Optimum Design of Transformers for Offshore Wind Power Generators Considering Their Behavior

Abstract

This paper presents a structural optimization technique based on precise heat transfer analysis for transformers. Mechanical and thermal stresses during operation are particularly critical in confined environments, such as within wind turbine generators. To optimize the transformer’s structure, the temperature distribution is first calculated using computational fluid dynamics (CFD) based on the finite volume method. Eddy current losses, necessary for thermal analysis, are determined through electromagnetic analysis. These CFD results are then used as input conditions for structural analysis. Finally, structural optimization is performed on critical areas, primarily the radiation fins and panels. All models are analyzed in 3D, and the simulation results are validated through comparison with experimental data. Consequently, the optimized design results were validated against simulations, exhibiting a maximum deviation of 4%. These findings confirm the success of the proposed transformer design optimization.

1. Introduction

Structural optimization is essential in the design of transformers used in offshore wind turbine generators. These transformers are frequently subjected to vibration, salt, and moisture exposure due to their maritime environment. To mitigate deterioration from such contaminants, a hermetically sealed transformer design is preferred over the conventional conservator type. In this hermetically sealed design, there is no air buffer between the enclosure tank and the insulating oil, which presents unique thermal and mechanical challenges [1,2].

Without a conservator, higher internal pressure develops within the transformer, requiring a robust structural design to withstand mechanical stresses. Simultaneously, the temperature rise within the transformer is exacerbated due to the confined space, leading to increased thermal loads. While existing studies have individually explored these thermal and mechanical challenges, few have attempted to address them in an integrated manner, leaving an opportunity for more holistic design approaches [3,4].

Numerous studies have addressed the thermal and mechanical optimization of transformers, focusing on either performance improvement or reliability enhancement. Ortiz et al. [5] evaluated the natural convection cooling performance of dry-type transformers using computational fluid dynamics (CFD), emphasizing the importance of precise thermal modeling in managing overheating. Similarly, Kim et al. [6] employed coupled magneto-thermal finite element analysis (FEM) to predict temperature rise and heat distribution in gas-insulated transformers, combining electromagnetic and thermal characteristics for more accurate analyses. These studies demonstrated the critical role of thermal modeling in managing overheating, but often overlooked the mechanical consequences of thermal stresses.

On the other hand, structural stress analysis has been investigated to enhance mechanical robustness. Liu et al. [7] analyzed stress distribution in transformer tank walls under internal pressure, proposing designs to enhance structural integrity using FEM. Silva et al. [8] optimized the mechanical strength of corrugated radiator fins, addressing weaknesses caused by internal pressure variations and thermal expansion. While these studies improve structural integrity, they typically treat thermal and mechanical analyses as independent problems, failing to capture their interdependent nature in confined transformer systems.

A limited number of studies have attempted to integrate thermal and structural analyses. For example, Wang et al. [9] combined electromagnetic and thermal simulations to analyze core insulation performance, but the work did not extend to structural optimization. Similarly, Nie et al. [10] developed thermal network models for improved transformer cooling but lacked experimental validation, limiting the practical applicability of their findings.

To address conflicting design goals, multi-objective optimization techniques have been employed. Hong et al. [11], for instance, used response surface methods to enhance both efficiency and mechanical performance in electrical machines. However, there remains a lack of comprehensive studies that integrate thermal analysis, structural stress optimization, and experimental validation, particularly for transformers operating in offshore environments.

This paper presents an integrated optimization framework tailored specifically for offshore transformer applications. Unlike previous studies, it addresses the interplay between thermal and mechanical factors through a unified analysis and validation process. First, thermal behavior is precisely assessed by calculating ohmic losses through electromagnetic analysis and performing a 3D heat transfer analysis to evaluate temperature distribution and thermal impacts. Second, structural analysis is conducted to evaluate mechanical stresses, taking into account the thermal behavior obtained from the first step. Finally, optimization is applied to structurally weak areas based on these analyses. The proposed method was validated on a 3 MVA transformer, with mechanical stress testing conducted under pressure-rise conditions. Test results showed that cracks appeared at 0.75 bar, surpassing the yield strength, which closely matched the predicted values from our analysis. The optimization of the corrugated fins resulted in a more robust structural design, improving both thermal management and mechanical stability, as presented in this study.

2. Temperature Behaviors

2.1. Test Transformer

The specifications and figures of the test transformer used for the temperature-rise and mechanical tests are presented in

Table 1. Transformer Specifications.

Specification	Details
Model	Rated Power: 3 MVA
Rated Voltage	13.2 kV/400 V
Phase	3
Insulation Oil	Vegetable oil (FR3)
Cooling Method	Natural Convection

and Figure 1, respectively. The transformer utilizes vegetable oil (FR3) as the insulation oil, and the cooling method is based on natural convection.

