Structural Analysis and Lightweight Optimization of a Buoyant Rotor-Type Permanent Magnet Generator for a Direct-Drive Wind Turbine

Abstract

This study presents a structural analysis and optimization for the lightweight design of a buoyant rotor-type permanent magnet (BRPM) generator, which was first presented in Bang (2010), and compares its structural performance to that of a conventional generator with a spoke arm-type rotor and stator. The main benefit of a BRPM generator is that it can be constructed as a bearingless drive system, free from the mechanical failure of rotor bearings, by using a buoyant rotor. Additionally, the deformation of the generator by blade vibration can be effectively suppressed using joint couplings between the blades and the rotor. For design optimization, the objective is set as the mass of the rotor and the stator, and the maximum deformation of the airgap clearance between the rotor and the stator by external forces is constrained below 10% of the gap width. The commercial software OptiStruct is used for the analysis and optimization. In this investigation, the analysis and optimization are conducted for a 10 MW wind turbine generator. However, the proposed methods can be extended to larger generator designs without requiring considerable modification. The mass of the optimized 10 MW BRPM generator is 160.7 tons (19.3 tons for the rotor and 141.4 tons for the stator), while that of an optimized conventional spoke arm-type generator is 325.6 tons.

1. Introduction

Direct-drive generators have attracted increasing interest for large-sized wind turbines due to the low structural failure ratio and ease of maintenance [1,2]. Stronger wind at higher altitudes for a large-sized wind turbine facilitates a higher energy density per unit area. However, as the size of a wind turbine increases, the mass of the generator increases with a larger proportion. This is because the mass of the structural part of a generator is dominant over the active (or electromagnetic) part in a large-sized wind turbine. The larger mass of a generator requires stronger supporting structures in the nacelle and the tower, which results in a significant increase in the overall mass of the wind turbine structure. Thus, the lightweight design of a generator is important to lower the manufacturing and maintenance costs of large-sized wind turbines.

Considerable research has been conducted to achieve lightweight designs for permanent magnet direct-drive generators. Shape optimization for mass minimization using the dimensional parameters of structural or electromagnetic generator components as optimization variables has been extensively discussed in prior research [3,4,5,6,7,8,9]. Zavvos et al. [6,7] used an analytical approach to minimize the mass of a permanent magnet direct-drive generator by optimizing both the electromagnetic and structural dimension parameters simultaneously. They used a genetic algorithm for optimization and presented optimized designs for various diameter-to-axial-length ratios. In their works, the elastic deformations of a generator due to the electromagnetic force, gravitational force, and angular acceleration were constrained during optimization, for which the simplified equations for displacement calculations in [8] were used to deal with the heavy computational cost of the genetic algorithm. However, it is worth noting that the performance of optimization in these studies can be significantly influenced by the shape parameterization methods that are employed.

In order to enhance the efficiency of optimization while eliminating the constraints of shape parameterization, topology optimization has also been considerably employed [10,11,12]. Jaen-Sola et al. [10] minimized the structural mass of the rotor and stator of a direct-drive wind turbine generator using topology optimization. Starting from a disk-type rotor (or stator) with no holes, spiral-shaped (or circular-shaped) holes were removed from the disk through topology optimization for a mass-minimized design. They also optimized the thickness and topology of a conical rotor by considering dynamic responses as well as static stiffness to maintain the airgap clearance [11]. However, in their research, the main structure of the rotor had a predefined shape, and topology optimization was subsequently performed as a secondary design process. As a result, the potential for achieving significant mass savings through optimization was limited in their study. In the work of Hayes et al. [13,14], periodic repetitions of lattice structures manufactured by additive techniques were employed for the lightweight design of a 5 MW generator. In their research, the geometric dimensions of the unit lattice of a stator and a rotor were optimized for a mass minimization problem, while the structures had enough stiffness in the radial, torsional, and axial directions. In [15], a carbon/epoxy composite material was utilized to achieve mass savings in generators, ranging from 100 kW to 3 MW. However, for the manufacturing of large direct-drive generators exceeding 10 MW, the implementation of additive manufacturing and composite materials may necessitate additional development and advancements before becoming feasible.

