Hot-Spot Temperature Reduction in Oil-Immersed Transformers via Kriging-Based Structural Optimization of Winding Channels

Abstract

Winding hot-spot temperature (HST) is a key factor affecting the insulation life of transformers. This paper proposes an optimization method based on the Kriging response surface model, which minimizes HST by adjusting the key structural parameters of the number of winding zones, vertical oil channel width, and horizontal oil channel height. First, a two-dimensional axisymmetric temperature–fluid field coupling model is established, and the finite volume method is used to solve the HST under the actual structure, which is 92.59 °C. A total of 50 sample datasets are designed using Latin hypercube sampling, and the whale optimization algorithm (WOA) is used to determine the optimal kernel parameters of Kriging with the goal of minimizing the root mean square error (RMSE) under 5-fold cross-validation. Combined with the genetic algorithm (GA) global optimization of structural parameters, the Kriging model predicts that the optimized HST is 89.77 °C, which is verified by simulation to be 89.79 °C, achieving a temperature drop of 2.80 °C, proving the effectiveness of the structural optimization method.

1. Introduction

Power transformers are vital components of electrical systems and must be safe and reliable during operation [1]. In transformer research, the high HST of transformer windings is an important cause of transformer insulation aging and failure. It follows the 6 °C principle, that is, in the temperature range of 80~140 °C, the transformer life is reduced by half for every 6 °C increase in temperature [2]. Reference states that when the hotspot temperature exceeds 122 °C, the relative thermal aging rate can be as high as 16 [3]. The transformer achieves voltage conversion through the electromagnetic coupling of primary and secondary windings. During operation, the windings and the core generate copper loss and iron loss, which are converted into heat and cause the equipment to warm up. The insulating oil transfers the heat from the windings through thermal convection to the oil tank wall, then through thermal conduction to the radiator, and finally dissipates as heat radiation into the environment [4,5]. Among all the losses generated by the transformer, copper loss is the main loss, and the highest temperature point occurs in the windings. Therefore, in order to maximize the service life of the power transformer and enhance its economic and practicality, it is necessary to minimize the hotspot temperature of the windings as much as possible. Using appropriate methods to predict the hotspot temperature of the transformer and reasonably distributing the transformer load, it is possible to ensure the safe operation of the transformer while increasing the load utilization rate. However, this method has its limitations in ensuring the safe operation of the transformer and improving its load capacity [6]. Optimizing the internal structure design of the transformer is beneficial for reducing the hotspot temperature of the transformer and is very important for ensuring the safe operation of the transformer and improving its load capacity [7].

In terms of optimizing the internal structure design of power transformers to reduce the temperature of the winding hotspots, many studies have been conducted both domestically and internationally. Currently, the research mainly focuses on two aspects: optimizing the electromagnetic design to reduce losses and improving the cooling system to enhance heat dissipation. In terms of optimizing electromagnetic design, reference [8] optimized the materials of the transformer to achieve the lowest copper loss and iron loss. Reference [9] conducted three-dimensional transient electromagnetic field simulation on transformer cores with mixed different types of laminations, providing suggestions for laminated mixtures to minimize core loss. Reference [10] proposed an optimal magnetic shielding scheme for a 400 MVA single-phase autotransformer based on an extreme learning machine.

In terms of improving the cooling system, reference [11] used the response surface method with the average Nu number as the objective function to optimize the key parameters of raised structures on the inner walls at the top and bottom of the transformer. Reference [12], based on the response surface method, optimized radiator design using the height (L) of the radiator, the spacing (D) of the fins, and the number (N) of the fins as the optimization variables, thereby determining the cost-optimal radiator layout for a specific cooling capacity of the radiator. Reference [13] developed an analytical model for rapid iterative optimization of the radiator in the oil-immersed air-cooled (ONAF) mode, optimizing the diameter of the fan blades. Reference [14] constructed a two-dimensional closed-loop CFD model, and showed, based on the results of the transformer’s thermal–flow field distribution, that installing an oil-blocking plate and increasing the height of the radiator could both improve oil flow circulation in the “dead oil area” and reduce the average or hotspot temperature rise of the windings. Reference [15] optimized the size of the winding coil core based on electromagnetic–thermal bidirectional coupling, ultimately achieving a significant reduction in the hotspot temperature of the winding and the amount of conductive material used. Reference [16] proposed an optimization strategy for the size of the transformer oil-blocking plate based on the dynamic radial basis function response surface model.