A structural strength test was conducted to verify the mechanical integrity of the 3 MVA transformer under a test condition of 0.75 bar for 2 cycles. As shown in Figure 1, failure occurred in the form of cracks in the welded joints between the panel and the fin. This failure is primarily attributed to stress concentration at the welded joint, exacerbated by pressure fluctuations and the thermal expansion and contraction of the transformer components during operation. These factors weaken the structural integrity of the welded area, making it susceptible to cracking under mechanical stress [12]. This highlights the importance of optimal design to address structural strength issues caused by such failures. The detailed specifications for the test transformer are summarized in Table 1.

2.2. Electromagnetic Analysis

In this work, the eddy current losses, which are mainly on the panel, fin, and the clamp, are obtained through an electromagnetic analysis. The eddy current induced in conductor can be calculated from following equation,

$$▿\times {\displaystyle \frac{1}{\mu }}\left(▿\times \overrightarrow{A}\right)={\overrightarrow{J}}_t={\overrightarrow{J}}_s+{\overrightarrow{J}}_e$$

(1)

where ${\overrightarrow{J}}_t$ is the total current density, ${\overrightarrow{J}}_s$ is the source current density, and ${\overrightarrow{J}}_e$ is the eddy current density.

The power losses including source and eddy current losses are obtained by ohmic loss, the current density squared divided by the electrical conductivity, J²/$\mathsf{\sigma }$W/m³ [13]. Because the transformer has symmetric geometries only, a quarter analysis model of the real transformer is used for eddy current losses and computational fluid dynamics (CFD).

To calculate the power losses, 3D quasi-static electromagnetic analysis is used with the finite element method. The maximum eddy current density at panel and clamp are 2.2 and 0.5 × 105 A/m², respectively, as shown in Figure 2.

2.3. Heat Transfer Analysis

To analyze natural convection flows in the test transformers, Boussinesq approximation is used. The heat transfer coefficients for the panel and the fin are taken into account with an empirical formula in order to consider heat exchange between the panel and the surrounding air [14].

Winding and eddy current losses, as mentioned above, are inputted as heat sources for the heat transfer analysis. Other required models (radiation model, conduction model, turbulent model) are driven by computational fluid dynamics using the commercial CFD code Fluent. The ambient temperature is 300 K. Figure 3 shows the temperature contours of the panel, winding, and core. The hottest temperature of the winding is 404 K, in which the experimental value is 395 K, and the maximum temperature of the panel is 357 K [15].

3. Optimum Design

This section outlines the methodology for optimizing the transformer’s design, focusing on structural strength and stiffness. The optimization process includes defining design variables, analyzing structural performance, and identifying the optimal design solutions using response surface analysis.

3.1. Design Variables, Levels, and Samplings

The optimization process focuses on minimizing the maximum membrane stress in welded panels composed of steel plates [16]. To achieve this, careful selection of design variables is crucial for establishing an effective optimization framework.

Table 2. Design variables and their levels.

Level	Radiation Fin Thickness (mm), ${\mathit{T}}_{\mathit{fin}}$	Panel Thickness (mm), ${\mathit{T}}_{\mathit{panel}}$	Reference Image
1	1	8
2	2	12

summarizes the design variables and their respective levels, with $T_{fin}$ and $T_{panel}$ identified as key parameters influencing mechanical strength.

A central composite design (CCD) approach is adopted in order to efficiently explore the design space while minimizing computational cost. This method reduces the number of required simulations while maintaining the accuracy of the results. Each “experiment” in the CCD corresponds to a simulation case involving structural strength and natural frequency analyses, performed using a 3D finite element method (FEM). Although these analyses are computationally time- and resource-intensive, they provide critical insights into the behavior of the transformer prototype under various design conditions [17,18,19].

Table 3. Central composite design (CCD).

No.	${\mathit{T}}_{\mathit{panel}}$ (mm)	${\mathit{T}}_{\mathit{fin}}$ (mm)	Maximum Membrane Stress (MPa)	First Natural Frequency (Hz)
1	7	1	481.34	27.681
2	10	1.5	258.58	37.224
3	13	2	175.08	45.829
4	13	1.5	194.26	47.604
5	7	1.5	360.22	25.158
6	10	2	213.04	34.875
7	10	1	336.84	39.555
8	7	2	292.18	23.195
9	13	1	250.20	46.310

presents the sampling data, CCD configuration, and corresponding analysis results. The sensitivity of the design variables is assessed by examining their primary effects on the maximum membrane stress and the first natural frequency, ensuring a comprehensive understanding of their impact on structural performance.

3.2. Optimization Formulation

The optimization formulation of 3 MVA transformer can be defined as follows:

• CASE I: structural strength optimization
      Minimize: membrane stress ($T_{panel}$, $T_{fin}$)
• CASE II: stiffness optimization
      Maximize: First natural frequency ($T_{panel}$, $T_{fin}$)
• CASE III: structural strength and stiffness optimization
      Minimize: membrane stress ($T_{panel}$, $T_{fin}$)
      Maximize: First natural frequency ($T_{panel}$, $T_{fin}$)
      8 ≤$T_{panel}$ ≤ 12, 1 ≤ $T_{fin}$ ≤ 2

