Predicting the Temperature Rise in Oil-Immersed Transformers Based on the Identification of Thermal Circuit Model Parameters

Abstract

The temperature rise test for transformers is time-consuming, energy-intensive, and has low detection efficiency. To improve the efficiency of the temperature rise test and reduce energy consumption, this paper proposes a temperature rise prediction method for oil-immersed transformer windings. This method is based on identifying the parameters of a thermal circuit model. Firstly, a fifth-order thermal circuit model of oil-immersed transformers is put forward. Then, based on a two-hour temperature rise curve, the thermal capacity and resistance model is identified through genetic algorithms. The obtained parameters are used to compute the temperature rise curve, steady-state average temperature rise, and top oil temperature rise. The results show that the heat capacities of the low-voltage (LV) winding, high-voltage (HV) winding, oil tank, and oil of a 400 kVA transformer are approximately 50 kJ/K, 75 kJ/K, 320 kJ/K, and 90 kJ/K, respectively. Additionally, the thermal resistances from the LV winding to oil, HV winding to oil, oil tank, and air are about 8 mK/W, 5 mK/W, 1 mK/W, and 11 mK/W, respectively. When the transformer capacity increases, the heating power of the windings escalates, and the oil resistance of HV windings decreases from 8 mK/W for a 400 kVA capacity to 5 mK/W for an 800 kVA capacity. The absolute prediction error for transformers of 400 kVA, 630 kVA, and 800 kVA is 2.9 °C. These findings can facilitate the swift detection and assessment of the winding temperature rise in oil-immersed transformers.

1. Introduction

The temperature rise test, as an important means to evaluate thermal stability and insulation life, is of great significance in transformer design and test assessment. However, the traditional temperature rise test usually lasts for dozens of hours, takes a long time, and consumes a lot of energy. It is urgent to establish a more efficient temperature rise test and thermal performance evaluation method to accelerate product verification and technical iteration.

Currently, the evaluation methods for the temperature rise performance of oil-immersed transformers are primarily categorized into Computational Fluid Dynamics (CFD) simulation, thermal path modeling, and direct measurement methods. Wen et al. utilized CFD simulation to calculate the detailed temperature distribution of oil-immersed transformer windings for the rapid solution of temperature rise [1,2]. However, such methods require the discretization of high-order nonlinear N-S equations. A single transient temperature rise solution takes several hours to compute. Additionally, the computational results depend on the transformer’s structure and material parameters, necessitating the establishment of simulation models for different transformers [3,4,5]. Equivalent thermal circuit models are widely used for steady-state temperature calculations in transformers due to their physical interpretability and computational efficiency. The core principle of the thermal circuit is to characterize the internal heat transfer process of the transformer through a thermal resistance–thermal capacitance network and deriving the calculation method for temperature nodes via a thermoelectric analogy. Low-order thermal circuit models simplify the transformer by treating oil and air as a lumped thermal resistance to establish the thermal circuit model. These models have two or three parameters and fast computation speeds, but they cannot calculate winding temperatures [6,7]. Third-order thermal circuit models, which consider both winding and top-layer oil temperatures, have better performance, but involve complex calculations of thermal capacitance and thermal resistance. The thermal network models perform distributed calculations on high- and low-voltage windings, enabling the determination of winding hotspot temperatures and their specific locations. However, such methods involve numerous parameters, result in significant computational loads, and are susceptible to variations in transformer winding configurations and cooling methods, leading to large computational errors that limit their engineering applications [8,9,10,11]. Direct temperature measurement is only applicable in simulated devices or in the oil flow at the top and bottom layers of the transformer, and cannot be directly applied to measuring the temperature of the transformer windings [12,13,14,15].

Based on this, the study developed a fifth-order thermal circuit model that considers the core and winding losses of transformers at high and low voltages. Genetic algorithm r is used to identifying the thermal capacity and thermal resistance parameters of the thermal circuit model using two-hour temperature rise curve data. Then identified parameters are used to predict the steady-state transformer’s temperature rise. This yielded three key technical parameters are the temperature rise in the high-voltage winding, the temperature rise in the low-voltage winding, and the temperature rise in the top layer of oil in oil-immersed transformers. These parameters enable the evaluation of transformers’ thermal performance. The research methods presented in this paper can be used to rapidly detect and evaluate the winding temperature rise in oil-immersed transformers.