During the convective heat exchange process between transformer oil and winding coils, oil channels play a very important role. The performance of oil channels in guiding oil flow is related to factors such as the size of the transformer’s winding coils, the size of the oil channels, the position of the oil retaining plates, and the number of winding coils between the horizontal oil-retaining plates. Relatively few studies have focused on the optimization design of key parameters of the winding structure, such as the size of the oil channels, the number of oil-retaining plates, and the number of winding coils between the horizontal oil-retaining plates. However, the parameter design of the number of winding zones, the width of the vertical oil channels, and the height of the horizontal oil channels is very important for the study of the heat dissipation performance of the internal windings of the transformer.

This paper proposes an optimization method based on the Kriging response surface model for the structural optimization problem of oil-immersed power transformer windings, aiming to reduce the HST of the windings and improve the service life of the transformer. This paper first constructs a two-dimensional axisymmetric winding model of an oil-immersed power transformer with natural oil circulation [17,18]. Considering the comprehensive effects of multiple factors such as winding structure and cooling oil channel parameters, the number of winding zones, vertical oil channel width, and horizontal oil channel height are used as optimization variables. Subsequently, the temperature–fluid field simulation calculation is performed on the actual transformer winding structure to obtain the winding temperature field distribution results. The Latin hypercube sampling (LHS) method is used in the sample space to collect the HST data of the example and calculate the corresponding response values. Finally, in order to solve the optimal winding structure, this paper combines WOA to optimize the automatic relevance determination (ARD) regression kernel parameters and establishes the nonlinear relationship between winding structure parameters and HST through the Kriging response surface model method [19]. By optimizing the GA, the structural parameters such as the number of winding zones, vertical oil channel width, and horizontal oil channel height are optimized, aiming to minimize the HST and verify the feasibility of the optimization results.

2. Numerical Calculation of Transformer Temperature–Fluid Field

The multi-physics coupling analysis method can be divided into direct coupling and indirect coupling according to the coupling mode between the fields. Indirect coupling solves each physical field in a sequential manner and only considers the influence of the former on the latter in the coupling calculation. Direct coupling not only needs to consider the influence of the former on the latter but also the influence of the latter on the former [20]. In the transformer temperature–fluid field calculation involved in this paper, fluid movement affects the temperature field distribution inside the transformer, and the temperature change in turn affects the transformer oil material properties, thereby affecting the oil flow. Therefore, the direct coupling method is used for the solution.

2.1. Constitutive Relations

The object of optimization in this paper is an oil-immersed power transformer with natural oil circulation, whose capacity is 50 MVA, including high-voltage (HV), medium-voltage (MV), and low-voltage (LV) windings, and the operation mode adopts the HV–MV minimum tapping mode. The main nameplate parameters of this transformer are shown in

Table 1. Main nameplate parameters of a 50 MVA oil-immersed power transformer.

Nominal Ratings	Numerical Value
Rated capacity (kVA)	50,000
Rated Voltage (kV)	(110 ± 8 × 1.25%)/38.5/10.5
Rated Current (A)	262.4/780.2/2749.3
No-load current	0.24%
Short-circuit impedance	18.4%
No-load loss (kW)	40.5
Load loss of the HV-MV minimum tapping mode	256.5

. The main structural parameters of this type of transformer are shown in

Table 2. Main structural parameters of a 50 MVA oil-immersed power transformer.

Structural Parameters	Structural Dimensions (mm)	Structural Parameters	Structural Dimensions (mm)
LV winding inner diameter	722	Regulating winding outer diameter	1422
LV winding outer diameter	856	Winding body height	1000
MV winding inner diameter	900	Horizontal oil channel height	3
MV winding outer diameter	1042	Vertical oil channel width	8
HV winding inner diameter	1138	Number of LV winding sections	77
HV winding outer diameter	1270	Number of MV winding sections	58
Regulating winding inner diameter	1338	Number of HV winding sections	68

. Figure 1 shows a cross-sectional diagram of the HV winding, and 6 oil baffles divide the HV winding into 6 zones along the axial direction. The height of the horizontal oil channel is 3 mm, and the width of the vertical oil channel is 8 mm. The HV winding has a total of 68 coils, each coil consists of 7 turns of wire, and each turn of wire is made up of two combined wires wound in parallel.

According to the heat source distribution, cooling method, and structure of the transformer, the following assumptions are made for the model:

(1) The three-phase winding structure of the transformer is symmetrical, and the thermodynamic conditions are the same, so the three-phase problem can be converted into a single-phase problem.

(2) The transformer windings are geometrically symmetrically distributed along the circumferential direction, and the temperatures do not have gradient changes along the circumferential direction, so the three-dimensional calculation can be converted into a two-dimensional axisymmetric calculation.