Lightweight designs can be also achieved by using effective structural system layouts. Alstom used the so-called pure torque concept to achieve a lightweight and reliable structural design; bending loads on blades imposed by wind were not transmitted to the rotor by using separate bearings and flexible couplings between the blade hub and the rotor [16,17]. Shrestha et al. [18] proposed the employment of a magnetic bearing for the support of the rotor, where the air gap between the rotor and the stator was maintained by the magnetic floating effect. The effectiveness of a lightweight design by the use of the magnetic bearing support system was remarkable, with a 45% mass reduction in the case of a 5 MW generator compared to a conventional direct-drive generator. However, it can be less reliable for a heavy dynamic load and needs more complex system configuration and higher manufacturing costs than those of conventional ones. In [19,20], the bearingless system with a buoyant rotor-type permanent magnet (BRPM) generator shown in Figure 1a, which is buoyed by the fluid of the stator, was presented. Note in Figure 1a that the rotor is completely enclosed by the stator and is connected to the blades through joints without the use of support bearings. Seals should be used in the stator to prevent fluid leakage through the joint connection.

In this investigation, the BRPM generator in Figure 1a is employed for the design of a 10 MW direct-drive generator system. The BRPM generator offers several advantages over conventional generators for the lightweight design of large-sized generators:

(1)

It benefits from a bearingless drive, eliminating wear and fractures in the rotor bearing.

(2)

Joint couplings between blades and the rotor effectively prevent the transmission of vibrations and impulsive forces from the blades to the generator.

(3)

It experiences smaller deformations caused by the bending vibrations of the tower.

(4)

The rotor’s lightweight nature enables a more efficient gap control.

To achieve mass-minimized designs, CAE-based structural optimization methods are employed to determine the structural layouts and dimensions for both the buoyant rotor and the stator. However, a significant challenge arises in the case of the stator’s wall on the blade side, which requires disconnection for joint connection (see the inset of Figure 1a), leading to a notable reduction in the bending stiffness of the stator that supports the attractive magnetic force. Consequently, the most critical design issue for the BRPM generator is how to reinforce the stator wall with a minimum increase in mass. Furthermore, it is essential to highlight that the structural layout of the BRPM generator differs entirely from conventional types, necessitating the discovery of an effective layout from scratch, rather than modifying the existing designs. To address this issue, an array of ribs (or reinforcing plates) is introduced in the circumferential direction as stiffening members. Additionally, to maximize the stiffness-to-mass ratio of the ribs, topology optimization is employed. By solving the optimization problem, the layout of the mass-minimized rib is designed, while its stiffness satisfies a given constraint. For the design of the rotor, thickness optimization based on a surrogate model approach is used to determine the plate thicknesses. The commercial software OptiStruct [21] is used for the optimization.

To show the effectiveness of a BRPM generator from a structural viewpoint, the dynamic performance of the optimized BRPM generator is calculated for external vibratory loads. Load cases for blade vibration and tower vibration are analyzed by using finite element analysis. The results of the optimized BRPM generator are compared with those of a conventional spoke arm-type generator, which is also optimized by using the surrogate-model-based optimization method. The mass of the final design of the BRPM generator for a 10 MW wind turbine is obtained as 160.7 tons (19.3 tons for the rotor and 141.4 tons for the stator), significantly lower than the mass of the optimized conventional generator, with 325.6 tons (97.5 tons for the rotor and 228.1 tons for the stator).

