A Comprehensive performance assessment of the integration of magnetic bearings with horizontal axis wind turbine

Accepted Manuscript A Comprehensive performance assessment of the integration of magnetic bearings with horizontal axis wind turbine M. Fekry, Abdelfatah M. Mohamed, Mohamed Fanni, S. Yoshida

PII: DOI: Reference:

S0378-4754(18)30168-X https://doi.org/10.1016/j.matcom.2018.06.011 MATCOM 4606

To appear in:

Mathematics and Computers in Simulation

Received date : 6 November 2017 Revised date : 23 June 2018 Accepted date : 23 June 2018 Please cite this article as: M. Fekry, A.M. Mohamed, M. Fanni, S. Yoshida, A Comprehensive performance assessment of the integration of magnetic bearings with horizontal axis wind turbine, Math. Comput. Simulation (2018), https://doi.org/10.1016/j.matcom.2018.06.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A Comprehensive Performance Assessment of the Integration of Magnetic Bearings with Horizontal Axis Wind Turbine M.Fekrya,b,∗, Abdelfatah M. Mohameda,c , Mohamed Fannia,d , S. Yoshidae a Dept. of Mechatronics and Robotics Engineering Egypt-Japan University of Science and Technology New Borg El-Arab City, Alexandria, 21934, Egypt b On leave from Dept. of Electrical Power and Machine Engineering, Zagazig University, Zagazig, Egypt c On leave from Dept. of Electrical and Electronics Engineering, Assiut University, Assiut, Egypt d On leave from Dept. of Production Engineering and Mechanical Design, Mansoura University, Mansoura, Egypt. e Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580, Japan

Abstract This paper presents a comprehensive performance assessment for the use of Magnetic Bearing (MB) with Wind Turbine (WT) . It is reported that this integration aims to improve the WT performance by eliminating fractional losses, mitigating vibration, reducing cut-in speed and prolonging the life span. However, there is no thoroughly studies regarding this integration in the literature to show in what extent the power generation will be affected in comparison with the use of conventional mechanical bearing (CB). This task constitutes the aim of this work. First, the main shaft of an existing conventional WT is redesigned to match the assembly with the MBs. Second, the design of two Radial Homopolar Pole Biased Hybrid Magnetic Bearings (RHPBHMB) and one Axial Active Magnetic Bearing (AAMB) is introduced. The MB design is analyzed using nonlinear magnetic circuit analysis and FEM. The MSC Marc software is employed to build 3D and 2D FEM models for the two radial and axial MBs respectively. Thirdly, the mathematical dynamic equations for the overall system is derived and an elaborated multi-disciplinary dynamic model is built using Simscape package. In addition, a robust intelligent TSK fuzzy Q-parameterization controller is synthesized to stabilize the RHPBHMB in order to achieve robust stability, overcome model nonlinearity and reject the step and sinusoidal imbalance disturbance at any rotational speed. Finally, an extensive comparison between the performance of the WT supported with Conventional mechanical Bearing (WT-CB) and the WT supported with Magnetic Bearing (WT-MB) is provided. The results shows the ability of MB to defeat the disadvantages of the Conventional mechanical Bearing (CB), as well as, enhance the WT performance without decreasing the power generated. Keywords: Wind Turbine, Magnetic bearing design, Homopolar magnetic bearing, Hybrid magnetic bearing, Magnetic circuit analysis, FEM, Pole biased magnetic bearing, Parallel distribution control, Intelligent control, Robust control, Robust stability, Q-parameterization, Wind turbine performance, Extensive comparison, Comprehensive assessment. 2010 MSC: 00-01, 99-00 1. Introduction

5

10

The MB is a suspension system to levitate a rotating object freely without any physical contact. It enables a ro- 15 tating shafts to attain very high speed without any friction or mechanical wear [1]. The operation of MB depends on harnessing magnetic fields to control the position of a ferromagnetic object (typically rotor) precisely. These forces may be attractive or repulsive according to the suitability 20 of application. The integration between MB and WT helps in eradicating the frictional losses throughout the turbine. Further∗ Corresponding

author Email addresses: [email protected] (M.Fekry), [email protected] (Abdelfatah M. Mohamed), [email protected] (Mohamed Fanni), [email protected] (S. Yoshida)

25

more, this will result in decreasing cut in speed, reducing noise, increasing efficiency, rationalizing energy consumption, eliminating lubricant contaminations, being maintenance - free and prolonging turbine design lifetime [2, 3, 4, 5, 6, 7, 8]. Also, the inherent instability of MB is a challenging problem which must be overcome via an efficient controller. This controller should be able to reject disturbances, compensate for vibrations and overcome model nonlinearity and uncertainties. The WT is a multi-disciplinary complex system that includes aerodynamics, mechanical and electrical subsystems. As a matter of fact, to provide comparative study between using WT-CB and WT-MB, the study should be multi-disciplinary which incorporates all these subsystems and consider the interactions between them. The authors in [9] have investigated the possibility of applying MB to direct drive wind turbine in order to re-

Preprint submitted to Journal of Mathematics and Computers in Simulation

June 24, 2018

30

35

40

45

50

55

60

65

70

75

80

85

duce the weight of generator and increase the structure flexibility. Firstly, they have designed the MB for a WT called Harakosan Z72 (1.5MW @ 18.5rpm). Based on the analytical and numerical design, they estimated that the losses in the MB is below 0.5% of the rated power which 90 is typically the same magnitude of loss for CB. After that, a new concept depending on using the CB to isolate wind loads was applied in designing 5MW generator in [10]. As a result, the rotor weight is reduced by 45%( from 50 tons to 28 tons). In addition, they have introduced three con- 95 cepts of using MB in direct drive machine in [11]. The first concept is a single MB concept which relies on replacing the CB with a magnetic one, the second concept is the flexible rotor concept which depends on design magnetically levitated rotor made of a thin cylindrical ring. The100 third concept is the hybrid magnetic bearing which depends on isolating the wind load by CB and magnetically levitated stiff rotor ring. The later concept is the best one because it combines the ease of control and structure flexibility. Later in [12] they have introduced a complete105 design for 5 MW direct drive generator with the hybrid magnetic bearing concept. However, the authors of these papers focused only on the MB design process regardless of controller design or aerodynamic model which means the lack of considering the interactions between the differ-110 ent subsystems. Moreover, his comparison didn’t include accurate estimation for the CB losses. The authors in [13] have studied the effect of shaft eccentricity of a small WT on the generator performance levitated by Passive Magnetic Bearing (PMB). They used115 CB in the generator to stabilize the axial DOF. The proper stiffness to compensate this eccentricity is determined to meet the generator requirements. After that, in [7], they have experimentally introduced a complete design of the the magnetically levitated WT with PMB and studied120 the radial forces and natural frequencies of the rotor system. Later in [8], they have incorporated the aerodynamic loads of the impeller using Computational Fluid Dynamics (CFD) method. However, the use of CB at any stage degrades the MB performance to some extent because of125 friction. In [14], a design and implementation of a novel realtime controller for Active Magnetic Bearing (AMB) equipped for WT has been presented. This controller is implemented using a floating-point digital signal processor (DSP) and on a Field-Programmable Gate Array (FPGA). Furthermore, the controller has high computing capacity, low system costs and inherent flexibility which result in an improvement of the control performance and a remarkable reduction in software development time. However, the author focused only on the controlling of AMB regardless of the others subsystems. The authors in [6] have introduced the design and control for AC radial-axial Hybrid Magnetic Bearing (HMB) which has been used in wind energy generation system. The experimental results showed that the presented displacement/current measuring method and control system 2