(3) Based on the above assumptions, the three-dimensional problem is converted into a two-dimensional axisymmetric problem, which reduces the complexity of the calculation. The simplified two-dimensional model of the transformer is shown in Figure 2, where the number of coils of the LV winding is 77, the MV winding is 58, and the HV winding is 68. Since this paper needs to optimize the structure of the winding, a large number of sample sets need to be obtained. In order to improve the efficiency of modeling and simulation, the method of thermal equivalence for winding material parameters in the literature [21] is used to calculate the equivalent thermal conductivity and equivalent specific heat capacity of the winding along the axial and radial directions.

The thermodynamic parameters of the transformer oil are shown in

Table 3. Transformer oil thermodynamic parameters.

Physical Parameters	Values
Density/kg∙m−3	880
Specific heat/J∙kg−1∙K−1	1433 + 5.233 × (T − 273.15)
Thermal conductivity/W∙m−1∙K−1	0.105
Dynamic viscosity/Pa∙s	$e^{51.88}\times {\left(T\right)}^{-9.86}$

, and other physical parameters of the solid materials are shown in

Table 4. Other physical parameters of solid materials.

Material	Thermal Conductivity/W∙m−1∙K−1	Density/kg∙m−3	Specific Heat Capacity/J∙kg−1∙K−1
Copper	401	8933	395
Insulating cardboard	0.2	1400	1200
LV winding	Axial: 3.59	8933	395
	Radial: 1.67
MV winding	Axial: 5.28
	Radial: 1.69
HV winding	Axial: 4.51
	Radial: 0.94

[22]. The material of the transformer winding conductor is copper, and the material of the winding end insulation and oil baffle is insulating paperboard.

2.2. Temperature–Fluid Field Control Equation

For transformers with natural oil circulation heat dissipation, the natural convection process of transformer oil realizes the transfer and transfer of heat inside the transformer. The solution of the convection heat transfer process follows the three basic principles of physics: conservation of mass, conservation of momentum, and conservation of energy [23]. The specific description of the three in the Cartesian coordinate system is as follows:

(1) Mass conservation equation (continuity equation)

The fluid mass conservation equation is also called the continuity equation, which means that the increase in the fluid mass in the fluid microelement per unit time is equal to the net mass flowing into the fluid microelement. The specific differential form of the continuity equation is shown as follows:

$$\nabla \cdot \rho \mathit{v}=0.$$

(1)

In the formula, ∇ is the Hamiltonian operator, e_x, e_y, and e_z are the unit direction vectors on the x, y, and z coordinate axes, respectively; t is the time quantity; v is the fluid velocity vector; $\mathrm{v} = u{\mathrm{e}}_x+v{\mathrm{e}}_y+w{\mathrm{e}}_z$, where u, v, and w are the flow velocities of the fluid in the x, y, and z directions, respectively; ρ is the fluid density, kg/m³.

(2) Momentum conservation equation

Fluid particles are subject to mass force and surface force during motion. The mass force is a non-contact force, including gravity and electromagnetic force, while the surface force is a contact force, including fluid viscosity and pressure. The physical meaning of momentum conservation is that the rate of increase in fluid momentum in a fluid element is equal to the sum of various forces acting on the fluid element. The specific differential form of the momentum conservation equation is shown in Equation (2):

$$\rho \cdot \nabla \mathit{v}=\mathit{F}-\nabla p+\eta {\nabla }^2\mathit{v}.$$

(2)

In the formula, p is the fluid pressure, in Pa; f is the external force per unit volume of the fluid, N/m³; η is the dynamic viscosity of the fluid, kg/(m·s).

(3) Energy Conservation Equation

The energy conservation equation states that the rate of change of energy in a fluid element is equal to the sum of the work done by the volume force and the surface force on the element and the net heat flow entering the element. The differential form of the energy equation is shown in Equation (3):

$${\displaystyle \frac{\partial (\rho T)}{\partial t}}+\nabla \cdot (\rho \mathit{v}T)=\nabla ({\displaystyle \frac{k}{c_P}}\nabla T)+\mathsf{\Phi }+S_h.$$

(3)

In the formula, T is the fluid temperature, K; c_p is the specific heat capacity of the fluid, J/(kg·K); k is the thermal conductivity of the fluid, W/(m·K); Φ is the heat source in the fluid; S_h is the part of the fluid’s mechanical energy converted into thermal energy under the action of fluid viscosity.

Equations (1)–(3) contain the quantities v, p, T, and ρ to be solved. In order to make the equations closed, a state equation that relates p and ρ needs to be added, namely

$$\rho =f(p, T).$$

(4)

The finite volume method discretizes the field to be solved into a finite number of non-overlapping control volumes. It then integrates and discretizes the differential control equation for each control volume, resulting in a set of linear equations based on the field node variables and calculates and solves them. Since the finite volume method integrates and discretizes the control equations within each control volume, the conservation property of the equation is satisfied in a single control volume and the entire solution domain and does not require the solution in the calculation domain to be continuous everywhere. Compared with the finite element method, it has better convergence and solution accuracy [24,25]. Therefore, this paper uses the finite volume method to discretize and solve the above control equations and obtain the temperature field and fluid field distributions in the transformer.