2. Lightweight Structural Design of a BRPM Generator

2.1. Design Problem Definition and Load Cases

Figure 2 shows the stator and rotor of a BRPM generator. The buoyancy of the fluid inside the stator supports the rotor without contacting it. The active parts for power generation are not illustrated in the figure. The vertical plates of the stator and rotor are under attractive magnetic force, by which the bending deflection of plates arises as a dominant deformation. In Figure 2a, $w$ denotes the width of the gap for a joint coupling between the rotor and the blades. In order to reinforce the bending stiffness of the stator plates, ribs were employed in the circumferential direction of the stator as a circular array. The number of ribs and the depth (or thickness) of the rib were determined through numerical tests.

The overall size of the rotor is determined by considering the design conditions in Table 1. In Figure 2b, to determine the inner radius, $R_{ri}$, the height of the cross-section, $l_{rh}$, and the length, $l_{rb}$, of the buoyancy of the rotor should be also considered (see [19] for more details). Once the size of the rotor is determined, the size of the cross-section of the stator should be designed such that it can contain a volume large enough to provide buoyancy for the rotor support. The width of the stator, $l_{sb}$, in Figure 2a is expressed as:

$$l_{sb}=l_{rb}+l_{sa}+l_{ra}+2l_g+2t_{rb}$$

(1)

where $l_{sa}$ and $l_{ra}$ are the thicknesses of active materials on the stator and the rotor, respectively. In Equation (1), $l_g$ is the gap width between the rotor and the stator, which was set as $l_g=$ 8.9 mm in this study.

While the sizes of the cross-sections for the rotor and the stator can be calculated based on the generative power of the wind turbine, the plate thicknesses, such as $t_{sb1},t_{sh1}$, and the layout of the ribs are not pre-determined. Therefore, they should be optimized to obtain a mass-minimized rotor and stator, while the generator should have a large enough stiffness against external loads.

Table 2. Parameters for the design of the BRPM generator in Figure 2.

	Parameter	Value (mm)	Optimization
Stator	$R_{si}$	3757.3	-
	$l_{sh}$	1365.4	-
	$l_{sb}$	1978.1	-
	$t_{sb1}$	20	-
	$t_{sb2}$	20	-
	$t_{sh1}$	15	-
	$t_{sh2}$	20	-
	Rib	Design domain	Topology optimization
	$w$	10	-
Rotor	$R_{ri}$	3846.1	-
	$l_{rh}$	1187.8	-
	$l_{rb}$	1595.9	-
	$t_{rb}$	Design variable	Thickness optimization
	$t_{rh1},t_{rh2},\dots ,t_{rh9}$	Design variable	Thickness optimization

lists whether each parameter in Figure 2 is treated as a design variable for optimization or not. It also lists which optimization method is employed for each design variable.

Table 1. Design conditions of a BRPM generator.

Principal Factors	Values
Rated power	10 MW
Rated wind speed	12 m/s
Rated rotational speed	8.6 rpm
Maximum rotational speed	8.6 rpm
Cut in wind speed	3.2 m/s
Cut off wind speed	25 m/s
Airgap size	8.9 mm

Table 1. Design conditions of a BRPM generator.

Principal Factors	Values
Rated power	10 MW
Rated wind speed	12 m/s
Rated rotational speed	8.6 rpm
Maximum rotational speed	8.6 rpm
Cut in wind speed	3.2 m/s
Cut off wind speed	25 m/s
Airgap size	8.9 mm

Structural topology and thickness optimizations were performed by considering only the static deformations of the rotor and the stator. After optimizations, the dynamic responses of the structures were checked for verification purposes. A major deformation of the rotor and the stator was induced by attractive magnetic force. As additional external loads, the hydrostatic force by the fluid inside the stator, the shear force by the fluid due to the rotation of the rotor, and the gravitational force were considered.

The attractive magnetic force is estimated as:

$$F_{dn}=\frac{B_g}{2{\mu }_0}$$

(2)

where $F_{dn}$ is the normal force density in the airgap; $B_g$ is the magnetic flux density in the airgap; and ${\mu }_0$ is the permeability of air ($F_{dn}$ was calculated as 320 kPa in this investigation).

