value of
kpc,r Sf can be determined from the following relation between the cooling power, the temperatures of the insulation liquid circulating in closed loop (entering ϑ
f,h and exiting ϑ
f,c), and the outer water temperatures (entering ϑ
w,c and exiting ϑ
w,h) (these data are provided in the catalog for the rated conditions):
Factor
F can be determined using the graphs for the specific heat exchanger [
26] (a two-pass shell and tube is a typical construction used in LIPTs). Consequently, the cooling power can be determined based on to the following equation for the cooling power:
The ratios of the CHTCs at an arbitrary operating point and a rated operating point (αf/αf,r and αw/αw,r) might be obtained from the corresponding equations from the literature for more accurate calculations, or considered as equal to the rated values, meaning that the influence of the fluid flow and inlet fluid temperatures on the CHTCs is neglected. Similar is valid for the hydraulic resistance to the oil flow through the compact cooler.
For the application of the approach to the experimental set-up, the rated cooler operating point is set using the measurements in steady state for a heating power of 200 W. In this case, the inner and outer pipe surfaces were known and the CHTCs on both surfaces were calculated for the rated 200 W conditions.
7. Results
Two different tests were performed for the newly developed dynamic THNM.
The first test was a comparison of the steady-state temperatures available from the experiments (Exp.) from McGill University [
20], with the temperatures calculated using static THNM (Stat.) and the values reached in a steady-state condition with specified constant heating power delivered by the new dynamic THNM (Dyn.). This was the only possibility to make a comparison of static THNM/dynamic THNM and the measurements, since static THNM delivers the temperatures in steady states, i.e., does not calculate the temperatures during transient thermal processes. The dynamic simulations were performed using the same tuned coefficients as the ones for the static calculations (obtained by the calibration procedure on the data of the experiment with 200 W and specified in
Section 5.4). The results are presented in
Table 4. The differences between results of the static and dynamic THNM are small. The deviations for each of the calculated temperatures from those measured are similar—RMSD (root mean square deviation) values for all four temperatures are (the first number is for static THNM and the second for dynamic THNM): for 200 W 0.40/0.80 °C, for 125 W 0.29/0.24 °C, for 50 W 0.13/0.38 °C. As already explained, the mathematics of the static and dynamic THNMs are completely different. The presented results encourage further development and application of dynamic THNM, since the obtained results are close to the results of static THNM, which is implemented in the design software tool, applied for years in the transformer industry.
The second validation test is a comparison of the dynamic THNM simulation results with the values measured during a transient thermal process [
20].
Figure 10 compares the results for four step changes of the heating power (50–125 W, 125–200 W, 200–125 W, and 125–50 W). The initial conditions are the steady-state temperatures at a heating power of 50 W.
Table 5 shows the characteristic deviations of the calculated from the measured temperatures.
Figure 10 indicates a good agreement between the measured and calculated temperatures during all heating up and cooling down processes. The RMSDs (
Table 5) are around 1 °C.
Figure 11 shows the zoomed-in transient periods for temperatures ϑ
w,hp2 and ϑ
w,cp3 (for each transient period 1 h is presented). The differences in steady states are summarized in
Table 4—the maximum deviation is 2.06 °C, for ϑ
w,cp3 in steady state with 200 W.
Figure 12 presents the temperatures, the heat transferred from the heater to the water, and the water flow predicted by the dynamic THNM for a cold start scenario and a heating power of 200 W. The following can be observed: (1) There is “dead time” in the change of water temperature at the heater’s top and bottom. At the beginning of the heating process, the heat transferred from the heater is low due to the low heater block temperature. In the initial period with no water flow (ca. 21 s) the heat is transferred through the water by heat conduction, thus the heating of the water slowly propagates inside the heater case, around the heater element toward the measuring points 2 and 3 on the cold and hot pipes.
