Multi objective optimization on suspension characteristics of vertical axis wind turbine

Abstract

In order to improve the suspension characteristics of passive magnetic bearings in vertical axis wind power generation system, T-shaped group of passive magnetic bearing with three rings which owns high suspension characteristics are proposed. The working principle of the structure is described briefly. The multi-objective optimization is achieved with high bearing capacity and high magnetic stiffness by studying its impacts on the magnetic ring with variable geometric parameters: the height of the magnetic ring, the thickness of the magnetic ring and the air gaps between the magnetic rings. The research shows that suspension characteristics of the novel structure are 2 times of the traditional radial magnetic bearings. Its the bearing capacity is at least 1.75 times before optimization and the magnetic stiffness is at least twice as high as before optimization which provides a reference for the suspension characteristics of the vertical axis wind turbine suspension system.

Keywords

Vertical axis high magnetic stiffness high bearing capacity suspension characteristics

1 Introduction

Suspended vertical axis wind turbines are widely used due to their small starting speed, small aerodynamic noise and better wind performance [1]. To improve utilization rate of wind energy, the magnetic levitation technology is fully integrated with the vertical axis system. According to the Earnshaw theorem, passive bearings can not be suspended at all degrees of freedom. Therefore at least one degree of freedom it has to be mechanically, actively, electrodynamically controlled. Magnetic levitation technology is widely used in flywheel energy storage, high speed miniaturized motor, permanent magnet motor, maglev train with its advantages of friction-free, low cost and long service life [2–5]. The research of suspension system has become the hot trend in the present [6,7]. Domestic and foreign scholars have studied the magnetization direction of magnetic bearing and proposed a continuous halbach [8–10] structure with high magnetic Stiffness, but with high processing cost. The structure of three-ring is presented with high magnetic stiffness and bearing capacity [11,12], but its system stability is poor. For its analytical method, the semi-analytical method with a short computation time, maxwell’s stress method and virtual displacement method were proposed [13–15] and the corresponding mathematical calculation model were established. For its optimization algorithm, Maxence Van Beneden et al. proposed SCLs to estimate the performance of permanent magnet thrust bearings and compared the magnetic force produced by Colombian and 2D planar models. The modified coefficient method was used to optimize the performance of permanent magnet thrust bearings [16]. In this paper, a new T-shaped group of PMB with three rings for vertical axis is put forward to improve the bearing capacity, magnetic stiffness and the utilization ratio of wind energy of the passive bearing. Compared with the traditional radial magnetic bearings, the bearing capacity of the new three ring T magnetic ring group is increased by 1.75 times and the rigidity is increased by 2 times.

2 Principle of Maglev

Magnetic levitation bearing is a bearing which uses magnetic force to levitate the rotating shaft. According to its working principle, it can be divided into active magnetic bearing (AMB), passive magnetic bearing (PMB) and hybrid magnetic bearing (HMB). AMB uses a control system to generate a levitation force but with high charge. HMB uses electromagnets and permanent magnets to generate levitation forces with little charge. PMB has become more widely used in industrial applications than other structures due to its simplicity and the least economic charges, which has become the object of this study.

This paper presents a T-shaped group of PMB with three rings which is suitable for the vertical axis wind power generation system, including inner magnetic ring 1, 4, outer magnetic ring 2, 5 and lower magnetic ring 3 and 6. The magnetic ring 2, 3, 5 and 6 are static magnetic rings, and 1, 4 are moving magnetic ring which connects to the generator which is shown in Fig. 1. The gravity of the wind turbine and its components is 200 N, the magnetizing direction of the magnetic ring 1 is axially upward while the magnetization direction of the magnetic ring 3 is axially downward that can produce great repulsive force in order to balance the gravity of the wind power system. The distance between the upper magnetic ring group is 10 mm greater than that of the lower magnetic ring group which provides the 60 N repulsion in the stable state.

If the repulsive force is greater than the force of gravity due to external disturbances, it will cause the suspension system to be thrown. The magnetic ring 6 of the upper T-shaped group of PMB with three rings will provides a downward repulsion to keep the system in balance in the axial direction. The magnetization direction of the magnetic rings 2, 5 is magnetized radially to the center of the circle. When the radial disturbance,the magnetic ring 2 and 5 provide the radial balance magnetic force which makes the system suspended in a radial direction. The upper and lower magnetic ring groups are the same. The main research object is the lower magnetic ring group.