In this paper, two fitted second-order polynomials with three effective design variables for each objective function—structural strength and first natural frequency—are determined, as shown in Equations (2) and (3). The adjusted coefficients of determination, R, which indicate the reliability of the metamodel, are 99.8% and 97.9% for structural strength and first natural frequency, respectively.

$$\begin{array}{ll}\mathrm{Membrane}\mathrm{stress}=& 2121.79-161.15T_{\mathrm{panel}}-406.52T_{\mathrm{fin}}\\ & +3.45T_{\mathrm{panel}}^2+27.5T_{\mathrm{fin}}^2+14.75T_{\mathrm{panel}}·T_{\mathrm{fin}}\end{array}$$

(2)

$$\begin{array}{ll}\mathrm{First}\mathrm{natural}\mathrm{frequency}=& 6.39+3.37T_{\mathrm{panel}}-2.72T_{\mathrm{fin}}\\ & -0.039T_{\mathrm{panel}}^2-0.694T_{\mathrm{fin}}^2+0.317T_{\mathrm{panel}}·T_{\mathrm{fin}}\end{array}$$

(3)

4. Results and Discussion

This section presents the outcomes of the optimization process, comparing the initial and optimized designs in terms of structural strength and stiffness.

4.1. Structural Strength Results

The structural strength analysis was conducted to verify the membrane stress of the developing transformer under an internal pressure of 0.75 bar and thermal load, as determined by the CFD results. Figure 4 shows the contour and response surface of the maximum membrane stress. The maximum membrane stress is reduced from 332.86 MPa in the initial model to 175.18 MPa in the optimized model, representing a 47% reduction. This improvement is primarily due to the optimized panel and radiation fin thickness, which redistribute the stress more effectively.

Figure 5 shows the feasible region of the optimal solution and the solutions determined by the optimizer. The optimal model demonstrates better performance than the initial model. Minimizing the thickness of the radiation fin reduces both the maximum generated membrane stress and the radiation characteristics of the fin. Figure 6 presents the membrane stress distribution between the initial model and the optimized model.

4.2. Stiffness Improvement

The first natural frequency analysis is performed to prevent resonance in the transformer structure under development. Maximizing the first natural frequency is critical to ensure structural stability and avoid resonance, as it is prioritized among the various natural frequencies.

Figure 7 presents the contour plot and response surface of the first natural frequency. The results indicate that increasing the panel thickness while decreasing the radiation fin thickness leads to a significant increase in the first natural frequency. This outcome not only improves structural stiffness but also enhances the radiation characteristics of the fins.

The feasible region for the optimized solution and the results identified by the optimizer are illustrated in Figure 8. The optimization demonstrates that the first natural frequency of the optimum model is approximately 1.56 times higher than that of the initial design, highlighting the effectiveness of the proposed approach.

Figure 9 shows the first vibration mode shape for both the initial and optimized models. The differences in the mode shapes are primarily attributed to the variations in the panel and radiation fin thickness, reflecting the improvements in stiffness achieved through optimization.

4.3. Combined Optimization Results

The single-objective problem was addressed and examined previously. Here, a two-objective problem is considered. Figure 10 illustrates the feasible region of the optimal solutions determined by the optimizer, taking into account both structural strength and stiffness. As panel thickness increases and radiation fin thickness decreases, membrane stress decreases, and the first natural frequency increases.

Table 4. Comparison of original model and optimum model.

Model	Max. Membrane Stress (MPa)	First Natural Frequency (Hz)
Initial	332.86	34.852
Predicted optimization of structural and stiffness	177.5	46.217
Verification simulation result	175.18	46.431
Optimization error (%)	4.16	1.51

provides a comparison between the initial model and the optimized models. The structural stability of the initial model is inadequate due to exceeding the yield strength of the material, which is approximately 240 MPa. Therefore, design modifications should be considered.

Compared to the initial model, the structural and stiffness optimization model demonstrates superior performance in terms of maximum membrane stress and first natural frequency. A higher first natural frequency is advantageous, as it reduces vibration displacement. By maximizing the first natural frequency and avoiding resonance, the development of a low-noise transformer is anticipated.

4.4. Validation of Results

The results are compared with experimental data to verify the reliability of the optimized design. As shown in Figure 11, the membrane stress distribution and the first natural frequency mode shape of the optimized model align closely with the experimental results, with a maximum deviation of 4%. This demonstrates the accuracy and effectiveness of the proposed optimization framework.

5. Conclusions

This paper addresses the optimal design of a 3 MVA-rated transformer for wind turbine generators, with a focus on structural strength and stiffness considerations. Electromagnetic and 3D CFD analyses were conducted as part of the design process. A structural strength test was carried out on a prototype (initial model) that had not yet undergone optimization. Failures and issues were observed during testing, with cracks exceeding the yield strength occurring at an internal pressure of 0.75 bar.

Consequently, statistical optimization was applied to improve the design. Structural strength analysis was performed under an internal pressure of 0.75 bar and thermal loads, with temperature results obtained from CFD and natural frequency analysis. The optimized design results were compared with simulations and found to be within a maximum deviation of 4%. Based on these findings, the transformer’s design optimization was successfully achieved.