can drive the AC radial-axial HMB in a stable way with excellent performance. However, the authors neither consider the aerodynamic system nor provide any comparison with CB. The authors in [15] have designed a practical testbed for a WT with a PMB. The testbed provides a basis for the experimental determination of vibration, evaluation of bearing performance and estimation of efficiency of a WT under various wind conditions. The authors in [16] have presented a 5-DOF magnetically levitated WT with two radial PMB and one axial HMB. Additionally, they designed a fuzzy PID-like controller to stabilize the system. However, the PMB has limited ability to compensate aerodynamic disturbance. A new self-decoupling magnetically levitation generator has been proposed for WT in [17, 18]. The generator is able to control 2-DOF in the radial direction while the WT is suspended by 3-DOF HMB. Although this configuration is suitable for short shaft length, a portion of shaft weight is still supported by bias current of the generator bearing. Double Stage Cross Feedback Control and Cross-feedback-based Sliding Mode Control have been designed in [19, 20] to overcome the gyroscopic coupling effect and the imbalance forces generated under the high-speed situation of the magnetically levitated WT. However, this work concentrates only on control process. Table 1 summarizes the different aspects of the Magnetically Levitated Horizontal Axis Wind Turbine (MLHAWT) researches mentioned in above literature. It is obvious that, there is a still need for providing a fair comparison between the use of MB and CB with Horizontal Axis Wind Turbine (HAWT). So, a comprehensive modeling of HAWT which include the aerodynamic, mechanical and electrical model is necessary to consider the different subsystems effects on each other. As well, this comparative study will provide a clear pattern on how much the MB improves the performance of WT. Over the above, it is reported in [21, 22, 23] that the bearings and gearbox are responsible for 76% and 17% of wind turbine failure in 2015 respectively. Besides, the gearbox is replaced after five years of installation, although the typical design life for its utility is 20 years which results in doubling the capital cost.

Table 1: Research status of magnetically levitated Horizontal axis wind turbine Ref. [9] [10] [11] [12] [13] [7] [8] [14] [6] [15] [16] [19] [20] *

Year 2007 2008 2009 2010 2012 2013 2014 2010 2012 2012 2012 2014 2015

Suspension HMB HMB& Mech. B HMB & Mech. B HMB & Mech. B PMB & Mech. B PMB & Mech. B PMB & Mech. B AMB HMB PMB & Mech. B PMB & HMB AMB AMB

+:included, -:not included o:not rquired

DOF 5 1 1 1 2 2 2 5 2 5 4 4

Aerodynmic + + + -

MB design + + + + + + + + + + -

Control o o o + + + + + +

Comp. + -

Exp. + + + + + -

130

135

140

145

150

In this paper, a comprehensive comparison between the performance of WT-CB and WT-MB is conducted. So, a multi-disciplinary elaborated model for the WT including aerodynamic, mechanical, and electrical subsystems is developed using Simscape Simulink package. In section 2, the main shaft of an existing WT is redesigned to match the assembly with two RHPBHMB and one AAMB. The MBs are designed and analyzed using magnetic circuit and FEM. In section 3, the overall mathmatical dynamic equation for the whole system is derived. Besides, the frictional moment of CB is calculated. In section 4, the TSK fuzzy Q-parametrization control with PDC is synthesized for the two RHPBHMB and Q-parametrization is synthesized for the AAMB. Furthuremore, the controller performance is evaluated to prove that the controller achieves the desired objectives. In section 5, an extensive comparison is evolved between the performance of WT-CB and WT-MB. The results depict the ability of MB to improve the performance of WT by gaining the well-known merits of MB such as eliminating the frictional losses, decreasing cut in speed, minimizing vibration, prolonging the lifespan and increasing efficiency without decreasing the extracted power from the wind.

Figure 1: Magnetically Levitated Wind Turbine

Table 3: Mechanical properties of S45C [24]. Property Density Youngs Modulus Tensile Strength Yield Strength Poisson’s ratio

Unit (kg/m3 ) (GPa) (Mpa) (Mpa)

Value 7700-8030 190-210 569 (Standard) 686 (Quenching, Tempering) 343 (Standard) 490 (Quenching, Tempering) 0.27-0.30

2. Design of Overall System

155

This section presents the design of main shaft of the WT and the MBs that is necessary to position the WT along its five axes. The parameter of system under study and loads are given in Table 2 . Two RHPBHMB and one AAMB are designed to levitate the WT. Fig 1 shows the different components of the overall system. Table 2: Loads of wind turbine components

Load types Static

Dynamic

Componant Blades Hub Shaft and thrust disk of axial AMB Generator Axial forces Tilt moments

Parameter Designed Power Rotor Radius of Wind Turbine (Propeller) Rated rotational speed

160

165

Load 3×2.75 kg 7.2 kg

Figure 2: Free body diagram of shaft

65 kg 85 kg 1554.49 N 250 N.m Value 3 kW 1.25 m 300 rpm

170

175

2.1. Shaft Design: The shaft is designed iteratively until reaching the desired requirement such as withstanding the static and dy-180 namic loads and matching the dimensions of the two RHPBHMB and one AAMB. The shaft material is made of carbon steel S45C whose parameter is given in Table 3. The shaft diameter is obtained using Goodman equation 3

[25]. Fig. 2 and 3 show the free body diagram and the dimensions of the designed shaft respectively. In addition, the static analysis of the shaft is performed by FEM using CATIA software. The shaft is studied under six loading conditions described in Table 4. The stress and deflection analysis can be found in Fig.4 & 5 respectively. The max. stress is 65.3 MPa which is five times less than the designed material yield strength. Furthermore, the max. deflection is 0.39 mm. Besides, the frequency analysis is performed to determine the first natural frequency of the modified shaft and it was found to be 62.596 Hz. This value is much higher than the rated speed of wind turbine which is 300 rpm (5 Hz). In addition, the fatigue strength is conducted to determine the minimum shaft diameter, d, at the most critical location using Goodman equation. The calculated diameter is 50 mm which is less than the designed one (55 mm).