2. Hot Circuit Model Parameter Identification and Winding Temperature Rise Prediction

2.1. Internal Heat Conduction Process of Transformers

In oil-immersed transformers, the primary heat sources are windings and the core. Copper losses in the windings are attributed to the Joule heat generated when currents flow through the windings [16,17,18]. The core’s hysteresis and eddy current losses stem from the alternating magnetic field. During operation, the heat generated by losses in silicon steel sheets and conductors is transferred to their surfaces via solid heat conduction, causing an increase in the surface temperature [19,20,21]. The surface heats the oil that transfers heat from the transformer interior to the oil tank and radiator fins via thermal convection. The oil tank and external radiators then dissipate heat to the surrounding air through convective cooling and radiation [22,23,24,25,26]. Hence, the transformer’s internal heat conduction process is characterized by a sequence of events that includes winding (core) → oil → oil tank (radiator fins) → environment (air).

The above heat dissipation is often analyzed through the thermal circuit model. Such models are similar to the electric circuit model. The definitions and corresponding relationships of parameters in the thermoelectric analogy method are shown in

Table 1. Parameter comparison in thermoelectric analogy method.

Thermal Parameters	Electrical Parameters
Heat output q/W	Current i/A
Temperature T/K	Voltage u/V
Thermal resistance Rth/(K·W−1)	Resistor Rel/Ω
Thermal capacitor Cth/(J·K−1)	Capacitance Cel/F

. The heat sources can be regarded as a current source. Therefore, the temperature is similar to the voltage on the resistor, whose value is determined by the thermal resistance. The thermal capacitor works as the capacitance to absorb and release the heat from circuits.

2.2. Fifth-Order Thermal Circuit Modeling of Oil-Immersed Transformers

Based on the above physical processes, the losses in the high-voltage and low-voltage windings and the no-load losses of the transformer are treated as three ideal heat sources; the ambient temperature is analogized as an ideal temperature source and the heat transfer between the transformer core, windings, oil, and external air is analogized to the resistance and capacitance in a circuit [27,28,29]. The five-order thermal circuit model of the transformer is shown in Figure 1.

The model parameters are defined as shown in

Table 2. Hot circuit model parameter comparison table.

Parameters	Notation	Unit
High-voltage winding loss	QWH	W
Low-voltage winding loss	QWL	W
Core loss	QC	W
Core thermal capacity	CC	J/K
High-voltage winding thermal capacity	CWH	J/K
Low-voltage winding thermal capacity	CWL	J/K
Oil thermal capacity	Co	J/K
Tank thermal capacity	CT	J/K
Core to oil thermal resistance	RC-O	K/W
High-voltage winding to oil thermal resistance	RWH-O	K/W
Low-voltage winding to oil thermal resistance	RWL-O	K/W
Oil to tank thermal resistance	RO-T	K/W
Tank to ambient thermal resistance	RT-A	K/W
Core average temperature	TC	K
High-voltage winding temperature	TWH	K
Low-voltage winding temperature	TWL	K
Top oil temperature	TO	K
Tank temperature	TT	K
Ambient temperature	TA	K

. Q_C, Q_WL, and Q_WH represent the losses of the transformer core, low-voltage winding, and high-voltage winding, respectively, with units in W; C_C, C_WL, C_WH, C_O, and C_T represent the equivalent thermal capacities of the core, low-voltage winding, high-voltage winding, oil, and oil tank, respectively, with units in J/K; R_C-O, R_WL-O, R_WH-O, R_O-T, and R_T-A represent the thermal resistance between the core and oil, low-voltage winding and oil, high-voltage winding and oil, oil and oil tank, and oil tank and environment, respectively, with units in K/W; T_C, T_WL, T_WH, T_O, T_T, and T_A represent the average temperature of the transformer core, low-voltage winding, high-voltage winding, top oil layer, oil tank, and environment, respectively, with units in K.