3 Design of magnetic bearing

3.1 Basic dimensions

The excellent suspension characteristics of bearings affect the wind energy utilization of the wind power system, which means that the magnetic bearing should have high bearing capacity and high magnetic stiffness. The geometrical parameters of bearing have great influence on the magnetic stiffness and capacity of passive magnetic bearing.

Bearing capacity is an important parameter of bearing. The difference of geometric parameters fundamentally affects B _g and A _g, thus affecting the bearing capacity. The calculation formula of the bearing capacity is as follows: $\begin{eqnarray}\displaystyle F=\frac{B_{g}^{2}A_{g}}{u_{0}}. & & \displaystyle\end{eqnarray}$ (1)

Where B _g is the magnetic induction intensity in the air gap, A _g is the sectional area of air gap and u ₀ is the vacuum permeability.

Therefore, T-shaped group of PMB with three rings is proposed in this paper. Compared with the traditional radial magnetic bearings, It added one ring under the original double ring magnetic ring group which is consistent with the internal magnetic ring specification. The T-shaped group of PMB with three rings increases the magnetic density and area near the inner ring, which greatly increases the capacity and magnetic stiffness of the bearing.

In this paper, the permanent magnet material used for bearing is NdFeB and the brand name is N42M. In order to ensure the maximum working capacity of the permanent magnet, the static analysis working point of the permanent magnet is stabilized near the maximum energy product and the proper bearing size is calculated by analyzing the demagnetization curve and the anti-magnetic curve of the NdFeB material.

The radial magnetization length of the permanent magnet is: $\begin{eqnarray}\displaystyle L_{m}=fkH_{g}L_{g}\sqrt{\displaystyle \frac{B_{r}}{H_{C}(BH)_{\max }}}. & & \displaystyle\end{eqnarray}$ (2)

The f stands for magnetic resistance, usually between 1.1–1.5, k is the correction factor of the length of magnetic circuit in air, which is equivalent to the length of air gap magnetic circuit, L _g is working air gap length, this paper takes 0.5 mm, H _g is the air-gap magnetic field intensity, Br is remanent magnetization, H _C is the coercive force, BH is the maximum magnetic energy product.

In accordance to Earnshaw’s theorem, sum of stiffnesses must be equal to zero, which leads to the radial stiffness K _r is half of axial stiffness K _z. Magnetic stiffness is the increment of the external disturbance force required by the radial displacement of a radial magnetic bearing in one direction. The mathematical expression is as follow [8]: $\begin{eqnarray}\displaystyle K_{r}=\displaystyle \frac{dp_{x}(t)}{dx(t)}. & & \displaystyle\end{eqnarray}$ (3)

According to the size of the generator shaft and the NdFeB in optimal working point, knowing Inner ring of magnetic bearing L _m × 𝛷_b1 × h is 10 × 36 × 10, Outer diameter of inner ring 𝛷_b2 = 𝛷_b1 + 2L _m =55 mm, Inner diameter of outer ring L _m × 𝛷_b3 × h is 10 × 56 × 10, Outer diameter of outer ring 𝛷_b4 = 𝛷_b3 + 2L _m = 56 and one side air gap is 0.5 mm.

3.2 The comparison of bearing capacity

When the system is disturbed downward, the inner magnetic ring will shift in the axial direction. The magnetization directions of the inner and lower magnetic rings are axially opposite. When the distance becomes smaller, the lower magnetic ring will give the repulsive force on the inner magnetic ring.

In Fig. 2, it is known that the structure of three rings increases the magnetic density among rings. When the inner ring is axially offset, the bearing capacity of the two structures varies with the displacement are shown in Fig. 3 which shows that the bearing capacity of T-shaped group of PMB with three rings is 2 times of the traditional radial magnetic bearings. But in actual operation, the magnetic ring needs to provide 260 N repulsion to balance the gravity of the wind generator system and its components, so its structural parameters need to be optimized.

4 Optimization of bearing

4.1 Optimization of magnetic ring height h and air gap g1

Geometric parameters affect the suspension characteristics of magnetic bearings greatly. In this paper, four parameters of magnetic ring thickness A (l), magnetic ring height B (h) and air gap C (g1) and D (g2) between rings are taken as the factors and the multi-objective optimization design of magnetic bearing is carried out.

The influence of the four parameters on the bearing capacity F and the magnetic stiffness K of the passive magnetic bearing is studied and the quality characteristic of the magnetic bearing is optimized through optimization. That is to say, it has high bearing capacity and high stiffness.

In the orthogonal test, the optimized target is called the quality characteristic, the condition that affects the quality characteristic is called factor and the value of the factor is called the level of the factor. The factor and the level of the factor are shown in Table 1. Each optimization variable takes 4 factor levels, and combination of the orthogonal experiment was carried out.