The upper bound of the hydrostatic force by a fluid acting on the surface of the plates of the rotor and the stator is:

$$p^H=\rho gh\le 1000{\mathrm{kg}/\mathrm{m}}^3\times 9.8{\mathrm{m}/\mathrm{s}}^2\times 10.25\mathrm{m}=100.5 \mathrm{kPa}$$

(3)

where $\rho$ and $g$ are the fluid density and gravitational acceleration, respectively. In the above, $h$, the depth from the free surface of the fluid, is lower than $h\le 10.25 \mathrm{m}$, which was estimated using the dimensions in Table 2.

Due to the rotation of the rotor, shear stress is induced on the plates of the rotor and the stator. If the fluid is assumed to be Newtonian, the maximum shear stress is:

$$\tau =\mu \frac{u}{w}=0.001\mathrm{Pa}/\mathrm{s}\times \frac{4\mathrm{m}/\mathrm{s}}{0.0089\mathrm{m}}=0.449\mathrm{Pa}$$

(4)

where $\mu$ is the dynamic viscosity, $u$ is the rotor velocity, and $w$ is the gap width between the rotor and the stator (8.9 mm in this work). In Equation (4), the velocity, $u=4\mathrm{m}/\mathrm{s}$, was calculated by using the rated rotational speed and the radius of the rotor.

A gravitational force acting on the stator includes the weights of the rotor and the fluid as well as the weight of the stator. For comparison purposes, the total gravitational force was converted to a surface force by dividing the total weight by the inner surface area of the stator:

$$f=\frac{W_r+W_s+W_f}{S}=\frac{0.491\mathrm{MN}+1.988\mathrm{MN}+0.210\mathrm{MN}}{186{.91\mathrm{m}}^2}=14.4\mathrm{kPa}$$

(5)

Because the masses of the rotor and stator are not pre-determined, their estimated values, by following those in [19], were used in Equation (5). A gravitational force acting on the rotor can be ignored due to the cancellation effect by the buoyancy.

Compared to the attractive magnetic force in Equation (2), the magnitudes of hydrostatic force in Equation (3) and the gravitational force of the stator in Equation (5) cannot be regarded as negligible. Thus, the static deformations of the BRPM generator by these three external loads were considered for problem formulations of structural topology and thickness optimizations. Dynamic responses by blade vibration and tower vibration were also checked for the final optimized design in Section 3.

2.2. Structural Optimization of a Stator

In Table 2, the parameters concerning the overall sizes of the stator and rotor were determined based on the design conditions in Table 1, as well as the necessary buoyancy [19]. Consequently, only the design densities in the stator rib and the plate thicknesses of the rotor were considered as design variables for topology optimization and thickness optimization, respectively.

Because the bending deformation of stator plates by external loads is mainly supported by the ribs, the design of ribs is crucial to obtain a lightweight stator with high stiffness. An optimal rib layout was found by using topology optimization for the gray region in Figure 2a. Topology optimization is a gradient-based optimization method that finds an optimum structural layout (or topology) under a limited resource of mass. Because topology optimization presents a totally new layout starting from a very simplified design domain, it can be efficiently applied to design problems with little previously known designs as in the present problem [22].

The gray region in Figure 2a is discretized by finite elements, each of which is parameterized with a design variable, $x_i$, for optimization:

$$E_i=E_0{x}_i^n (0<x_i\le 1)$$

(6)

where $E_i$ is Young’s modulus for element i (i = 1, 2, …, N; N: total number of elements in the gray region in Figure 2a), and $E_0$ is Young’s modulus for a rib material. In Equation (6), if the design variable is optimized as 1, the associated element is as stiff as a rib material, so the element is in a solid state. If the design variable is close to 0, the element has very little stiffness, i.e., in a void state. The exponent n in Equation (6) is a penalty parameter that pushes $x_i$ toward 0 or 1 during optimization (n = 2.5 was used in this work). Because the design variable in Equation (6) is continuous, the maximum or minimum of the objective can be found rapidly by using a gradient information. In Figure 2a, the wall thicknesses, $t_{sb1}$, $t_{sb2}$, $t_{sh1}$, and $t_{sh2}$, were set with low values, as shown in Table 2. If thicker wall thicknesses are preferable, topology optimization would add more material around plates, resulting in thicker plates at the end of optimization.