Figure 3: The upper half of shaft dimensions

Load conditions

1

2

3

4

5

6

Table 4: Load cases of the shaft

Load case forces Blade+ Hub Shaft Generator Mtilt M B1 rotor M B2 rotor Aux. bearing Axial Loads R1 R2 Blade+ Hub Shaft Generator Mtilt M B1 rotor M B2 rotor Aux. bearing Axial Loads R1 R2 Blade+ Hub Shaft Generator Mtilt M B1 rotor M B2 rotor Aux. bearing Axial Loads R1 R2 Blade+ Hub Shaft Generator Mtilt M B1 rotor M B2 rotor Aux. bearing Axial Loads R1 R2 Blade+ Hub Shaft Generator Mtilt M B1 rotor M B2 rotor Aux. bearing Axial Loads R1 R2 Blade+ Hub Shaft Generator Mtilt M B1 rotor M B2 rotor Aux. bearing Axial Loads R1 R2

Value 15.45 kg 65 kg 85 kg 250 N.m 5 kg 8.5 kg 0.4 kg 1500 N 960 N 800 N 15.45 kg 65 kg 85 kg -250 N.m 5 kg 8.5 kg 0.4 kg 1500 N 340 N 1420 N 15.45 kg 65 kg 85 kg 250 N.m 5 kg 8.5 kg 0.4 kg -1500 N 960 N 800 N 15.45 kg 65 kg 85 kg -250 N.m 5 kg 8.5 kg 0.4 kg -1500 N 340 N 1420 N 15.45 kg 65 kg 85 kg 0 5 kg 8.5 kg 0.4 kg 1500 N 650 N 1110 N 15.45 kg 65 kg 85 kg 0 5 kg 8.5 kg 0.4 kg -1500 N 650 N 1110 N

Max. Stress

Max. Deflection

Figure 4: The shaft stress analysis (Load case: 1)

6.53 × 107 Pa

0.388 mm

6.36 × 107 Pa

0.36 mm

6.41 × 107 Pa

0.39 mm

Figure 5: The shaft deflection analysis (Load case: 3)

2.2. Magnetic Bearing System Analysis: 6.3 × 107 Pa

0.36 mm

185

190

6.86 × 106 Pa

0..0233 mm

195

7.33 × 106 Pa

0.0236 mm

200

4

The MB system is composed of two radial Hybrid Magnetic Bearing (HMB) and one axial Active Magnetic Bearing (AMB) 2.2.1. Radial HMB Analysis: The design of Radial HMB is divided into two parts. The first part is the design of permanent magnet (PM) to compensate the static loads passively in vertical direction. The second part is the design of electromagnet to compensate and control the aerodynamic forces along vertical and horizontal direction. The design of the first and second parts depend on the calculated reactions which obtained from the shaft free body diagram as in Table 5. The configurations of the proposed RHPBHMB for both MB1 and MB2 and the forces direction are shown in Figs. 7 and 6 respectively. Each MB consists of four identical PM pole pairs and four identical electromagnetic pole pair. Furthermore, each pole pair has its separated magnetic path and isolated from the others by a nonmagnetic material.

Table 5: Bearing design load

Bearing MB1 MB2

205

210

Static Load (N) 650 1110

Dynamic Load (N) 310 310

So, it is preferable for the magnetic flux to confine the low reluctance iron path rather than high reluctance path of nonmagnetic material. The PM material is NdfeB-N40 grade 40 and the steel material is Silicon Iron M-19. The equivalent magnetic circuits of the bias and control flux are shown in Fig. 8. This analysis assumes that there is no leakage flux or fringing at air-gap. In addition, The flux density is assumed to be constant around the magnetic core.

(a) MB1

(b) MB2

(c) Legend Figure 7: Proposed design of MB1 & MB2 (mm).

Figure 6: Force directions of MB

Let FP M i is the MMF of the ith PM pole. 2FPM1 σP 2FPM2 σP 2FPM3 σP 2FPM4 σP

= φP M 1 (2

(1)

= φP M 3 (2

By ignoring the core reluctance, the eqn. 1 can rewritten as:

= φP M 4 (2

2FPM1 σP 2FPM2 σP 2FPM3 σP 2FPM4 σP

where φP M i Flux of ith PM. φP M t Summation of all fluxes of all PM poles. 215


= φP M 1 (2

2FC 1 σc 2FC 2 σc 2FC 3 σc 2FC 4 σc

= φP 1 (2

(5)

= φP 3 (2

where (a) Magnetic circuit of Bias Flux

φP i Flux of ith electromagnetic pole. φP t Summation of all fluxes of all electromagnetic poles. 230


(b) Magnetic circuit of Control Flux Figure 8: Equivalent magnetic circuits.

At balance condition, the PM poles fluxes can be written as follows:
220

    φP M 1 FP M 1 φP M 2  FP M 2  1     φP M 3  = σP (

(3)

FP M 1 cos(30)+FP M 2 cos(30)+FP M 3 cos(60)+FP M 4 cos(60) ∵ FP M = FP M 1 = FP M 2 = FP M 3 = FP M 4 T otalP M √ (4) ∴ FP M = (1 + 3)FP M

and

FP2 M µo σP2 Ag (

= φP 4 (2
    φC1 FC1 FC2  φC2  1     φC3  = σc (
(7)

Bg2 φCi 2 (N i)2 µo Ag Ag = = 2µo 2µo Ag 2lg2

(8)

The magnetic force FCi for ith electromagnetic pole can be calculated according to: FCi =

Z−axis

FP M =

(6) = φP 3 (2

where
Where:

= φP 2 (2
At balance condition, the electromagnetic poles fluxes can be written as follows:

Z−axis

225

= φP 1 (2
Where 235

φCi The magnetic flux of ith electromagnetic pole. Bg The flux density of air gap.

Similarly, Let FCi is the MMF of the ith coil pole.

µo The permeability of free space (4π × 10−7 H/m). Ag Cross section area of air gap. 6

lg Length of air gap. 240

N Number of coil’s turns. 250

i Coil current. The position stiffness of the PM poles along z-axis can be calculated as follows: √ −2(1 + 3)FP2 M ∴ KP M Z = 2 2 2 (9)255 µo σP Ag (

245

The forces of electromagnetic poles can be deduced as follows: T otal = FC2 − FC1 FC Z−axis T otal = FC3 − FC4 FC T otal Y −axis FP 2 FP 2 = σ2 µo A ∴ FC 2 − σ 2 µ A (< 2 g (
c

2.2.2. Axial AMB Analysis: The configuration of the AAMB is shown in Fig. 9. The inner and outer air gap is 1 mm. It has two symmetrical parts to generate a force along the positive and negative x-axis. The stator and rotor are made of Silicon iron M-19. Both of electromagnetic coils has 100 turns of copper with ampacity Jmax = 6A/mm2 and the bearing is designed to compensate axial force up to 4000 N. The dimension and the equivalent magnetic circuit of the Axial AMB is shown in Fig. 9. The fluxes can be obtained as follows: FA1 σA (
φA1 = φA2

Using equation 8, the force of the axial bearing are given by:

c

The position stiffness of electromagnetic poles is: 2

KCY =

4FP σc2 µ2o A2g (
Similarly, KCZ =

4FP σc2 µ2o A2g (
(10)

The current stiffness of electromagnetic poles can be calculated as follows:

265

Similarly, KCZ

4N 2 i = 2 σc µo Ag (
MB2 1.25 1.15 8961426.97 397887.35 2000 1 11500 100

φ2A2 φ2A2 + 2µo Agin 2µo Agout

(14)

φAi Flux of ith half of AAMB. FAi MMF of ith half of AAMB. FAi Force of ith half of AAMB. FAt Net force of AAMB.
Lg Air gap length. The position stiffness can be calculates as follows: Kx =

Table 6: The parameter of radial MB

MB1 1.2 1.11 16293503.59 723431.55 1100 1 11500 100

FA2 =

σA Leakage coefficient of AAMB.