2.3. Heat Sources and Boundary Conditions

2.3.1. Heat Source Loading

The total loss of the transformer includes no-load loss and load loss. The no-load loss is mainly caused by the hysteresis effect and eddy current effects in the iron core during the operation of the transformer. The load loss is mainly composed of winding copper loss and eddy current loss in the metal structural parts and the oil tank. The copper loss in the transformer winding is caused by its own resistance and the leakage magnetic field near the winding wire. The eddy current loss in metal structural parts and the oil tank is also caused by the leakage magnetic field of the transformer.

In this paper, a two-dimensional model of the transformer winding is established, and only the copper loss of the winding is loaded as the heat source. The DC resistance loss is calculated using the formula $P_{dc}=I^2{R}_{dc}$, and the eddy current loss in the winding is calculated through eddy current field numerical calculation. Considering transformer operation under high and medium voltages and the rated most negative tap operation conditions, the total loss of the three-phase overall model, including the low-, medium-, and high-voltage and voltage-regulating windings, is calculated, as shown in

Table 5. Total loss of three-phase overall model.

Winding Position	Three-Dimensional Model Loss (kW)
LV winding	0.23
MV winding	81.49
HV winding	135.68
Regulating winding	23.57

. In the simulation, the loss is evenly applied in combination with the two-dimensional structure size of the winding.

2.3.2. Boundary Conditions

(1) Inlet boundary conditions: It is stipulated that the inlet oil flow is evenly distributed, and the velocity direction is perpendicular to the inlet boundary. From the three-dimensional simulation results, the oil flow velocity is 0.042 m/s, and the oil flow temperature is 67.2 °C.

(2) Outlet boundary conditions: The pressure is set to 0 Pa at the oil flow outlet.

(3) The thermal conductivity of the leftmost insulating paper tube and the upper and lower end rings of the LV winding is small. Therefore, heat exchange between these components and the outside is not considered, and the temperature boundary conditions for this part are set as adiabatic boundary conditions.

(4) When the oil flow moves, the oil flow is set to a non-slip wall boundary condition at the surface boundary of the solid material.

(5) Flow pattern: Since the flow velocity in the natural oil circulation transformer is very low, the Reynolds number is calculated using Formula (5):

$$\mathrm{Re}={\displaystyle \frac{\rho vL}{\mu }}.$$

(5)

The Reynolds number is less than 2100, so the flow pattern of transformer oil is laminar [26].

The setting of the model boundary conditions is shown in

Table 6. Boundary condition settings.

Winding Part	Properties	Size
Oil inlet	Oil flow rate	0.042 m/s
	Oil flow temperature	67.2 °C
Oil outlet	Outlet pressure	0 Pa
LV winding insulation left paper tube	No heat exchange	Adiabatic boundary conditions
Upper and lower insulation end rings

. Figure 3 is a schematic diagram of the model boundary conditions setting.

2.4. Grid Independence Verification

The number of grids has an important influence on the accuracy of the solution of the transformer temperature fluid field. Too few grids cannot guarantee the solution accuracy, while too many grids increase the solution amount and reduce the solution efficiency. For the two-dimensional winding model in this paper, it is important to encrypt the boundary layer grid of the oil channel near the winding coil to fully capture the boundary layer effect of the fluid. The calculation results of the HST of the winding under different boundary layer grid division numbers set in this paper are shown in

Table 7. Calculation Results of Winding Hotspot Temperature under Different Grid Partition Levels.

Model	Number of Horizontal oil Channel Divisions	Number of Vertical Oil Channel Divisions	Total Number of Grids (10,000)	Winding HST (°C)
A	4	4	11.28	89.06
B	8	8	16.99	91.10
C	8	16	22.85	91.83
D	16	24	41.32	92.59
E	24	32	65.22	92.90
F	32	40	94.55	93.04

. The temperature difference between model D and model F is within 0.5 °C, which can be considered that the grid is basically converged [27]. Considering calculation accuracy and solution efficiency, this paper selects the division method in model D as shown in Figure 4.