The problem formulation for the topology optimization of the rib is:

$$\mathrm{Find} x={\left\{x_1,x_2,\dots ,x_N\right\}}^T \mathrm{that}$$

$$\mathrm{minimize} f_s(x)=\mathrm{total} \mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{rib}$$

(7a)

$$\mathrm{subject} \mathrm{to} g_s(x)=\frac{u_s^{\max }}{{\overline{u}}_s}-1\le 0$$

(7b)

where $u_s^{\max }$ is the maximum displacement of the stator in the axial direction by external loads. The objective to be minimized in Equation (7a) is the mass of the rib, and the design constraint in Equation (7b) is to restrain the maximum axial displacement of the stator under a prescribed value ${\overline{u}}_s$.

Figure 3 shows the results of topology optimization. In Figure 3a, the mass of the rib decreases as the optimization proceeds. Although there is a bump in the middle of Figure 3a due to the constraint violation, the topology shows a stable convergence. The optimized rib layout consists of truss-like slender members, showing that the bending deformation of the stator plates is suppressed by the high axial stiffness of truss-like members. Figure 3b,c illustrate the von Mises stress distribution of the stator walls without and with the rib, respectively. As evident from the results, the inclusion of the rib effectively suppressed the bending of the walls, which was observed as the dominant deformation in Figure 3b.

Once the rib layout was obtained, through numerical tests, the depth (or thickness) of the rib and the number of ribs were determined. A total of 120 ribs with 40 mm depth positioned at every 3 degrees in the circumferential direction of the stator were found to satisfy the design constraint. The stator with the optimized ribs had a mass of 141.4 tons (without active parts), and its maximum axial displacement was calculated to be 0.786 mm.

2.3. Structural Optimization of a Rotor

The layout of the rotor in Figure 2b proposed in [12] has high stiffness for the attractive magnetic force so that the force can be efficiently supported by the axial stiffness of partitioning plates inside. Bang [12] also presented minimum thicknesses for partitioning plates by using the Euler beam theory. However, he did not consider the curvature effect of the rotor, and all the plates were modeled to have the same thickness. In this work, the plate thicknesses of the rotor in Figure 2b were optimized separately by solving a two-dimensional elastic problem, for which axisymmetric shell elements were used for the finite element model. The problem formulation for the thickness optimization is:

$$\mathrm{Find} x={\left\{t_{rb},t_{rh1},t_{rh2},\dots ,t_{rh9}\right\}}^T$$

(8a)

$$\mathrm{minimize} f_r(x)=\mathrm{total} \mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{rib}$$

(8b)

$$\mathrm{subject} \mathrm{to} g_r(x)=\frac{u_r^{\max }}{{\overline{u}}_r}-1\le 0$$

(8c)

$$8\le t_{rb}\le 16 \mathrm{and} 2\le t_{rh1},t_{rh2},\dots ,t_{rh9}\le 8$$

(8d)

where $u_r^{\max }$ and ${\overline{u}}_r$ are the maximum axial displacement of the rotor and its constrained value, respectively. Because the vertical plates in Figure 2b underwent bending deformation, their thicknesses were set to be larger than those of the partitioning plates in Equation (8d). The thickness of the internal plates, $t_{rh2}$, $t_{rh3}$, …, $t_{rh8}$, were set to have the same thickness during optimization.

The optimization problem in Equation (8a)–(8d) was solved by approximating the objective and constraint functions using Kriging surrogate models, for which sample points were selected following the CCD (central composite design) [23]. All the optimization process was performed within the framework of OptiStruct [21]. The optimized plate thicknesses are listed in

Table 3. Results of the thickness optimization for the rotor (unit: mm).