Traditional analysis variables need to do 4⁴ = 256 times experiments,while Taguchi method establishes the experimental analysis matrix, which requires only 16 times finite element analysis and its specific orthogonal results are shown in Table 2.

4.2 The proportion of factor level on quality characteristics

Analyzing the influence of 4 factors on quality characteristics of proportion, such as factor A in level factor 2 effect on the F can be solved as follow [17]. $\begin{eqnarray}\displaystyle F_{A2}=\displaystyle \frac{1}{4}\times \left(F_{A25}+F_{A26}+F_{A27}+F_{A28}\right) .& & \displaystyle\end{eqnarray}$ (4)

The effect of each factor level on factors A, B, C and D is shown in Table 3:

4.3 The proportion of factors affecting the quality characteristics

The average values of bearing capacity F and magnetic stiffness K can be obtained from the results of 16 groups of orthogonal experiments in Table 2. The effects of four variables A, B, C and D on multi-objective optimization are analyzed which shown in Table 4.

The influence of four factors A, B, C, D on the bearing capacity F and magnetic stiffness K is analyzed, such as the influence of factor A on the bearing capacity F, and the formula is as follows [18]: $\begin{eqnarray}\displaystyle S_{SB}=4\mathop{\sum }_{i=1}^{4}\left(F_{Ai}-\overline{F}\right)^{2}. & & \displaystyle\end{eqnarray}$ (5) Where F _Ai is the average of the bearing capacity F under the influence factor A and the Factor level i; F is the average value of the bearing capacity.

4.4 Analysis of results

In order to know the influence of each level factor on the bearing capacity F and the magnetic stiffness K intuitively, Table 3 is used for graphic processing which is shown in Fig. 4.

It can be seen in Fig. 4 that the maximum combination of the bearing capacity is A4B4C2D3, while the maximum combination of the magnetic stiffness K is A3B4C2D1. It can be seen in Table 4 that the influence factor B has the most influence and factor D has the least influence on the magnetic force while the influence factor D has the most influence and factor C has the least influence on the magnetic Stiffness.

To achieve multi-objective optimization, the final value of B4C2 can be determined by Taguchi algorithm. Since the choices of A or D have a great impact on the bearing capacity F and the magnetic stiffness K, D has the greatest influence on K but least influence on F, so D is to be adopted D1. The specific values of A and D are verified and analyzed by the following optimization methods.

4.5 Optimization of magnetic ring thickness l and magnetic ring gap g2

The second order model of the response output and the variable response factor were proposed by selecting the experimental design scheme which using graphical method and analysis method to find out the optimization setting of independent variables. This method is a traditional statistical method to solve multivariate problems [19]. An appropriate mathematical model is firstly established by experimental data. The function between objective function and variable is usually established by using second order function: $\begin{eqnarray}\displaystyle y={\beta}_{0}+{\beta}_{1}x_{1}+{\beta}_{2}x_{2}+{\beta}_{11}x_{1}^{2}+{\beta}_{22}x_{2}^{2}+{\beta}_{12}x_{1}x_{2}+{\varepsilon}. & & \displaystyle\end{eqnarray}$ (6)

Where y is the response function of magnetic bearing bearing capacity F or magnetic stiffness K, 𝛽 is the undetermined coefficient, ϵ is its fitting error, A and B are the thickness of magnetic ring A (l), gap between the upper and lower magnetic D (g2), respectively. The thickness A (l) affects the magnetic density of the magnetic ring, and the amount of magnetic flux and magnetic flux leakage depends on the size of the air gap. It can be seen from formula 1 that the magnetic density and air gap area plays a decisive role in the bearing capacity of the passive suspension bearing, so the optimal solution can be obtained through scientific and rigorous optimization.

From the above optimization analysis, it can be seen that when A belongs to A1–A3 and D belongs to D1–D3 can achieve the optimal target, the range of independent variables are: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle 12∼\text{mm}\lt A(l)\lt 14∼\text{mm}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \displaystyle 0.8∼\text{mm}\lt D(g2)\lt 1.2∼\text{mm}.\nonumber\end{eqnarray}$

By using the central composite design method, the values of independent variables are encoded separately. The center point is set to ensure the uniform accuracy of the predicted values of the entire experimental area. The experimental scheme and results are obtained as follows in Table 5.