2.5. Simulation Results Analysis

Using the 2D axisymmetric model of the transformer winding in Figure 2, along with the material properties and boundary conditions in Table 2 and Table 3, the steady-state distribution of the temperature field under the actual structure of the winding (6 zones, vertical oil channel width 8 mm, horizontal oil channel height 3 mm) was calculated, as shown in Figure 5, and the temperature distribution along the center line of the HV winding was further extracted, as shown in Figure 6. As seen in Figure 6, the temperature distribution of the winding coil is a cycle of “oil baffle–winding coil–oil baffle”, and it shows a trend of first rising and then falling within the same zone. From the overall temperature distribution, the temperature of the winding coil generally shows an upward trend with the increase in height. The HST of the winding is 92.59 °C, which is located at the first coil of the sixth zone of the HV winding. This is because the existence of the oil baffle causes poor heat dissipation conditions at this location. The flow velocity distribution of the coil in zone 3 is further extracted, as shown in Figure 7. The flow velocity is large near the oil inlet and oil outlet, and the maximum flow velocity is 0.09 m/s. Figure 8 shows the distribution of winding coil temperature and horizontal oil channel flow velocity along the axis of HV winding zone 3. Near the oil baffle, the oil flow is almost stagnant, resulting in a high coil temperature, which shows that the number of winding coil zones has a great influence on the winding temperature distribution. The coil temperature along the axis shows a trend of low at both sides and high in the middle, and the oil flow velocity in the horizontal oil channel also shows a trend of low at both sides and high in the middle, which shows that the density of the oil flow boundary layer near the coil in Section 2.4 fully captures the boundary layer effect of the fluid.

In conclusion, this section introduces the numerical calculation method for calculating the two-dimensional temperature–fluid field of the transformer. On one hand, it obtains the temperature distribution of the windings under the actual structure, where the maximum temperature is 92.59 °C. On the other hand, this calculation method is the cornerstone for optimizing the winding structure. The output response values under different winding structures are all calculated using this method. In the following text, the number of winding coil zones, the vertical oil channel width, and the horizontal oil channel height will be optimized to reduce the HST of the winding.

3. Kriging Response Surface Model Training

Response surface methodology (RSM) is a classic optimization method that can significantly reduce the computational cost of experiments or numerical simulations by establishing an explicit mathematical model between design variables and target responses while quickly determining the optimal combination of design parameters to optimize the objective function [28].

The RSM optimization process is mainly divided into two stages: (1) experimental design stage: using scientific sample design strategies (such as Latin hypercube design) to obtain system response data under different level combinations of design variables; (2) modeling and optimization stage: building a high-precision response surface model based on sample data, using explicit mathematical expressions to approximate the complex nonlinear relationship between design variables and target responses. In this study, the number of winding zones, vertical oil channel width, and horizontal oil channel height are used as design variables, and the winding HST is used as the target response. A Kriging response surface model is used to establish the nonlinear relationship between the two.

3.1. Kriging Response Surface Model

The Kriging Model is an interpolation method based on a Gaussian process, which is used to approximate complex nonlinear functional relationships. It predicts the response value at unknown points through known sample points and can provide uncertainty estimates of the predicted values. The following is a brief introduction to the principle of Kriging method [29,30].

A Kriging model usually consists of two parts: a global polynomial regression model (used to describe deterministic trends) and a local random deviation model (used to describe nonlinear or noise parts). Assuming that m sample points $\mathit{X}={[x_1,x_2,\dots ,x_m]}^T$ and response vectors $\mathit{T}={[T_1,T_2,\dots ,T_m]}^T$ are collected, the mathematical model of Kriging is as follows:

$$T(x)=\mu +Z(x),$$

(6)

where $T(x)$ is the response value at the sample point x, μ is a parameter model, generally a constant, and $Z(x)$ is a random process with a mean of zero, and its correlation function is generally a Gaussian function. For any sample point, the Kriging model predicts its response value as follows:

$$\widehat{T}(x)=\widehat{\mu }+{\mathit{r}}^T(x){\mathit{R}}^{-1}(\mathit{T}-\mathit{I}\widehat{\mu }),$$

(7)

where $\widehat{\mu } ={\mathit{I}}^T{\mathit{R}}^{-1}\mathit{T}/{\mathit{I}}^T{\mathit{R}}^{-1}\mathit{I}$, $\mathit{I}={[1,1,\cdots ,1]}^T$, $\mathit{T}={[T_1,T_2,\dots ,T_m]}^T$, and $\mathit{R}$ is the correlation matrix, which is determined using Formula (8).

$$\mathit{R}=\left[\begin{array}{ccc}R(x_1,x_1)& \cdots & R(x_1,x_m)\\ ⋮& \ddots & ⋮\\ R(x_m,x_1)& \cdots & R(x_m,x_m)\end{array}\right],$$

(8)

where $\mathit{r}(x) ={[R(x,x_1),\cdots ,R(x,x_m)]}^T$ is the correlation vector between the prediction point and the sample point.

