Research on Real-Time Calculation Method of Transformer Temperature Field Based on Reduced Order Model and Sparse Measurement

The lifespan of transformers is closely related to their operating temperature. Among the existing temperature acquisition methods, the measurement method cannot obtain the complete temperature field temperature, while the numerical calculation method presents the challenge of significant computational expenses. This study theoretically derived how to determine the real-time temperature distribution of transformer windings based on sparse measurement data and reduced order models and solved sensor layout issues, including the number and placement position of sensors. In this study, the maximum calculation error of the temperature distribution is 1.8 K, and the calculation time is 0.38 seconds. The research findings suggest that multiple modes can contribute to the temperature distribution of a transformer. Under a constant load condition, the calculation of transformer temperature can be achieved by utilizing a temperature sensor positioned at the hot spot in conjunction with the first mode.


Introduction
The operating status of transformers is closely related to their winding hot spot temperature (HST) and overall temperature rise level [1][2][3].Therefore, the methods for obtaining the HST and temperature field distribution of transformer windings have received widespread attention.
The early methods for obtaining transformer winding temperature were based on the measured top oil temperature to infer the HST [4,5].In the later stage, the working current of the transformer was converted into the corresponding temperature through the electric heating element, and the top oil temperature was superimposed to obtain the HST [6,7].Since indirect winding temperatures are always obtained [8], it is recommended to use fiber optic sensors to directly measure winding temperature rise in "Load Guidelines for Oil Immersed Power Transformers".Researchers also introduced model reduction methods to increase computational speed.Liu et al. combined the finite element method with the invasive model order reduction method to establish a reduced order model (ROM) for calculating the transformer temperature [9].The field calculation efficiency is increased by approximately 45 times.This method requires modification of the discretized control equation of the studied problem, which is difficult to implement.Liu et al. established the relationship between transformer operating conditions and POD mode coefficients based on the response surface method [10].In 100 test conditions, the error compared to full-order calculation was not greater than 2.0 K, and the time was only 3.13 seconds.However, the non-invasive model reduction method has the problem of high computing costs in the offline stage.
The model order reduction method provides a solution for obtaining the transformer temperature field.However, the existing invasive and non-invasive model reduction methods have the problems of numerical instability and high offline calculation cost, respectively.Hence, this study introduces a calculation approach for estimating the temperature distribution in transformer windings.This method relies on sparse sensing data and utilizes the model order reduction technique.This method determines mode coefficients through sparse measurement data, so there is no numerical instability problem, and the computational cost is small in the offline stage.

Inverse solution process of complete field distribution
Assuming that the unknown transformer temperature is x , sensors are used to measure it and get sparse measurement results y .The above process can be mathematically expressed in the form of Equation ( 1).y = Cx ( 1 ) where , and n is the number of mesh grids; where is the truncated POD modes; r is the order of truncation.
After the mode coefficients a are obtained through Equation ( 4), x can be obtained by using Equation (5).
 r x a  ( 5 ) However, it should be noted that when the measurement matrices C are different, different matrices s  will be obtained, thus different POD mode coefficients a can be solved, and different temperature fields x are obtained.Therefore, it is necessary to determine the optimal measurement matrix C .

Sensor placement strategy
2.2.1 Determination of the number of sensors used.When using the measured values of p sensors and the reduced-order model formed by r order POD modes to calculate the complete field distribution, the entire process is shown in Equation ( 6).
where  ,  ,  are the cropped POD modes.Therefore, from the perspective of equation solving, it can be seen that the number p of sensors used needs to satisfy p r  , otherwise, Equation ( 6) is underdetermined.Therefore, the solution of the mode coefficients a can be expressed in the form of Equation (7).

Sensor placement.
In Equation ( 6), a can be obtained, thereby achieving the solution of the complete field distribution.For Equation (3), when y changes slightly, if the condition number of the equation is small, the solution a will only change slightly.On the contrary, if the condition number of the equation is large, the solution a will change significantly.Therefore, our goal is to minimize the condition number when determining the placement position of the sensor.The definition of the condition number is shown in Equation (8).
where max  is the largest singular value and min  is the smallest singular value.
As can be seen from Figure 1, the placement of the sensor actually determines which rows are selected from the matrix r  to form the matrix s  .According to Equation ( 8), we need to select the specific rows of r  to form the matrix s  with the smallest condition number, so we perform QRpivoting transformation on T r  as shown in Equation ( 9).

Results & Discussion
A simulation model as shown in Figure 2 is established, and then the rated load rate is applied.The entire temperature rise process lasts for 7 hours and reaches thermal equilibrium.where  is the singular value corresponding to the POD mode, also called energy.
From Figure 3, the first-order POD mode captures 99.9% of the energy.We directly use the firstorder mode to construct the ROM.At the same time, according to the sensor placement strategy in Section 2.2.2, the placement position of the sensor is obtained at the 62760th grid point of the model,

Accumulated energy proportion
Mode order located at the hot spot position.Therefore, at 4-7 hours, the simulated values at this location are used to simulate the sensing measurements to obtain the field distribution.
Figure 4 shows the difference between calculated results using the proposed method and simulation model for 4.5 -7 hours.The maximum calculation error is 1.8 K with a calculation time of 0.38 s.

Conclusion
The paper concludes as follows: (1) When the load factor is constant, the first-order POD mode can capture 99.9% of the information regarding the temperature distribution in the winding.
(2) Using the proposed method, the winding temperature is obtained within seconds by using the measurement values of the sensor placed at the hot spot and the first-order POD mode, with a maximum error of 1.8 K and a calculation time of 0.38 s.
(3) When the load rate changes, how to use the order of the POD mode and the placement scheme of the sensor need to be further studied in the future.
Orthogonal Decomposition (POD) mode is used to express the unknown x . r y C a  and the coefficients a of POD modes are unknown.Furthermore, Equation (3) can be obtained.To express Equation (3) more clearly, Figure1is used to illustrate it.

Figure 1 .
Figure 1.The schematic of Equation (3).As shown in Equation (4), the parameter can be calculated.†  s a y 

Figure 2 .
Figure 2. Transformer winding temperature field simulation model.We export 140 snapshots and use 80 snapshots corresponding to the first 4 hours to extract the POD modes, which are used to build the reduced-order model (ROM).The singular value decomposition (SVD) of the snapshot set is performed in the first 4 hours, as shown in Figure 3.

Figure 3 .
Figure 3. Accumulated energy proportion of POD modes.The calculation method of the cumulative energy proportion is shown in Equation (10).

Figure 4 .
Figure 4. Calculation error.4.ConclusionThe paper concludes as follows:(1) When the load factor is constant, the first-order POD mode can capture 99.9% of the information regarding the temperature distribution in the winding.(2)Using the proposed method, the winding temperature is obtained within seconds by using the measurement values of the sensor placed at the hot spot and the first-order POD mode, with a maximum error of 1.8 K and a calculation time of 0.38 s.(3)When the load rate changes, how to use the order of the POD mode and the placement scheme of the sensor need to be further studied in the future.