(12)

The parameters of radial HMB is given in Table 6.

Parameter σP σc

φ2A1 φ2A1 + 2µo Agin 2µo Agout

Agin , Agout Cross section area of inner and outer air gap respectively.

2

4N i σc2 µo Ag (
FA1 =

FAt = FA2 − FA1

The total position stiffness along z-axis is equal to the summation of both position stiffness of PM poles and electromagnetic poles: T otal 260 FP 2 FP 2 = F − 2 2 σc µo Ag (
KCY =

(13)

2Agin Agout µo FA 2 (Agin +Agout )2 2 (A 3 σA gin Lg +Agout Lg +Agin Agout Rsvin µo +Agin Agout Rsvout µo )

(15)

The current stiffness can be calculated as follows: Ki =

270

7

2N 2 i(Agin + Agout ) 2 (R 2 Agin Agout µo σA svin + Rsvout + Rgin + Rgout ) (16)

The parameters for the AAMB are given in Table 7.

Table 7: The parameter of AAMB

Parameter µr (steel)
275

(a) Diamension of the upper half of axial MB (mm).

280

Value 8000 24616.419 127.7 100 190342

Parameter σA
Value 1.1 4180.76 21.7 32327

2.2.3. Finite Element Analysis: The proposed designs of RHPBHMB and AAMB are analysed by 3D FEM and 2D FEM analysis respectively. All these FEM models are constructed using MSC Marc package as shown in Fig. 10. The position and current stiffness of electromagnetic poles of the MB1, MB2 and AAMB are shown in Fig. 11. It is clear that there is a great consistent between the analytical and FEM results. A comparison between the analytical and FEM results is given in Table 8 3. Mathematical Dynamic Model of Overall System

(b) Legend

285

This section presents the mathematical model for each component of MLWT and describe the interaction between them. The system is composed of three coupled models: aerodynamic model, mechanical model, permanent magnet synchronous generator model. The mechanical model includes the MB and CB model. 3.1. Modeling of Aerodynamic:

290

295

(c) The direction of flux through the Axial MB

This model aims to find the relationship between the shape of airfoil and its aerodynamic properties. The aerodynamic axial forces, spining torque and bending moment can be determined in terms of the aerodynamic lift (FL ) and drag (FD ) forces. So, the Blade element theory can be used to obtain these forces depending on the two-dimensional aerofoil characteristics of NACA 0015 as in [26]. The wind turbine aerodynamic parameters can be summarized in Table 9.

Table 8: Comparison between analytical and FEM results MB

MB1

Parameter KCY = KCZ (N/mm) KCY = KCZ (N/A) T otal FP M (N ) Z−axis

MB2

FC1 = FC2 = FC3 = FC4 (N ) Max. Capacity at 6A (N ) KCY = KCZ (N/mm) KCY = KCZ (N/A) T otal FP M (N ) Z−axis

(d) Magnetic circuit Axial

Figure 9: Equivalent magnetic circuit of the axial MB.

8

FC1 = FC2 = FC3 = FC4 (N ) Max. Capacity at 5A (N ) Kx (N/mm) Ki (N/A) FA1 = FA2 = (N )

Analytical 403886.288 67.314

FEM 396666.66 67.33

Designed

626.84

617.13

650

403.88 1030.72 684142.71 114.024

394.7 1029.9 675000 110.66

310 960

1139.7

1138.9

1110

475.09 1614.8 4263636.441 1408 4236.9

443.1 1551.3 4900000 1520 3795.6

310 1420

4200

(a) Force displacement stiffness of MB1

(b) Current force stiffness of MB1

(c) Force displacement stiffness of MB2

(d) Current force stiffness of MB2

(e) Force displacement stiffness of AAMB

(f) Current force stiffness of AAMB

(a) The FEM model of MB1 with coil current of 6 A

(b) The FEM model of MB2 with coil current of 5 A

Figure 11: The comparison between FEM and magnetic circuit analysis

Table 9: Wind turbine aerodynamic parameter

Parameter Rblade ρ

Value 1m 1.22 kg/m3

cl cd

0.8445 0.008871 0.208 m

Lchord

(c) The FEM model of AAMB with coil current of 5 A

305

Figure 10: The FEM models of the three MB.

3.2. Modeling of RHPBHMB: 300

310

The magnetic bearing is considered to be a rigid floating body. So, the theory of flight dynamics is proper to describe its dynamic model.

Value 20o 7.677o 11 m/s 16 m/s

3.2.1. Rotor Motion Equation: Fig. 12 shows a simplified horizontal shaft suspended by MB. In order to fully position the shaft, one has to apply forces along five axes. Two radial and one thrust MB are needed in this case. In such a system we have five degrees of freedom (DOF): three linear motion: vertical, horizontal, and axial, and two rotational motions: pitching and yawing. The remaining DOF, which is rolling motion, is controlled by load torque. The equations which describe the 5-DOF motion can be written as follows: • Axial direction:

9

Parameter β α Rated wind speed Max. wind speed

- Linear motion mx¨o + βxo + 2γ x˙o = • Radial direction: 315

X

Fa

(17)

- Linear motion 1. Horizontal my¨o − αyo = 2. Vertical

X

X

X

Fy2

(18)

Fz1 −

X

Fz2

(19)

X

Fz1 − l2

X

Fz2

(20)

X

Fy1 − l2

X

Fy2

(21)

mz¨o − αzo = − - Rotational motion

Fy1 +

(a) 3D

1. Pitching Jy θ¨ + wr Jx ψ˙ = l1 2. Yawing Jy ψ¨ − wr Jx θ˙ = l1

3.2.2. Gap Deviation: The ith gap deviation, gi0 , between the stator and rotor of MB can be expressed in terms of yo , zo , θ, ψ as follows:

where 320

gi = Do + gi0 i = 11, 13, 21, 23  0   0    g11 g12 zo − l1 θ 0  g21  0   zo + l2 θ   0  = − g22    0 = g13  g14 −yo − l1 ψ  0 0 g23 −yo + l2 ψ g24

(b) 2D

(22)

(23)

(c) Gap deviation of MB1 (d) Gap deviation of MB2

0 0 g11 , g12 The vertical gap deviation of MB1 in +ve and -ve direction respectively.

Figure 12: Active magnetic bearing system

0 0 g13 , g14 The horizontal gap deviation of MB1 in +ve and -ve direction respectively.

3.2.3. Magnetic Force: The forces in HMB is induced from the biased and control fluxes. The total forces of biased fluxes has been derived in equation 4 and can be rewritten as follows: √ T otal 4(1 + 3)FPM 2 FP M = 2 σP µo AP M (2

0 0 The vertical gap deviation of MB2 in +ve and -ve , g22 g21 direction respectively. 325

0 0 g23 , g24 The horizontal gap deviation of MB2 in +ve and -ve direction respectively.

zo The rotor centre of mass coordinate along z axis.