Using the ARD Gaussian correlation function, $R(x_i,x_j)=\exp \left[-{\displaystyle \sum _{k=1}^n{\theta }_k}{(x_i^k-x_j^k)}^2\right]$, n is the number of levels of the design variable. The unknown parameter vector $\mathit{\theta }$ can be obtained by maximizing the likelihood function $\underset{\theta }{max}\left[-{\displaystyle \frac{m}{2}}\ln ({\widehat{\sigma }}^2)-{\displaystyle \frac{1}{2}}\ln |\mathit{R}|\right]$. The larger the ${\theta }_k$ is, the more significant the influence of the corresponding variable on the response quantity is.

Specifically for the Kriging response surface model in this paper, the design variables are defined as three-dimensional variables, the number of zones V, the vertical oil channel width W₁, and the horizontal oil channel height W₂, and the HST response value at the sample point. For the Kriging model, the parameters that need to be determined are mainly the relevant parameter vector $\mathit{\theta }=[{\theta }_V,{\theta }_{W1},{\theta }_{W2}]$, where θ_V is the relevant parameter of the zone number V, θ_W1 is the relevant parameter for the vertical oil channel height W₁, and θ_W2 is the relevant parameter for the horizontal oil channel height W₂.

3.2. Experimental Design

This section introduces the LHS method to obtain the sample dataset and uses the temperature–fluid field calculation method in Section 1 to obtain the numerical calculation value of the winding HST for each sample. The LHS method is a commonly used experimental design technique that can evenly distribute design variables in a smaller sample space, ensuring that the combination of each parameter can fully represent the characteristics of the entire design space, thereby improving the efficiency and accuracy of the optimization process. Through LHS sampling, different combinations of the number of winding zones, vertical oil channel width, and horizontal oil channel height can be systematically selected, and the response value corresponding to each set of design variables (i.e., the winding HST) can be numerically calculated.

In the experimental design process, the design variables of the winding structure include three main parameters: the number of winding zones, the width of the vertical oil channels, and the height of the horizontal oil channels. Through LHS sampling design, 50 sets of calculation examples were generated, as shown in

Table 8. Sample dataset.

No.	Number of Zones	Vertical Oil Channel Width (mm)	Horizontal Oil Channel Height (mm)	Winding HST (°C)
1	4	4	3	103.70
2	4	4	4	106.30
3	4	6	3	94.32
4	4	7	2	94.23
5	4	9	4	96.12
6	4	10	6	96.92
7	4	11	6	96.80
8	4	12	4	90.89
9	4	12	6	100.80
10	5	6	3	94.31
11	5	6	5	98.26
12	5	8	2	94.73
13	5	8	4	93.94
14	5	10	5	94.83
15	5	10	6	96.25
16	5	12	4	90.21
17	5	12	5	92.77
18	6	4	6	111.00
19	6	6	4	94.97
20	6	6	5	96.51
21	6	7	4	92.79
22	6	8	2	99.00
23	6	8	3	92.59
24	6	8	4	91.65
25	6	8	5	95.21
26	6	9	6	95.72
27	6	10	3	91.58
28	6	10	5	93.54
29	6	11	2	102.09
30	6	12	2	105.30
31	7	4	3	105.30
32	7	4	5	109.20
33	7	4	6	111.00
34	7	6	5	96.23
35	7	8	3	93.58
36	7	8	5	93.18
37	7	10	3	92.16
38	7	10	4	91.03
39	7	12	4	90.47
40	8	6	3	97.36
41	8	6	4	95.83
42	8	6	6	97.59
43	8	7	6	95.22
44	8	8	2	102.80
45	8	8	4	92.97
46	8	9	2	102.61
47	8	10	2	103.40
48	8	11	4	91.46
49	8	12	2	109.20
50	8	12	4	91.12

. The number of zones is controlled between 4 and 8, the width of the vertical oil channels is controlled between 4 and 12 mm, and the height of the horizontal oil channels is controlled between 2 and 6 mm. These values were determined based on the actual limitations of the transformer design. Since the actual number of partitioned windings for this type of transformer is six zones, the width of the vertical oil channels is 8 mm, and the height of the horizontal oil channels is 3 mm, the design variables in this article are set within the range of fluctuations based on the actual structure. At the same time, considering the practical issues of production and manufacturing, all values are rounded to integers. The LHS sampling design ensures that the distribution of sample points is highly representative, covering various structural configurations that may affect the winding temperature. This provides sufficient experimental data for the subsequent establishment of the response surface model.