The impact of core temperature on lifespan is negligible, since the core’s losses are lower than those of the windings and the core has better heat dissipation than the windings. Consequently, when solving the thermal circuit model equations in practice, the thermal resistance between the core and oil is ignored. According to the aforementioned thermal circuit model, the subsequent system of differential equations can be formulated:

$$Q_{WL}=C_{WL}{\displaystyle \frac{\mathrm{d}{T}_{WL}}{\mathrm{d}t}}+{\displaystyle \frac{T_{WL}-T_O}{R_{WL-O}}}$$

(1)

$$Q_{WH}=C_{\mathrm{WH}}{\displaystyle \frac{\mathrm{d}{T}_{WH}}{\mathrm{d}t}}+{\displaystyle \frac{T_{WH}-T_O}{R_{WH-O}}}$$

(2)

$$Q_C+{\displaystyle \frac{T_{WL}-T_O}{R_{WL-O}}}+{\displaystyle \frac{T_{WH}-T_O}{R_{WH-O}}}=C_O{\displaystyle \frac{\mathrm{d}{T}_O}{\mathrm{d}t}}+{\displaystyle \frac{T_O-T_{\mathrm{T}}}{R_{O-T}}}$$

(3)

$${\displaystyle \frac{T_O-T_T}{R_{O-T}}}=C_T{\displaystyle \frac{\mathrm{d}{T}_T}{\mathrm{d}t}}+{\displaystyle \frac{T_T-T_A}{R_{T-A}}}$$

(4)

Since the ratio of resistance loss, eddy current loss, and stray loss to the total loss of each winding is approximately 85%, 10%, and 5%, respectively, the losses in the high-voltage and low-voltage windings are directly proportional to the direct resistance loss. The proportion of these losses to the total load loss is as follows:

$$Q_{WH}={\displaystyle \frac{P_{WHDC}}{P_{WHDC}+P_{WLDC}}}{Q}_W,Q_{WL}={\displaystyle \frac{P_{WLDC}}{P_{WHDC}+P_{WLDC}}}{Q}_W$$

(5)

Among them, Q_W is the load loss and P_WHDC and P_WLDC are the direct resistance losses of the high- and low-voltage windings, respectively.

From the above equation, it can be seen that the temperature rise in an oil-immersed transformer can be regarded as a multi-input, multi-output response system. If Q_C, Q_WL, Q_WH, and the ambient temperature T_A are taken as input variables, and the parameters of the thermal circuit model are known, the temperature rise and steady-state temperature can be obtained by solving the above differential equations. Although references [30,31,32,33] provide formulas for calculating the thermal capacity and thermal resistance in the model using heat transfer theory, the complex internal structure of transformers brings difficulty in directly obtaining them.

2.3. Parameter Identification Methods

The top-layer oil temperature can be measured directly. Therefore, this paper uses the measured top-layer oil temperature as the parameter identification object, the power loss as the model input, and the calculated top-layer oil temperature as the output.

Genetic algorithms [34,35] exhibit strong global search capabilities, enabling them to avoid being trapped in local optima. The genetic algorithm parameter settings used are shown in

Table 3. Genetic algorithm parameter settings table.

Parameter Type	Parameter Value/Method Name
Population size	150
Crossover probability	0.6
Mutation probability	0.1
Number of generations	200
Operator selection method	Roulette wheel selection
Operator crossover method	Uniform crossover
Operator mutation method	Adaptive mutation

. Through approximation with the measured top-layer oil temperature, the thermal capacity and thermal resistance parameters are continuously assessed for fitness, propagated to offspring, mutated, and re-adapted to continuously optimize the population until the optimal thermal capacity and thermal resistance parameters are obtained.