Furthermore, the force due to control flux for the proposed homopolar MB arrangement is equal to:

yo The rotor centre of mass coordinate along y axis. l1 The distance between rotor centre of mass and MB1. 330

Homo (2N i)2 φ2 Fm = = µo Ag µo Ag (2
l2 The distance between rotor centre of mass and MB2. θ The pitch angle around y axis ψ The yaw angle around z axis. 10

(24)

3.2.4. Electric circuit equation: The electric circuit equation for any electromagnetic pole can be specified in terms of current as follows: E = Ri + L

di dt

335

where
u2 = e12

u3 = e21

u4 = e22

u5 = e13

u6 = e14

u7 = e23

u8 = e24

0 x1 = g11 x5 = g 0˙

0 x2 = g21 x6 = g 0˙ 21

0 x3 = g13 x7 = g 0˙ 13

0 x4 = g23 x8 = g 0˙

x10 = i12 ,

x11 = i21 ,

x12 = i22

x14 = i14 ,

x15 = i23 ,

x16 = i24

11

x9 = i11 x13 = i13 ,

,

1 (u7 − Re2 x15 ), L23

L23 =

x˙16 =

1 (u8 − Re2 x16 ), L24

L24 =

F11 = k1 F12 = k1 F21 = k2

x˙9 =

x˙10 =

1 (u1 − Re1 x9 ), L11

1 (u2 − Re1 x10 ), L12

L12 =

F23 = k2

FP M 1

(26)

FP M 2

N o +x1
x˙12 =

1 (u4 − Re2 x12 ), L22

L22 = L13

N o −x2
     

x9 o −x1
2

2

2

2

2

2

1 m 340

+

l12 Jy ,

(28)

2



4N 2 4N 2 µo Ag1 , k2 = µo Ag2 , FP M = Hm lm , l2 1 1 Hs22 = m + J2y , and Hd12 = m − lJ1 ly2 .

Where k1 =

Hs11 =

The HMB parameters can be summarized in Table 10: 3.3. Modeling of AAMB: The axial equation of motion (Eqnation 17) is independent and simpler than the other radial motion equations. It can be rewritten as follows:

N2 = o −x3
L14 =



2 x16 o +x4
F24 = k2

2

L21 =

1 (u6 − Re1 x14 ), L14

F14 = k1

2

1 (u3 − Re2 x11 ), L21

x˙14 =

F13 = k1

N o −x1
x˙11 =

1 x˙13 = (u5 − Re1 x13 ), L13

F22 = k2

2

L11 =

N2 o +x4
The magnetic forces are expressed as follows:

23

x˙1 = x5 , x˙2 = x6 , x˙3 = x7 , x˙4 = x8   α l2 x1 + l1 x2 w r l1 J x x˙5 = (x7 − x8 ) − m l1 + l2 Jy (l1 + l2 ) X X − Hs11 Fz1 − Hd12 Fz2   α l2 x1 + l1 x2 w r l2 J x x˙6 = (x7 − x8 ) − m l1 + l2 Jy (l1 + l2 ) X X − Hd12 Fz1 − Hs22 Fz2   α l2 x3 + l1 x4 w r l1 J x x˙7 = (x5 − x6 ) − m l1 + l2 Jy (l1 + l2 ) X X − Hs11 Fy1 − Hd12 Fy2   α l2 x3 + l1 x4 w r l2 J x (x5 − x6 ) x˙8 = − m l1 + l2 Jy (l1 + l2 ) X X − Hd12 Fy1 − Hs22 Fy2

N2 o −x4
The summation of forces along the horizontal and vertical directions can be expressed as follows: X Fy1 = F13 − F14 X Fy2 = F23 − F24 (27) X Fz1 = F11 − F12 − F R1 + FP M 1 X Fz2 = F21 − F22 − F R2 + FP M 2

(25)

(2N )2 L=
x˙15 =

N2 o +x3
mx¨o = Fa1 − Fa2 11

(29)

Table 10: The HMB parameters. Parameter l1 m Jx Do
Value 674.5 mm 179.35 kg 7.66 kg.m2 1 mm 30625.948 AT/wb 32613111.68 AT/wb 0.55776 Ω 650 N 25 mm

Parameter l2 h Jy N
Table 12: The generator parameters.

Value 132 mm 20 mm 39.32 kg.m2 100 turns 38697.114 AT/wb 17959064.27 AT/wb 0.86016 Ω 1110 N 2000 mm2 1100 mm2

Parameter Rs Vrated N 350

assuming β and γ are zero. The state space representation 355 can be written as follows: ua1 = ea1 ,

ua2 = ea2

0 ga1 ,

x2 = ga1 ˙ 0

x3 = ia1 ,

x4 = ia2

x1 =

360

Value 2.4Ω 220 V 375 rpm

Parameter Ls Prated poles

Value 51 mH 3 kW 16

3.5. Analysis of Conventional Mechanical Bearing (CB) The CB is considered here to provide a fair comparison between the performance of WT-CB and WT-MB. The frictional moment for the CB is obtained from the SKF Rolling bearings catalog [28]. In this work, two conventional SKF bearings, CB1 and CB2, are located in the same position of MB1 and MB2 respectively. Their designation number is 22211EK. The grease is used as lubricant at operating temperature 50C o . The grade viscosity for the two selected conventional bearing CB1 and CB2 are ISO VG 100 and ISO VG 150 respectively. 3.6. Simulation of Multi-domain Physical Systems:

x˙1 =x2

1 x˙2 = (Fa1 − Fa2 ) m

x˙3 =

1 (u1 − Raxial x3 ), LA1 = LA1
x˙4 =

1 (u2 − Raxial x4 ), LA2 = LA2
N2 Do −x1 µo Agin 2

+

Do −x1 µo Agout

Do +x1 µo Agin

+

Do +x1 µo Agout

365

N

(30) 370

The magnetic forces are expressed as follows: Fa1

N2 = 2µo Agin

Fa2

N2 = 2µo Agin

 

x3 Do −x1 µo Agin

x4 Do +x1 µo Agin

2 2

N2 + 2µo Agout N2 + 2µo Agout

 

x3 Do −x1 µo Agout

x4 Do +x1 µo Agout

2

2 (31)375

where
Parameter m Agin N Raxial

Value 179.35 kg 32327 mm2 100 turns 0.007389 Ω

Parameter
Value 149.4 At/wb 190342 mm2 1 mm

4. Controller Design

385

3.4. Modeling of Generator 345

The wind turbine is a multi-disciplinary system. As aforementioned, it includes aerodynamic, mechanical and electrical systems. Simscape is able to create models consist of multi-physical domain under the Simulink environment which helps to provide more realistic simulation and contributes in studying the interactions between the different physical systems comprehensively. Fig.13 shows the sub-systems of the wind turbine and the interaction between them. The aerodynamic model convert the wind speed into three components: spin torque, bending moment and axial forces. All these components are applied to the shaft which is magnetically levitated. According to the shaft position, the air-gaps are measured by the MB1, MB2, and AAMB to generate the required forces that compensate the reactions, R1 , R2 , and Faxial . The generator whose rotor is fixed on the shaft rotates with speed w caused by aerodynamic spinning torque. As well, the electromagnetic torque which is induced due to electrical load on the generator is reflected to the shaft and subsequently to the aerodynamic subsystem. Fig.14 shows the visualization of the whole system whose animation can be seen in [29].