3.3. WOA Optimization

This section combines the WOA to optimize the ARD kernel parameters in the Kriging response surface model and establishes a nonlinear relationship between the winding structural parameters and the HST of the winding [31]. First, the calculation results of the HST of the transformer are preprocessed, and then the WOA is used to simulate the encirclement mechanism, bubble net attack, and random search strategy of the whale’s predation behavior in the range of [0.1,10]. The fitness function is the minimum RMSE under 5-fold cross-validation, and the optimal kernel function scale parameter of the Kriging model is automatically searched. The Kriging model is constructed based on the optimized parameters. The model can not only accurately predict the HST but also quantify the importance of each structural parameter through the kernel function scale parameter.

WOA is used to optimize the ARD kernel parameters $\mathit{\theta }=[{\theta }_V,{\theta }_{W1},{\theta }_{W2}]$ in the Kriging response surface model. Compared with the traditional grid search with large computational complexity, the meta-heuristic algorithm can be used for efficient search. WOA establishes a mathematical model based on the three predation behaviors of humpback whales, including the encirclement of prey stage, the bubble-net attack stage, and the random search stage. In the encirclement of prey stage, we assume that the current best whale position (i.e., the current optimal solution) is the target prey, and other whales update their positions towards this position.

In each iteration, an individual whale selects an update strategy according to probability p, that is, the algorithm chooses between prey encirclement or search and spiral update with a probability of 50%. When p < 0.5 and $\left|\mathbf{A}\right|<1$, it enters the prey encirclement stage, and the whale approaches the optimal solution to achieve local exploration. When p < 0.5 and $\left|\mathbf{A}\right|\ge 1$, the prey search phase begins. The whale moves away from the current optimal solution and move to a randomly selected whale position to achieve global exploration. When p ≥ 0.5, the whale updates its position in a spiral manner, thus simulating the bubble-net attack behavior.

In the 5-fold cross-validation, the optimal kernel function parameter of the established Kriging model is $\mathit{\theta }=[{\theta }_V,{\theta }_{W1},{\theta }_{W2}]=[9.5176, 3.6796, 6.8859]$. This shows that the vertical oil channel width has the greatest impact on the HST of the winding, and the number of winding zones has the least impact on the HST of the winding. The RMSE under the 5-fold cross-validation is 1.1023 °C, and the RMSE of the Kriging response surface model finally constructed is 0.3422 °C, and the goodness of fit is 0.9963. The specific prediction results are shown in Figure 9.

4. Winding Structure Parameter Optimization

In this section, the established Kriging response surface model is optimized using a GA to find the structural parameters that minimize the HST of the transformer winding.

After constructing the Kriging response surface model, the GA is used to optimize the structural parameters globally. Constraints are set according to the actual needs of the project: the number of zones is controlled at 4~8, the vertical oil channel width is controlled at 4~12 mm, and the horizontal oil channel height is controlled at 2~6 mm to ensure that the model is suitable for practical applications. By evolving 50 populations over 100 generations, combined with selection, crossover, and mutation operations, the optimal structural combination that minimizes the HST of the transformer winding is finally found.

4.1. Structural Optimization Using Genetic Algorithm

The core parameters affecting the transformer’s heat dissipation performance include the number of zones V, the vertical oil channel width W₁, and the horizontal oil channel height W₂. The design variables are defined as three-dimensional variables $x={[V,W_1,W_2]}^T$, $\widehat{T}(x)$ and the HST is defined as a function of x, thereby constructing the Kriging response surface model:

$$\begin{array}{l}\min \widehat{T}(x)=\widehat{\mu }+{\mathit{r}}^T(x){\mathit{R}}^{-1}(\mathit{T}-\mathit{I}\widehat{\mu })\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}V\in Z,V_{\min }\le V\le V_{\max }\\ W_1\in Z,W_{1\min }\le W_1\le W_{1\max }\\ W_2\in Z,W_{2\min }\le W_2\le W_{2\max }\end{array}\right.\end{array}.$$

(9)

Considering the actual situation, the number of transformer zones is usually set to 4~8. If the number of zones is too small, the cooling oil flow cannot be guided to fully dissipate the heat of the winding. If the number of zones is too large, the cooling oil is strongly hindered, the oil flow rate becomes slow, the heat is removed slowly, and the heat dissipation efficiency is low. The vertical oil channel width is set to 4~12 mm, and the horizontal oil channel height is set to 2~6 mm. Considering the simplicity of production, the oil channel size is designed in 1 mm intervals.