The specific implementation steps for identifying transformers’ thermal circuit parameters using genetic algorithms are as follows:

Step 1: Input parameters and initial values, including high- and low-voltage winding load losses, core losses, initial values of high- and low-voltage winding temperatures, initial value of top oil temperature, initial value of tank wall temperature, and ambient temperature value. Determine the initial thermal capacitance and thermal resistance values of the fifth-order thermal circuit model based on the transformer’s structural parameters. The formula for calculating these initial values is as follows:

$$C_{th}=m\cdot c_p$$

(6)

$$R_{th}={\displaystyle \frac{L}{\lambda A}}$$

(7)

where m (kg) is the mass; c_p (J/(kg·K)) is the specific thermal capacity; L (m) is the heat transfer path length; λ (W/(m·K)) is the thermal conductivity; and A (m²) is the heat transfer cross-sectional area. Due to the presence of contact thermal resistance between structures, the results calculated from Equations (6) and (7) are not absolute accurate values, but rather estimated initial values. These values facilitate parameter convergence, enhance computational efficiency, and reduce computational time;

Step 2: Use the Runge–Kutta method to solve the system of differential equations defined by Equations (1)–(5), obtaining the top-layer oil temperature curve;

Step 3: Use the error between the model’s response and the measured temperature as the objective function. The top-layer oil temperature at time n is related to the temperature at time n – 1 and the input power at time n, and can be expressed as follows:

$$T(n)=f_n(T(n-1),T_A,Q_C,Q_{WL},Q_{WH},x)$$

(8)

Then, the target error function can be expressed as follows:

$$R(\mathit{x})=\sum _{n=1}^N\parallel T(n)-T_O(n){\parallel }^2$$

(9)

The root mean square error is expressed as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}R(x)}$$

(10)

where T_O(n) represents the measured temperature of the top-layer oil at time n, and x denotes the target vector parameters to be identified, where x = [C_WH, C_WL, C_O, C_T, R_WH-O, R_WL-O, R_O-T, R_T-A]. If a vector x whose elements are all non-negative exists, such that the objective function R(x) is minimized, i.e., the root mean square error (RMSE) is minimized, then the elements of x are equivalent thermal path parameters. The influence of contact thermal resistance between structures on the parameters is implicitly reflected in the identification results. Due to differences in the structural components of different transformers, the actual thermal path parameters may vary within a certain range. Therefore, the identification parameter range is set to 80–120% of its initial value to accelerate the convergence process of the algorithm;

Step 4: Output the identified thermal capacity and resistance.

2.4. Temperature Rise Calculation Method

Based on the identified thermal capacity and resistance, input power, and initial temperatures at each node, the ODE45 solver in MATLAB R2024a is used to calculate the temperature. The gradient (dT/dt) of top-layer oil temperature rise is calculated. When the temperature changes to lower than 1 K/h in three hours, the steady-state termination condition is triggered, and the calculation is stopped.

Based on the aforementioned parameter identification and temperature rise calculation methods, the temperature response curves and steady-state temperature rise in the transformer under any power input can be calculated.

3. Test Validation and Results

3.1. Temperature Rise Test Platform and Test Method

The diagram of the temperature rise test platform is shown in Figure 2.

In the temperature rise test, the LV winding of the tested transformer was short-circuited, and the high-voltage side was supplied with power. The voltage regulator T1 adjusted the voltage and current of the test transformer. The measurement accuracy of the voltage and current transformers (TA, TV) is both 0.2%. The power measurement error is <±0.5% (CosΦ > 0.1) and ±1.0% (0.05 < CosΦ < 0.1). In the test transformer, A, B, and C are the high-voltage side input terminals, while a, b, and c are the low-voltage side input terminals. The low-voltage side is in a short-circuited state.

K-type thermocouple temperature sensors were used to measure temperature. The measurement accuracy of the thermocouples is ±0.2 °C. The location of the thermocouples is shown in Figure 3. Four environmental temperature measurement points were distributed around the test transformer. Their measured temperatures were averaged as the environmental temperature. Four measurement points were placed at the upper and lower ends of the radiators on the front and back of the transformer. Their measured temperatures were averaged as the oil temperature. One measurement point was located at the oil inlet of the transformer, as shown in Figure 4. Its measured temperature was taken as the top oil temperature. To guarantee the accuracy of the parameter identification results, the temperature measurement interval was set to 1 min.