${\mathit{t}}_{\mathit{r}\mathit{b}}$	${\mathit{t}}_{\mathit{r}\mathit{h}1}$	${\mathit{t}}_{\mathit{r}\mathit{h}2}$	${\mathit{t}}_{\mathit{r}\mathit{h}3}$	${\mathit{t}}_{\mathit{r}\mathit{h}4}$	${\mathit{t}}_{\mathit{r}\mathit{h}5}$	${\mathit{t}}_{\mathit{r}\mathit{h}6}$	${\mathit{t}}_{\mathit{r}\mathit{h}7}$	${\mathit{t}}_{\mathit{r}\mathit{h}8}$	${\mathit{t}}_{\mathit{r}\mathit{h}9}$	Total Mass (Tons)	${\mathit{u}}_{\mathit{r}}^{\mathbf{max}}$
15.6	7.3	2.7	2.7	2.7	2.7	2.7	2.7	2.7	6.2	19.3	0.087

, from which the mass of the optimized rotor was calculated as 19.3 tons. As a result, the total mass of the optimized generator without active parts was obtained as 160.7 tons (141.4 tons for the stator and 19.3 tons for the rotor). For the optimized stator and the rotor, the maximum deformation of the gap width was calculated as 0.873 mm (0.786 mm for the stator and 0.087 mm for the rotor), which is lower than the design constraint, 10% of the gap width (0.89 mm).

3. Structural Performance Comparison between BRPM and Conventional Generators

The dynamic responses of the optimized BRPM generator were calculated for two scenarios: (1) the axial vibration of the rotor induced by the blade vibration and (2) the axial vibration of the rotor and the stator by tower vibration. Dynamic loads were not taken into account in the presented optimizations because their effect is not significant in the BRPM generator, as is shown in this section. For comparison, the dynamic stiffness of the conventional generator in Figure 1b was also calculated. Numerical analyses were conducted for the optimized BRPM generator and the conventional generator, which was also optimized using surrogate models.

3.1. Deformation of the Rotor by Blade Vibration

In Figure 1b, the rotor of a conventional generator is directly connected to the blade hub, so external forces imposed on blades can be transmitted to the rotor without attenuation. Thus, the rotor of the generator should be designed such that it can have enough structural stiffness to deal with blade vibration or impulses as well as external static forces, such as the attractive magnetic force and gravitational force. In this work, the effect of the axial vibration of the blade hub on the deformation of the rotor was investigated. It was assumed that the hub bearing of blades undergoes a rigid-body vibrating motion in the axial direction by the front wind. It was also assumed that the main shaft has little elastic deformation in the region between the hub bearing and rotor bearings. Therefore, contact surfaces of the rotor-on-rotor bearings vibrate synchronously with the hub bearing with the same amplitude.

Figure 4 illustrates the stator and rotor of a spoke arm-type generator whose dimensions are listed in

Table 4. Dimensions for the spoke arm-type generator shown in Figure 4 (dimensions were obtained by using the surrogate-model-based optimization method, which was also used for the rotor design of the BRPM generator).

Stator Parameter	Dimension (mm)	Rotor Parameter	Dimension (mm)
$L_w$	688	$G_i$	385
$L_h$	335	$R_i$	1970
$L_{th}$	38	$R_o$	3621
$O_{th}$	126	$W_i$	434
		$I_{th}$	69

. For the finite element analysis, the contact surface of the rotor indicated in Figure 4b was attributed a prescribed displacement in the axial direction, 5 mm in this problem, and a harmonic analysis was performed with a frequency range from 0.01 Hz to 1 Hz, with an increment of 0.001 Hz. This frequency range was selected based on the natural frequencies of blades of large-sized wind turbines [24]. The material for the generator was structural steel. For finite element discretization, 4-node shell elements were used (33,948 nodes for the rotor discretization).