The permanent magnet synchronous generator (PMSG) is used to supply an electrical load . The parameter of gen-390 erator is obtained from [27]. The Simulink PMSM block is operated as generator with the parameters given in Table. 12.

12

The design of the TSK fuzzy Q-parameterization controller is discussed in details in [30, 31, 32]. So, This section only focuses on the linearization process and the application of the controller to the previous MB dynamic model in eqn 26 . The controller objective are: achieve robust stability, overcome model nonlinearity by increasing the dynamic operating range of gap displacement, reject step and unbalance sinusoidal disturbance at different speeds.

and electromagnetic voltage of ith pole and i0i , e0i are the deviation of them from their nominal values. Then we can write: Ii = Ioi + i0i (32) ei = Eoi + e0i for i=11,13,21,23. Assume: 0 0 I11 = −I12 ,

e011 = −e012 ,

0 0 I13 = −I14 ,

e013 = −e014 ,

0 0 I21 = −I22 ,

0 0 I23 = −I24

e021 = −e022 ,

e023 = −e024 (33) So, the linearized model of each HMB subsystem described in equation. 26 can be written as follows: First subsystem (vertical motion of MB1): x˙1 = x5 x˙5 = Pˆ1 x1 + Gˆ1 x9

(34)

2Do −Re1 (
where 0 0˙ , x = i0 x1 = g11 , x5 = g11 9 11 √ 2
16(1+ 3)A F 4A N 2µ 2 2 1 µo +Io12 Pˆ1 = −Hs11 (2D+Ag1g1
Figure 13: The Interaction between the different sub-systems

−(2Ag1 Hs11 N 2 µo (Io11 +Io12 )) (2D+Ag1
P1

Second subsystem (vertical motion of MB2): x˙2 = x6 x˙6 = Pˆ2 x2 + Gˆ2 x11 −Re2 2Do x˙11 = (
(35)

where 0 0 0˙ , x x2 = g21 , x6 = g21 10 = i21 √ 2
2 16(1+ 3)A F 4A N µ 2 2 2 µo Pˆ2 = −Hs22 (2D+Ag2g2
−(2Ag2 Hs22 N 2 µo (Io21 +Io22 )) (2D+Ag2
P2

Third subsystem (horizontal motion of MB1): x˙3 = x7 x˙7 = Pˆ3 x3 + Gˆ3 x13 −Re1 2Do (
Figure 14: The Visualization of the whole system using Simscape.

4.1. System Decoupling and Linearization: 395

400

(36)

where 0 0 0˙ , x x3 = g13 , x7 = g13 13 = i13 2 2 2 −(4Ag1 Hs11 N µo (Io13 +Io14 )) Pˆ3 = 3

In order to simplify the controller design, the following assumptions are made to decouple the system into four subsystems: 1. The gyroscopic effect is neglected, hence, the vertical and horizontal motion are decoupled. 2. Assume the left and right bearing are located at their center of percussion, So, the left and right bearing are decoupled (Hd12 u 0) Then, each subsystem is linearized at different operating points. Let Ioi , Eoi are the nominal value of the current

Gˆ3 =

(2D+Ag1
Fourth subsystem (horizontal motion of MB2): x˙4 = x8 x˙8 = Pˆ4 x4 + Gˆ4 x15 −Re2 2Do x˙15 = (
(37)

420

(a) Gap Deviation (b) Current Devia- (c) Voltage Deviaof 1st subsystem tion of 1st subsys- tion of 1st subsystem tem

425

and second set of TSK fuzzy rules determines the most suitable Q-parametrization controller and the steady state input voltage (E1,E2) to drive the HMB system based on gap deviation and current deviation. The xi represent the gap deviation for i = 1, 2, 3, 4 and uj represent the voltage deviation for j = 1, 3, 5, 7. So, the rules can be written as follows: Plant Rule 1: If xi = ZE then x˙ = A1 x + B1 u and y = c1 x. Plant Rule 2: If xi = P OS and xj = ZE then x˙ = A2 x+B2 u and y = c2 x.

430

(d) Gap Deviation (e) Current Devia- (f) Voltage Deviaof 2nd subsystem tion of 2nd subsys- tion of 2nd subsystem tem

435

Plant Rule 3: If xi = N E and xj = ZE then x˙ = A3 x+B3 u and y = c3 x. Plant Rule 4: If xi = P OS and xj = N E then x˙ = A4 x+B4 u and y = c4 x. Plant Rule 5: If xi = N E and xj = P OS then x˙ = A5 x+B5 u and y = c5 x.

(g) Gap Deviation (h) Current Devia- (i) Voltage Deviaof 3rd subsystem tion of 3rd subsys- tion of 3rd subsystem tem

440

Plant Rule 6: If xi = P OS and xj = P OS then x˙ = A6 x + B6 u and y = c6 x. Plant Rule 7: If xi = N E and xj = N E then x˙ = A7 x+B7 u and y = c7 x. Where for the 1st subsystem:

(j) Gap Deviation of (k) Current Devia- (l) Voltage Devia4th subsystem tion of 4th subsys- tion of 4th subsystem tem

A1

Figure 15: Membership functions of HMB system

A3

405

where 0 0 0˙ , x x4 = g23 , x8 = g23 15 = i23 2 2 2 −(4Ag2 Hs22 N µo (Io23 +Io24 )) Pˆ4 = 3 Gˆ4 =

A5

(2D+Ag2
A7

415

0 1 = 1088.56 0 0 0  0 = 702662.13 0  0 = 4861026.12 −65891.74  0 = 268071.33 64880.97

 0 0.83  −20.6

1 0 0

1 0 0 1 0 0

For the HMB system described in equations 26. The controller is designed for the four subsystems described in linear equations 34, 35, 36, and 37. The HMB subsystems have the same sequence of rules sets to describe the non-linearity of each subsystem but with different membership functions, linearization values and applied input voltages. The membership functions of these rules at different operating points are defined in Fig.15. The first



 0 1 0 A2 = 728149.28 0 40.03  0 0 −4.46    0 0 1 0 39.35  A4 =  274229.65 0 −22.33 −36.74 −65891.74 0 −4.46    0 0 1 0 101.71  A6 = 4835090.3 0 101.44 −36.74 64880.97 0 −4.46  0 −22.06 −36.74

 0 Bi =  0  36.94

4.2. Application of TSK Fuzzy Q-parameterization Controllers to HMB: 410



for i=1,2,3,4,5,6,7. For the 2nd subsystem:   0 1 0 0.53  A1 = 714.3 0 0 0 −17.94 14



 ci = 1 0



 0

 0 1 0 A2 = 331419.16 0 20.41  0 0 −4.25



0 A3 = 315952.65 0  0 A5 = 2138818.85 −54952.86  0 A7 = 109282.09 53883.33

  1 0 0 0 19.94  A4 =  111901.72 0 −31.63 −54952.86   1 0 0 0 50.8  A6 = 2127382.3 0 −31.63 53883.33  1 0 0 −10.32 0 −31.63





0 Bi =  0  20.86

for i=1,2,3,4,5,6,7. For the 3rd subsystem: A1

A3

A5

A7

for i=1,2,3,4,5,6,7. For the 4th subsystem:

A5

A7

 0

 ci = 1

0

 0

475

   0 1 0 0 1 0 0.41  A2 = 315722.39 0 19.93  = 295.17 0 0 0 −17.94 0 0 −4.25     0 1 0 0 1 0 = 315722.39 0 19.93  A4 =  91962.78 0 −9.4  0 0 −31.63 −52220.02 0 −4.25     0 1 0 0 1 0 = 2009526.92 0 49.26  A6 = 2009526.92 0 49.26  −52220.02 0 −31.63 52220.02 0 −4.25   0 1 0 −9.4  = 91962.78 0 52220.02 0 −31.63

 0 Bi =  0  36.94

 ci = 1

0

for i=1,2,3,4,5,6,7. The following is the second set of rules:

Plant Rule 7: If xi = N E and uj = N E then E1 = em7 and E2 = en7 . The values of em and en for 1st, 3rd and 2nd, 4th subsystems are given in Tables 13 and 14 respectively. 4.3. Validation of HMB control system The proposed TSK fuzzy Q-parameterization controller is designed based on the linear system described in equations 34, 35, 36, and 37 then applied to a complete magnetically levitated system constructed by Simscape package. Simulation results are obtained at the rated speed of the system (31.41 rad/s = 300 rpm). 0 0 0 Fig. 16 shows the step response of g11 , g13 , g21 , and 0 g23 . The overshoot of these deviations are 14.4%, 14.7%, 13.3% and 13.7% respectively with zero steady state error. The settling time for all of them is less than 0.2s.

Table 13: Values of em and en for first and third subsystem of HMB system.

1st subsystem emi eni e11 = 0.6V e12 e11 = 2.27V e12 e11 = 2.27V e12 e11 = 0.45V e12 e11 = 0.45V e12 e11 = 4.065V e12 e11 = 4.065V e12 3rd subsystem Rules emi eni 1 e13 = 0.42V e14 2 e13 = 2.23V e14 3 e13 = 2.23V e14 4 e13 = 0.48V e14 5 e13 = 0.48V e14 6 e13 = 3.98V e14 7 e13 = 3.98V e14

Rules 1 2 3 4 5 6 7

 0

Plant Rule 1: If xi = ZE then E1 = em1 and E2 = en1 . 450

Plant Rule 3: If xi = N E and uj = ZE then E1 = em3 and E2 = en3 .

Plant Rule 6: If xi = P OS and uj = P OS then E1 = em6 and E2 = en6 .





445

460

Plant Rule 5: If xi = N E and uj = P OS then E1 = em5 and E2 = en5 .

   0 1 0 0 1 0 465 = 500.8 0 0.68  A2 = 702318.92 0 39.33  0 0 −20.6 0 0 −4.46     0 1 0 0 1 0 = 702318.92 0 39.33  A4 =  233538.04 0 −20.51 0 0 −36.74 −63222.58 0 −4.46     0 1 0 0 1 0 = 4619512.28 0 99.17  A6 = 4619512.28 0 99.17  470 63222.58 0 −4.46 −63222.58 0 −36.74   0 1 0 = 233538.04 0 −20.51 63222.58 0 −36.74

 0 Bi =  0  36.94

A3

0

Plant Rule 4: If xi = P OS and uj = N E then E1 = em4 and E2 = en4 .





A1

 ci = 1

 1 0 0 −10.46 0 −4.25  1 0 0 50.67  455 0 −4.25

Plant Rule 2: If xi = P OS and uj = ZE then E1 = em2 and E2 = en2 . 15

= 0.42V = 2.23V = 2.23V = 4.053V = 4.053V = 0.437V = 0.437V

= 0.42V = 2.23V = 2.23V = 3.98V = 3.98V = 0.48V = 0.48V

Table 14: Values of em and en for second and forth subsystem of HMB system.

Rules 1 2 3 4 5 6 7 Rules 1 2 3 4 5 6 7

480

485

490

495

500

2nd subsystem emi eni e21 = 1V e22 = 0.65V e21 = 3.52V e22 = 3.44V e21 = 3.52V e22 = 3.44V e21 = 0.76V e22 = 6.2V e21 = 0.76V e22 = 6.2V e21 = 6.23V e22 = 0.73V e21 = 6.23V e22 = 0.73V 4th subsystem emi eni e23 = 0.65V e24 = 0.65V e23 = 3.44V e24 = 3.44V e23 = 3.44V e24 = 3.44V e23 = 0.82V e24 = 6.07V e23 = 0.82V e24 = 6.07V e23 = 6.07V e24 = 0.82V e23 = 6.07V e24 = 0.82V

0 (a) g11

Fig. 17 shows that the proposed controller can overcome the non linear dynamics of HMB system by extending the gap deviation upto 800 µm. For brevity, we include 0 only the gap deviation of g11 of HMB1 as all the others have the same response. 0 0 0 0 Fig. 18 shows the response of g11 , g13 , g21 , and g23 due to step disturbance. The max. deflection of these deviations are 28, 35.5, 18, 31.6 µm due to step disturbance force of 100 N respectively. 0 0 0 Fig 19 shows the system response of g11 , g13 , g21 , and 0 g23 due to different vertical and horizontal initial conditions. These gap displacements (initial conditions) must be equally divided into small steps (about 50µm) in order to drive the HMB system to zero position in a stable way.

0 (b) g13

0 (c) g21

The proposed controller can generate a compensating forces to reject any imbalance sinusoidal disturbance up to 492 rpm as shown in Fig 20. The imbalance sinusoidal disturbance is generated by sticking a mass of 1 kg to the shaft at the location of each MB. 4.4. Controller Design of Axial AMB: The axial motion is independent of the radial motion. A Q-parameterization controller is synthesized to stabilize the axial motion described in equation 17 and it is controlled separately. For brevity, it is omitted in this paper. 0 (d) g23

505

510

5. Performance Comparison between WT-MB and WT-CB

Figure 16: The HMB System response due to a step at reference.

This section presents a comparison between the wind turbine performance under two different operating cases. In the first case, the wind turbine is supported by conventional mechanical bearing (WT-CB). While in the second

case, the wind turbine is supported by magnetic bearing (WT-MB). The wind turbine performance is evaluated un16

0 (a) g11

0 Deviation for HMB system Figure 17: The sinusoidal tracking of g11

515

der low wind speed and high wind speed assuming that the operating temperature of CB is 50C o . The steady state values of the WT performance are given in Table 15. 5.1. Wind Turbine Performance at Low Wind Speed:

520

525

530

535

540

0 (b) g13

The wind turbine performance is investigated under different low wind speeds (1, 2, 3, 5 m/s) with no electrical loads connected to the generator. The comparison includes different aspects such as spinning torque, rotational speed, consumed power and frictional moment. 1. Spinning Torque: The WT-MB required much lower aerodynamic spinning torque than WT-CB as clear in Fig. 21. The WT-MB can rotates with only 1 m/s wind speed compared with 2 m/s for WT-CB. This mean that the MB has reduced the starting torque and cut-in speed of wind turbine 2. Rotational Speeds: The WT-MB are higher than those of WT-CB for the same wind speeds and the difference between them decreases as the wind speed increases. This is shown in Fig. 22.