The test is divided into the total loss stage and the rated current stage [36,37]. When the rate of change in the top oil temperature remains below 1 K/h for at least three hours, the test transitions from the total loss stage to the rated current stage. After one hour in the rated current stage, the power supply was quickly disconnected, and the short-circuit connection was disconnected. The resistance of the high-voltage and low-voltage windings is measured, and the average temperature rise in the high-voltage and low-voltage windings of the tested transformer is calculated using the measured winding resistance data. The measurement accuracy of the DC resistance is as follows: high-voltage 0.2% ± 1 μΩ; low-voltage 0.2% ± 0.2 μΩ, with a minimum resolution of 0.1 μΩ.

Using the established platform, tests and predictions were conducted on the temperature rise in nine distribution transformers (3 units each of 400 kVA, 630 kVA, and 800 kVA) from different manufacturers but with the same capacity. Their winding direct resistances are shown in

Table 4. Test transformer parameters.

Model	Capacity/kVA	Load Loss/kW	No-Load Loss/kW	HV/LV Winding DC Resistance (75 °C)
S20-M-400/10	400	3.615	0.37	3.312 Ω/1.494 mΩ
S20-M-630/10	630	4.96	0.6	1.771 Ω/0.809 mΩ
S20-M-800/10	800	6.0	0.63	1.483 Ω/0.464 mΩ

. Both the high- and low-voltage windings use layered coils.

3.2. Model Parameter Identification Results

3.2.1. Thermal Capacity and Thermal Resistance of 400 kVA Transformers

Thermal path model parameter identification was performed based on data from the first 2 h of normal temperature rise. The curve showing the change in the RMSE was calculated according to Equation (10) as the number of evolutionary generations increases, as shown in Figure 5.

Figure 5 shows that as the number of evolutionary generations increases, the root mean square error (RMSE) of the fitted parameters exhibits a significant decreasing trend. After the offspring reach the 150th generation, the parameters gradually converge, and the RMSE of the fitted parameters for all three transformers is lower than 5, indicating excellent fitting performance.

Based on the identified parameters, the steady-state temperatures of the high-voltage and low-voltage windings and the top-layer oil were predicted using the calculation method described in Section 2.4. The predicted values and measured values are compared as shown in

Table 5. Comparison of steady-state values at each temperature node (All unit values below are in °C.).

Number	TA_Final/TA_Set	Temperature Node	TActual/TPredicted	∆TActual = TActual – TA_Final	∆TPredicted = TPredicted – TA_Set	Error = |∆TPredicted − ∆TActual|
400kVA-1	14.8/11.9	TO	58.8/55.1	44.1	45.0	0.9
		TWH	68.4/64.7	53.6	54.1	0.5
		TWL	69.8/66.8	55.0	57.9	2.9
400kVA-2	15.3/10.7	TO	57.9/53.7	42.6	43.0	0.4
		TWH	71.2/63.8	55.9	53.1	2.8
		TWL	70.1/67.3	54.8	56.6	0.8
400kVA-3	33.6/28.9	TO	79.8/73.1	46.2	44.2	2
		TWH	85.4/81.7	51.8	52.8	1
		TWL	92.5/85.7	58.9	56.8	2.1

.

The table shows that T_Actual and T_Predicted represent the measured and predicted steady-state temperature values, respectively; ∆T_Actual and ∆T_Predicted represent the measured and predicted steady-state temperature rise values, respectively; T_{A_Final} represents the ambient temperature at steady state; and T_{A_Set} represents the ambient temperature value set in the algorithm, which is also the average ambient temperature during the 2 h temperature rise test. Taking the average value reduces the impact of minor changes in ambient temperature within the 2 h period. The data in Table 5 show that there is a significant difference between the initial and final values of the environmental temperature during the actual measurement process, and the corresponding measured steady-state temperature values and predicted values also have significant errors, with the maximum error reaching 6.8 °C. In fact, the temperature rise values at the indicator temperature nodes do not change with the ambient temperature. The data in Table 5 also validate this conclusion: the measured and predicted errors for the steady-state temperature rise values of the top-layer oil in the three transformers are 0.9 °C, 0.4 °C, and 2 °C, respectively. In terms of winding temperature prediction, the study used an extrapolation method to calculate the winding temperature rise values at the moment of power disconnection. The results showed that the average temperature rise errors for the high-voltage windings were 0.5 °C, 2.8 °C, and 1 °C, respectively; the average temperature rise errors for the low-voltage windings were 2.9 °C, 0.8 °C, and 2.1 °C, respectively.