The GA is used to find the optimal structure of the transformer for the Kriging response surface model. The core process includes three core operations: selection, crossover, and mutation. Integer encoding is used to directly represent the structural parameters, that is, the three-dimensional variable parameters are integers. The population size is configured to be 50, the maximum number of generations is 100, and the parameter x to be optimized is standardized and input into the Kriging model to predict the temperature.

$$f(x)=\mathrm{Kriging}\left({\displaystyle \frac{x-\mu }{\sigma }}\right),$$

(10)

where $f(\mathit{x})$ represents the individual fitness, which is the predicted temperature value from the Kriging model. The lower the temperature, the better the fitness.

The final convergence mechanism and termination condition are $\left|f_{\mathrm{best}}^{(t)}-f_{\mathrm{best}}^{(t-k)}\right|<ϵ\left(ϵ={10}^{-3}\right)$, which means that the iteration stops when the optimal temperature change over consecutive k generations is less than 0.001 °C, and the optimal structure is found. The optimization process of winding structure using genetic algorithm is shown in Figure 10.

4.2. HST Comparison Before and After Optimization

The Kriging response surface model is globally optimized using the WOA–GA joint optimization algorithm, and the optimal parameter combination of the transformer winding structure is obtained, as shown in Table 8. The optimized winding coil temperature cloud map is shown in Figure 11. The temperature distribution of the HV winding coils along the axial line in the optimized and non-optimized cases is shown in Figure 12. As can be seen from the figure, after the baffles are used for partitioning, the transformer coil temperatures show an overall upward trend from low to high, consistent with the bottom-to-top flow trajectory of the transformer oil. Furthermore, the highest temperature is observed in the first coil of each zone. This is because the baffles both guide and obstruct the oil flow, resulting in poor heat dissipation and higher temperatures in the first coil near the baffles in each zone. In zone 1, due to its location at the oil inlet and the absence of baffles, heat dissipation is effective, and the initial temperature does not rise too high. After optimization, except for winding zones 1 and 2, where coil temperatures are slightly higher than those in the actual structure, the coil temperatures in the remaining zones are lower than those in the actual structure. In particular, in the high-temperature zones from zone 4 to 6, the overall temperature shows a significant drop, with the HST dropping by 2.807 °C, which meets the purpose of reducing the HST and proves the effectiveness of the winding structure optimization method. The comparison results of hotspot temperatures before and after optimization are shown in

Table 9. Comparison of HST before and after optimization.

Parameters	Actual Structure	Optimized Structure
Number of zones	6	6
Vertical oil channel width (mm)	8	11
Horizontal oil channel height (mm)	3	4
HST response value (°C)	92.38	89.77
HST simulation value (°C)	92.59	89.79
Error (°C)	0.21	0.02

.

4.3. Optimization Results and Verification

The winding structure optimization flowchart of this paper is shown in Figure 13. First, the transformer multi-physics field simulation calculation results are preprocessed, and WOA is combined with a fitness function to optimize the ARD kernel parameters with the minimum predicted RMSE under 5-fold cross-validation. Finally, the nonlinear relationship between the winding structure parameters and the HST is established. By optimizing the GA, the structural parameters such as the number of winding zones, vertical oil channel width, and horizontal oil channel height are optimized to obtain the optimal winding results. Finally, the optimization results are verified through multi-physics field simulation.

5. Conclusions

Based on the Kriging response surface model, this paper optimizes the key structural parameters of the number of zones, vertical oil channel width, and horizontal oil channel height of an oil-immersed transformer winding, effectively reducing the HST. The main research conclusions are as follows:

(1) A two-dimensional axisymmetric model of the transformer winding is established. By applying equivalent winding thermal parameters, the temperature field distribution under the actual structure of the transformer winding is obtained. The HST of the winding under the actual structure is 92.59 °C, located at the first pancake of the sixth zone of the HV winding.

(2) Based on the Kriging response surface model, a nonlinear relationship between the HST of the winding and the structural parameters of the winding is constructed. The RMSE of the Kriging response surface model finally obtained is 0.3422 °C, and the goodness of fit is 0.9963.

(3) A multi-objective optimization mathematical model is established to minimize the HST of the winding. Through a genetic algorithm, the optimal winding structure is obtained: the number of zones is six, the vertical oil channel width is 11 mm, and the horizontal oil channel height is 4 mm. Numerical calculation verification is carried out. After optimization, the HST of the winding is 89.77 °C, which is 2.80 °C lower than the actual structure of the winding.

This paper proposes a combined WOA–GA optimization strategy for transformer structural parameters. A two-dimensional axisymmetric model of the transformer windings is trained and optimized to obtain the optimal parameter combination, providing guidance for transformer design. However, a more complex three-dimensional model was not developed for discussion, and the steady-state analysis does not consider temperature fluctuations under dynamic loads. Future efforts will advance transformers towards high reliability and long life through multi-physics collaborative design and the development of intelligent algorithms.