0 (c) g21

3. Consumption Power: The WT-CB records lower losses than the MB input power below 3 m/s. After that the situation is reversed and the WT-CB has higher losses for wind speeds over 3 m/s because the increasing of rotational speeds as shown in Fig. 23. 4. Frictional Moment: The frictional moment of CB is proportional to the rotational speed and at wind speed of 1 m/s its value is bigger than aerodynamic spinning torque which block the WT-CB from rotation under this low wind speed (see Fig. 24).

0 (d) g23

Figure 18: The HMB System response due to a step rejection disturbance.

5.2. Wind Turbine Performance at high wind speed Speed:

In this section the wind turbine performance is evaluated at different high wind speed (9, 10, 11, 12, 13, 14,545 15, 16 m/s) with an electrical load connected to the generator. The comparison includes different aspects such as 17

(a) g11

(c) g21

(a) g11’ of MB1

(b) Net vertical force of MB1

(c) g13’ of MB1

(d) Net horizontal force of MB1

(e) g21’ of MB2

(f) Net vertical force of MB2

(g) g23’ of MB2

(h) Net horizontal force of

(b) g12

(d) g22

Figure 19: The HMB System response due to different initial conditions.

spinning torque, rotational speed, electromagnetic torque, electrical generated power, consumed power, efficiency and frictional moment. 550

555

1. Spinning Torque: The spinning torque in WT-CB is higher than that of WT-MB for all mentioned wind speed because of the presence of frictional moment in case of WT-CB as shown in Fig. 25 2. Rotational Speeds: The WT-MB achieves a little increase in rotational speed more than that WT-CB as shown in Fig. 26.

560

3. Electromagnetic Torque: The electromagnetic torque MB2 of WT-MB, which resulted from connecting electrical Figure 20: The HMB System response due to imbalance sinusoidal load to the generator, is higher than that of WT-CB disturbance rejection. as shown in Fig. 27.

565

4. Electrical Generated Power: The electrical net output power of the generator, after subtracting the MB input power in case of WT-MB, is shown in575 Fig.28. The WT-MB has higher electrical output power than WT-CB for the mentioned wind speeds.

570

7. Frictional Moment: The frictional moment of CB is proportional to the rotational speed as shown in Fig. 31).

6. Conclusions 5. Consumption Power: The MB of WT-MB has In this paper, a comprehensive assessment of the perlower input power than the bearing losses of WT-CB formance of WT-MB compared with WT-CB is provided. for all mentioned wind speed because the increasing 580 For this purpose, a design of two RHPBHMB and one of rotational speed. 29. AAMB is developed using magnetic circuit and FEM anal6. Efficiency: The WT-MB has little higher efficiency ysis. Furthermore, A TSK fuzzy Q-parameterization conthan that of WT-CB for all mentioned wind speed. troller is synthesised for RHPBHMB. As a result, this 30. study shows that the MB can successfully replace the CB 585 without decreasing the extracted power from the wind. 18

590

595

Figure 21: The wind turbine spinning torque under low wind speeds.

Figure 23: The bearings power consumption under low wind speeds.

Figure 22: The wind turbine rotational speed under low wind speeds.

Figure 24: The frictional moment of CB of wind turbine under low wind speeds.

Hence, the main benefits of MB can be gained such as avoiding wind turbine failure, increasing life span and re-600 sity. Authors sincerely acknowledge for this help and for ducing maintenance. Besides, the obtained results prove providing all necessary equipments to complete this work. the ability of MB to enhance the wind turbine speed, minimising starting torque, reducing cut-in speed and comReferences pensating of wind disturbances. Moreover, a robust gain scheduled TSK fuzzy Q-parametrization controller is de[1] H. Bleuler, M. Cole, P. Keogh, R. Larsonneur, E. Maslen, Y. Okada, G. Schweitzer, A. Traxler, G. Schweitzer, E. H. signed to overcome the nonlinear dynamics of HMB sys605 Maslen, et al., Magnetic bearings: theory, design, and applicatem, maximise the operating envelope up to 80% of gap tion to rotating machinery, Springer Science & Business Media, displacement and reject the step and imbalance sinusoidal 2009. disturbance at any operating speeds. [2] H. Wu, Z. Wang, Y. Hu, Study on magnetic levitation wind 610

Acknowledgment Wind Turbine information is provided by RIAM Lab., Research Institute for Applied Mechanics, Kyushu Univer19

turbine for vertical type and low wind speed, in: 2010 AsiaPacific Power and Energy Engineering Conference, 2010, pp. 1–4. doi:10.1109/APPEEC.2010.5448476. [3] B. M. Gonzlez, C. G. Garca, H. A. Coyotecatl, S. V. Limn, Characterization of a system suspended by permanent magnets,

Figure 25: The wind turbine spinning torque under high wind speeds.

Figure 27: The wind turbine electromagnetic torque under high wind speeds.

Figure 26: The wind turbine rotational speed under high wind speeds.

615

[4]

620

[5]

625

[6]

[7] 630

Figure 28: The wind turbine electrical net output power under high wind speeds.

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Figure 29: The bearings power consumption of wind turbine under high wind speeds.

Figure 31: The frictional moment of CB of wind turbine under high wind speeds.

Table 15: Comparison between WT-MB and WT-CB under different wind speeds.

Wind speed (m/s) 1 2 3 5 9 10 11 12 13 14 15 16

Figure 30: The wind turbine efficiency under high wind speeds.

650

655

660

665

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Wind speed (m/s) 1 2 3 5 9 10 11 12 13 14 15 16

TSpin (N.m) CB MB 0.1599 0 0.5344 0 0.5513 0 0.5778 0 43.09 42.07 49.18 48.76 55.18 54.73 61.02 60.55 66.67 66.17 72.09 71.56 77.23 76.68 82.07 81.49

MF ric (N.m) 0.473 0.5344 0.5513 0.5778 0.6121 0.6287 0.6455 0.6624 0.6795 0.6965 0.7133 0.7298

RPM CB MB 0 39 71.1 78 112.3 117 192 195 227.3 228.4 262.9 263.9 299.3 300.3 336.5 337.5 374.4 375.3 412.8 413.8 451.9 452.8 491.4 492.3

Pelectric (W) CB MB

961.3 1271 1625 2022 2460 2935 3443 3980

966.74 1277.74 1632.74 2030.74 2473.74 2944.74 3452.74 3990

Telecmag (N.m) CB MB

-42.48 -48.55 -54.53 -60.36 -65.99 -71.39 -76.51 -81.34

PConsumption (W) CB MB 0 4.26 3.99 4.26 6.498 4.26 11.65 4.26 14.61 4.26 17.35 4.26 20.29 4.26 23.4 4.26 26.71 4.26 30.19 4.26 33.84 4.26 37.66 4.26

-42.07 -48.76 -54.73 -60.55 -66.17 -71.56 -76.68 -81.49 η

CB

MB

93.7 93.9 94 94.04 94.1 94.15 94.22 94.24

94.69 94.79 94.87 94.8 95.22 94.96 94.9 94.99

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680

[20]

685

[21]

[22] 690

[23] 695

[24]

[25] 700

[26] [27] 705

[28] [29] 710

[30] 715

[31]

720

[32] 725

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