The 9 groups of orthogonal experimental results are obtained through the analysis of the finite element. The mathematical model of magnetic stiffness K and capacity F are estimated by the least square method as follow: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle y_{1}=-11972.8+1878.69x_{1}+143.65x_{2}-71.58x_{1}^{2}-75.22x_{2}^{2}+0.139x_{1}x_{2}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle y_{2}=-14588.5+2476.74x_{1}-1630.57x_{2}-93.24x_{1}^{2}+690.62x_{2}^{2}-29.11_{12}x_{1}x_{2}.\end{eqnarray}$ (8)

Figure 5 is a curved surface map and a contour map that responds to the target and variable factor which can solve the best combination of factors. The slope size of response surface reflects the significant influence of interaction between two variables on the response value. The steeper the slope is, the more significant the interaction effects on the response value. On the contrary, it is not significant.

Figure 5(a) is surface diagram formed by the magnetic force, magnetization thickness and air gap, 5(b) is the contour map of the magnetic force, magnetization thickness and air gap. In Fig. 5(b), the deepest part of the color indicates that the magnetic force is greater than 420 N (l is between 12.8 and 13.25 mm) and the g2 between 0.8–1.2 mm can reach the maximum capacity. Figure 5(c) is surface diagram formed by the magnetic stiffness, magnetization thickness and air gap, 5(d) is the contour map of the magnetic stiffness,magnetization thickness and air gap.

When the thickness is 12.46–13.75 mm and the gap is between 0.8–0.83 mm, magnetic stiffness can reach the maximum. Through the analysis of Minitab, the overall solution of the response optimization is (x ₁, x ₂) = (13.13, 0.8).

5 Comparative analysis

The selection of g2 by this optimization method is consistent with the value D1 to be adopted by g2 in the above optimization method. The parameters that affect the size of the magnet ring are optimized by two optimization methods.

The geometric parameters are finally determined as follows: the thickness of the magnet ring A (l) is 13.13 mm, the height of the magnet ring B (h) is 14 mm, and the air gap C(g1) is 0.5 mm and D (g2) is 0.8 mm. The suspension characteristics before and after optimization are shown in Fig. 6. The direction of gravity acceleration is positive, and the displacement reference direction is set to negative direction when the inner magnetic ring is shifted upwards.

As can be seen from the comparison of Fig. 6, the optimized bearing capacity of passive bearings is at least 1.75 times before optimization and the magnetic stiffness is at least two times higher. Therefore, the practicality of these two optimization methods can be verified. Due to the need to provide a total of 260 N for the gravity of the wind turbine system and its components and the repulsion of the upper magnetic ring, the optimization result meets the design requirements with a margin of 1.3 times.

6 Conclusion

This paper presents a T-shaped group of PMB with three magnetic rings bearing for vertical axis wind power generation system. The structure has high suspension characteristics through the increase of magnetic flux density and section area and optimization of geometrical parameters. The research shows:

(1) The T-shaped group of PMB with three magnetic rings provides enough axial repulsive magnetic force for the system with its novel structure and reasonable magnetization direction to balance the gravity and enough centripetal magnetic force of the system, so as to stabilize the system’s disturbance in the radial direction. The upper T-shaped magnetic ring group is used as a spare magnetic ring group which ensures the stability of the system. Its bearing capacity and magnetic stiffness are 2 times of the traditional radial magnetic bearings with better suspension characteristics and is more suitable for use in the small wind turbine generator system.

(2) The geometric parameters of the magnetic ring have great influence on the suspension characteristics of the magnetic ring group. The thickness of the magnetic ring h has the greatest influence on the bearing capacity. The air gap g2 between the upper and lower magnetic rings has the greatest influence on the magnetic stiffness. The research shows that the bearing capacity of T-shaped group of PMB with three rings is at least 1.75 times before optimization and the magnetic stiffness is at least twice as high as before optimization.

Footnotes

Acknowledgements

I would like to express my gratitude to all those who helped me during the writing of this thesis. I gratefully acknowledge the help of my stutor, Mr zhu, who has offered me valuable suggestions in the academic studies.

In the preparation of the thesis, he has spent much time reading through each draft and provided me with inspiring advice. Without he patient instruction, insightful criticism and expert guidance, the completion of this thesis would not have been possible. I also owe a special debt of gratitude to all the professors in Foreign Languages Institute, from whose devoted teaching and enlightening lectures I have benefited a lot and academically prepared for the thesis. This work was supported by the key research of development and promotion in henan province (182102210052), Special fund financing project for basic scientific research operation of henan university (NSFRF140115).

I should finally like to express my gratitude to my beloved parents who have always been helping me out of difficulties and supporting without a word of complain.

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