Since the environmental temperatures are different, this paper compares the measured and calculated temperature rise. The temperature rise in the three transformers is shown in Figure 6. The temperature rise change curves for the three transformers are shown in Figure 6. The blue curve represents the measured temperature rise curve, the red dashed line represents the temperature rise fitting curve obtained through parameter identification, and the black and pink dashed lines represent the predicted temperature rise curves for the high-voltage and low-voltage windings, respectively. The corresponding black and pink “X” points indicate the measured temperature rise values of the high-voltage and low-voltage windings at steady state.

The intuitive effect of the fit can be seen in Figure 6. Even with differences in the manufacturers and manufacturing processes of the three transformers, the top oil temperature curve, calculated from the temperature rise 2 h curve identification parameters, still maintains good agreement with the measured curve. Furthermore, when the high- and low-voltage winding temperature rise prediction curves reach steady state, they also closely approach the actual measured values. The error of oil temperature curve prediction is as follows:

$$R_{max}={\displaystyle \frac{|\Delta T_{oil-predicted}-\Delta T_{oil-actual}{|}_{max}}{\Delta T_{oil-actual}}}$$

(11)

The error calculation results for the three transformers were 4.84%, 5.68%, and 3.69%, respectively. These data indicate that the parameter identification method not only has high prediction accuracy for the top oil temperature, but also achieves good accuracy in predicting the winding temperature rise, fully verifying the reliability of the parameter identification results.

The thermal capacity and thermal resistance parameters, obtained from the 2 h temperature rise curve, are shown in Figure 7.

The identification results indicate that the fluctuation range of the equivalent thermal circuit parameters for transformers of the same model is <10%. The thermal capacity of the low-voltage winding ranges from 52.0 to 53.8 kJ/K, the thermal capacity of the high-voltage winding varies from 77.2 to 77.9 kJ/K, the thermal capacity of the oil is between 32.9 and 33.7 kJ/K, and the thermal capacity of the oil tank changes from 89.9 to 92.9 kJ/K. Thermal resistance parameters: R_T-A (10.7 to 10.9 mK/W) > R_WL-O (7.6 to 7.9 mK/W) > R_WH-O (4.8 to 5.2 mK/W) > R_O-T (0.90 to 0.94 mK/W). Since the low-voltage winding has poorer heat dissipation conditions than the high-voltage winding, the corresponding thermal resistance parameters are relatively higher. Additionally, since the winding is wrapped in insulating paper on the outer side, its corresponding thermal resistance is significantly greater than that between the oil and the tank.

3.2.2. Thermal Capacitance and Thermal Resistance of Transformers of Different Capacities

The average equivalent thermal capacitance and thermal resistance of transformers with different capacities, obtained by identifying the 2 h temperature rise curve, are shown in Figure 8.

As shown in the figure, as the transformer capacity increases, the thermal capacity increases. The oil thermal capacity rises from 350 kJ/K at 400 kVA to 800 kJ/K at 800 kVA. Thermal resistance decreases as the capacity increases. The thermal resistance of the high-voltage winding to oil decreases from 7.6 mK/W at 400 kVA to 4.6 mK/W at 800 kVA, while the thermal resistance of the oil tank to the environment decreases from 10.9 mK/W at 400 kVA to 6.2 mK/W at 800 kVA. The above patterns can be explained by the fact that as the capacity increases, the masses of copper, iron, and oil increase, thereby increasing the thermal capacity. As the heat generation power increases, thermal resistance decreases to constrain the temperature rise limits.

3.3. Temperature Rise Predictions with Different Transformers and Losses

3.3.1. Prediction of Temperature Rise for Transformers with Different Capacities

The typical temperature rise curves of transformer oil and windings for 630 kVA- and 800 kVA-capacity transformers obtained from experiments and predictions are shown in Figure 9, and the convergence process of the fitting error is shown in Figure 10.