The deformation of the rotor by the blade vibration is shown in Figure 5a. The most deformation is observed as bending deformation in the members connecting the inner and outer flanges of the rotor. Figure 5a also shows the maximum axial displacement of the rotor at point A for the excitation frequency range, where the maximum deformation significantly increases around 0.83 Hz. The peak in Figure 5a is 27.9 mm, 5.6 times larger than the displacement of the blade hub. This large deformation of the rotor can cause the unstable electromagnetic operation of the generator and can result in structural failure by inducing high stress on the support bearings. Figure 5b shows the mode shapes of the rotor of the spoke arm-type generator. The first mode in the figure has a natural frequency of 0.88 Hz, which is well matched with the peak frequency in Figure 5a.

In the case of the rotor of the BRPM generator, due to the usage of a bearingless drive, the vibration or impulsive forces of blades cannot be directly transmitted to the rotor from the blade hub. In Figure 1a, the torque of the blades is transferred to the rotor through joint couplings. The joint was designed such that it acts as a rigid connector for the torsional rotation of blades so that the rotor can rotate with the same angular speed as the blades. For the axial vibration of blades, the joint acts as a disconnector so that the axial vibration or impulsive movement of blades cannot be transmitted to the rotor. Figure 6a shows an example of the proposed joint coupling. In the figure, shackles are used to allow for the free axial translational movement while acting as rigid connectors for rotational movement. However, the joint needs further design modification. It requires damping to remove a blade vibration effect on the rotor. Nonetheless, it is obvious that the rotor of the BRPM generator takes no influence by the uniform axial vibration of the blades.

3.2. Deformation of the Rotor and the Stator by Tower Vibration

Because of the bending vibration of the tower of a large-sized wind turbine, a generator undergoes almost rigid-body translation with large displacement. The deformations of the BRPM and conventional generators under the vibration of the tower were analyzed by harmonically vibrating the generators 3000 mm in the axial direction. The frequency for the harmonic analysis was set as a low-frequency range, 0.01–0.3 Hz, which was determined based on the natural frequencies of towers for existing large-sized wind turbines [25]. The connection surfaces of the generators with the main shafts were excited by providing the axial displacement for the boundary condition.

Figure 7a shows the deformation of a spoke arm-type generator when the tower is excited with the frequency of 0.3 Hz. In the figure, bending deformations are noticeable in the outer flanges of the rotor and the stator. Because active parts are attached to the outer flanges, this deformation should be minimized for stable power generation. To this end, the cross-section of the outer flanges of the rotor and the stator should be designed to have large bending moments of inertia by using thicker plates, which might result in a large increase in the total mass of the generator. The displacements of the rotor and the stator in the radial direction on the nodes of line A and line B in Figure 7a are illustrated in Figure 7b,c, respectively. The increase (or decrease) in the gap width between the rotor and the stator is also plotted in the figures. In Figure 8, the maximum gap increase between the rotor and the stator is shown for the frequency range of 0.01–0.3 Hz. The maximum gap increase was calculated as 0.39 mm at 0.3 Hz, 4.4% of the total gap width. Considering that the upper bound of the design constraint for the gap increase was set as 10% of the total gap width in the optimization problems in Equations (7a), (7b) and (8a)–(8d), this is not a small value.

Because the rotor of a BRPM generator is buoyant in the fluid, the gap width should be maintained by using the magnetic control force. Through this gap control, the deformation of the generator by external loads can also be simultaneously considered. It is worth mentioning that the gap control can be conducted by using the active parts for power generation, and no additional system is required. This can be conducted by balancing the attractive forces that act on both sides of the airgap (see Figure 6b for the gap control).