Figure 13 presents zoomed-in initial period. The sharp peak shape of the transferred heat at the very beginning of the process is the consequence of the change of CHTC, which is very low at the beginning and starts to increase when the heater to water temperature gradient establishes (2). After this period, the water flow begins slowly. The temperature drop in the water at the heater bottom (ϑ
w,cp3) is due to the lower outer cooling water temperature (11.93 °C) compared to the initial temperature of circulating water, being equal to the ambient temperature (21 °C) (3). In the initial period, the heating power is mainly accumulated in the heater block, causing an increase of its temperature (4). With an increase of the heater block temperature, the heat transferred to the water increases and the slope of the heater block temperature decreases (5). During the initial period of low flow rate some time is needed for the water heated in the heater to reach the measuring point (for a velocity of 1 cm/s 156 s are needed—the lengths are presented in
Figure 2) causing prolonged “dead time” of the water at the heater top temperature (ϑ
w,hp2) (6). Further change of ϑ
w,hp2 is the result of the combination of the effects of the drop in ϑ
w,cp3, the increase of heat transferred from the heater block to the water, and the change of the water flow (7). As the consequence of the complex hydraulic and thermal phenomena there are peaks in the water flow and heat transferred from the heater to the circulating water. Please note that the heat transferred from the heater to the circulating water exceeds the heat power of the heater. This “paradox” is the consequence of the different inertia phenomena: the flow increase delays the increase of heat transferred to the water, the change of CHTC follows the water flow, causing that the temperature of the heater to exceed the steady-state value corresponding to the heat power (200 W). Similar, somewhat lower, local overshoots in transferred power are noticeable when the heating power is increased from 50 W to 125 W and from 125 W to 200 W (
Figure 14).
8. Conclusions
This paper describes the application of the convection−diffusion equation as the base for building a dynamic detailed THNM of LIPT. It has been explained that the heat equations have to be fundamentally changed in respect to static detailed THNM in order to solve the problem of the heat accumulated in insulation liquid. The “quasi-steady-state” model, as a small upgrade of the static THNM, is improper and a conceptual and huge change is needed. Contrary to this, the requested changes in the hydraulic model, defining the equilibrium of pressures in the insulation liquid circulating in closed loops, are small, where only the component of time change of the velocity has to be added. This extension was easy to implement. Although the convection−diffusion equation is basic theoretical knowledge, as far as we know it has not been successfully applied in the development of a dynamic thermal model of LIPT and we do believe the content presented in the paper points to the right research and development direction.
The model delivers the distribution of temperature over the volume of the elements of the active transformer part and oil circulating in the closed loop during transient thermal processes without involving any time constant. As an intermediate value in the calculations, the change of flow rate during the transient process is calculated. The heat accumulation in the components is determined as a distributed value over the volume. The calculation time for the developed software is not high, and thus the model can be used in real time. The simulation run time with 0.1 s time discretization for the experiment on a simple small-scale model of about 22 h was 17 min and 50 s. Based on these data and experience with the run time of detailed static THNM software used for transformer design, a realistic expectation is that the future method applied to a real transformer can satisfy online requirements. This is an essential requirement to apply the detailed dynamic THNM as an improved and more physics-meaningful alternative to the simple lumped models, commonly used in practice but with limited accuracy.
In solving the equations of static detailed THNM in general, there is the problem of convergence of the pressure equilibriums in closed liquid circulation loops. While implementing dynamic detailed THNM, convergence problems were faced and solved using the upwind interpolation approximation (UDS). The method is implemented in the developed software which, for now, models the fluid circulating in the single closed loop of a simple experimental small-scale model. The results of the calculations were compared with the results of measurements published in the literature. The presented results confirm the good agreement between the temperatures calculated by the proposed detailed dynamic THNM and the measurements (the RMSDs of calculated temperatures and the maximum difference in steady states are below 1 °C).
This paper should be understood as a proof of concept for the completely novel approach to the important technical issue of dynamic thermal modeling of LIMT. According to the authors experience in the development of static THNM and its application to transformer design, the key issues in dynamic detailed THNM were solved and the next step is to expand the new principles for real transformer parts, following the structure of the software developed for static detailed THNM. Measurements on a laboratory transformer winding model, and measurements during transformer extended temperature-rise tests, will be used to validate the complete detailed dynamic THNM.