As shown in Figure 8, Figure 9 and Figure 10, from the start of temperature rise to the steady-state temperature rise process, the predicted and measured temperature rise curves show a high degree of overlap, with the RMSE eventually stabilizing at a value below 5, indicating good fitting performance. The steady-state temperature rise and the absolute errors between the predicted and measured temperature of the high-voltage and low-voltage windings and top oil temperature of 630 kVA- and 800 kVA-capacity transformers are shown in Figure 11.

As shown in the figure, the prediction model demonstrates stable accuracy for the top oil temperature, with errors lower than 2 °C. The maximum prediction error for the average temperature rise in the high-voltage winding is 2.9 °C, while the maximum prediction error for the average temperature rise in the low-voltage winding is only 2.6 °C. Such accuracy can meet engineering requirements.

3.3.2. Impact of Losses on Prediction Results

The total losses that need to be applied during temperature rise tests increase with the capacity of the transformer. Conducting temperature rise tests on high-capacity transformers at low power and for short periods of time can save energy. To investigate the accuracy and feasibility of parameter identification under different loss power conditions, parameter identification was performed on the same 800 kVA transformer at 0.5 Q_W, 1.0 Q_W, 1.2 Q_W, and 1.5 Q_W. The curve of the root mean square error, calculated according to Equation (10) with the change in evolutionary generation, is shown in Figure 12, and the identification results of thermal capacity and thermal resistance under different power levels are shown in Figure 13.

Figure 12 shows that the parameter fitting results at different power levels gradually converge as the number of evolutionary generations increases. The RMSE at 200 generations is well below 5, indicating that the top-layer oil temperature fits well at all power levels. The results in Figure 13 show that the identified parameters vary within a small range, indicating that the proposed method is numerically stable for various power conditions.

The identification results were used to predict the temperature rise curve under 1 times the total loss, and the results are shown in Figure 14.

As shown by the results, in the 2 h temperature rise test, the temperature increases in the top-layer oil at four different power loss levels were 14.7 °C, 23 °C, 28.1 °C, and 35.4 °C. The normalized temperature rises to Q_W are 29.4 °C/Q_W, 23 °C/Q_W, 23.4 °C/Q_W, and 23.6 °C/Q_W. The above results indicate that measurement results at 0.5 times the total loss may be influenced by system nonlinearity and measurement errors under low temperature rise conditions. Temperature rises above 1 times the total loss are within the permissible error range and conform to the response characteristics of a linear system. The steady-state top-layer oil temperature prediction errors at four different power loss levels are shown in Figure 15.

Figure 15 shows that among the thermal circuit model parameters identified under different power loss conditions, the parameters identified using 1, 1.2, and 1.5 times the total power loss are accurate for predicting the temperature rise. This is due to the system nonlinearity at low temperature rises.

Based on the results shown in Figure 13, Figure 14 and Figure 15, a fundamental analysis of the cause reveals that the nonlinear temperature rise at low power levels caused a shift in the parameter identification results, thereby increasing the prediction error. Therefore, the temperature rise from tests conducted at 1 times the total loss can be used to predict the temperature rise in the windings and top-layer oil.

4. Discussion and Conclusions

This study proposes a fifth-order thermal circuit model, identifies the model parameters from the first two-hour temperature test, and predicts the steady-state temperature. The conclusions are as follows:

(1)

The thermal resistance of the low-voltage winding is higher than that of the high-voltage winding. The oil–solid thermal resistance is significantly lower than the solid–gas thermal resistance. The parameter variation range for transformers of the same capacity, but different individuals, is lower than 10%;

(2)

The results indicate a positive correlation between thermal capacity and losses. A negative relationship was found between the thermal resistance and capacity. This coincides with the necessity to enhance the efficiency of heat dissipation in transformer design as the capacity increases;

(3)

The maximum absolute error of the calculated temperature is 2.9 °C. When the loss power is only half of the rated losses, the prediction error increases substantially, reaching a maximum of 6.9 °C. This can be attributed to the nonlinear characteristic shift exhibited by the transformer’s thermal system under low loss power conditions.

The method proposed in this paper can reduce the temperature rise test to 2 h. It is a fast, energy-efficient approach for the rapid assessment of transformers’ thermal performance.