In this study, the control force that was required to maintain the gap width when the generator was under the condition of tower vibration was calculated by using finite element analysis. The control force can be indirectly calculated by constraining the deformation of the generator and then calculating the surface tractions required for this deformation constraint. The control force distribution was the same as the surface traction distribution. Because the gap was filled with fluid, the shear component of the surface traction could be ignored so that the control force distribution that was provided was opposite to the surface pressure with the same magnitude. To implement this in a finite element analysis, the fluid filled in the gap was modeled as an artificial material. To constrain the deformation of the gap, the elastic modulus of the artificial material was set to be very large, 1000 times larger than that of the generator. Other material properties of the artificial material were the same as the fluid. A harmonic analysis for the frequency range of 0.01–0.3 Hz was conducted for the BRPM generator in Figure 9, whose dimensions are listed in

Table 5. Dimensions for the BRPM generator in Figure 9.

Stator Parameter	Dimension (mm)	Rotor Parameter	Dimension (mm)
$s{t}_i$	3757.3	$R_i$	3846.1
$s{t}_o$	5122.7	$R_o$	5033.9
$s{t}_h$	1365.4	$l_{rh}$	1187.8
$s{t}_w$	1978.1	$l_{rb}$	1596.0
$r_{h1}$	1600	$t_{rh1}$	7.3
$r_{h2}$	1500	$t_{rh2}-t_{rh8}$	2.7
$r_{s1}$	550	$t_{rh9}$	6.2
$r_{s2}$	550	$t_{rb}$	15.6

. Figure 10 shows the control force distribution on the surfaces of the active segments. Because the same set of active segments was positioned every 6 degrees as a circular array, the results for one set of active segments are illustrated in the figure. The average control force (or surface pressure) was calculated as −24 kPa on Segment 1, 72 kPa on Segment 2, −85 kPa on Segment 3, −55 kPa on Segment 4, 114 kPa on Segment 5, and 18 kPa on Segment 6. Note that the directions of the control force were calculated as opposite on Segment 1 and Segment 2 due to the disconnection for the seal. The maximum gap increase was calculated as 0.06 mm, which is much smaller compared to that of the spoke arm-type generator in Figure 8.

Because the deformations of the spoke arm-type generator by the blade vibration and tower vibration are large, the optimization for the spoke arm-type generator should additionally consider deformation constraints for these dynamic load cases. In

Table 6. Performance comparison between the optimized BRPM generator and the optimized spoke arm-type generator.

	Maximum Displacement (mm)			Mass (Tons)
	Magnetic Force		Tower Vibration	Rotor	Stator	Total
BRPM	Rotor	0.078	0.12	19.3	141.4	160.7
	Stator	0.340
Spoke arm-type generator	Rotor	0.200	0.12	97.5	228.1	325.6
	Stator	0.125

, the performances of the optimized BRPM generator and the optimized spoke arm-type generator are compared. In the table, the displacement by a blade hub vibration is not compared because it cannot be calculated for a BRPM generator without considering the force control. The table shows that the BRPM generator has a similar deformation level to the spoke arm-type generator for dynamic excitation as well as static loads. However, a noticeable difference can be observed in the total mass between the two optimized generators.

4. Conclusions

This investigation demonstrated the significant advantages of a BRPM generator over conventional generators in terms of its lightweight design and structural efficiency. The optimized BRPM generator achieved an overall structural mass of 160.7 tons for a 10 MW capacity, which corresponds to only 49.3% of the mass of the optimized spoke arm-type generator. The key to achieving this mass-minimized design lies in the application of optimization techniques for structural layouts, stator wall reinforcement, and plate thickness determination. Unlike traditional approaches that modify the existing structural layouts, the BRPM generator’s design process maximizes the stiffness-to-mass ratio through topology optimization. It is important to note that the proposed design method, although applied to a 10 MW generator in this study, can be extended to larger-sized generators with minimal modifications. This scalability makes the BRPM generator concept highly promising for various power-generation applications. While this investigation focused primarily on lightweight design and stiffness, it is crucial to consider structural strength as well. Future design iterations should incorporate structural strength to thoroughly evaluate the structural integrity and ensure the generator’s long-term reliability.