energies Article
Optimal Design of Permanent Magnet Linear Generator and Its Application in a Wave Energy Conversion System Hong-wei Fang 1, * , Ru-nan Song 1 and Zhao-xia Xiao 2 1 2
*
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China;
[email protected] Tianjin Key Laboratory of Advanced Electrical Engineering and Energy Technology, Tianjin Polytechnic University, Tianjin 300387, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-138-2103-3642
Received: 9 October 2018; Accepted: 7 November 2018; Published: 10 November 2018
Abstract: The emerging global wave energy industry has great potential to contribute to the world’s energy needs. However, one of the key challenges in designing a wave energy converter (WEC) is the wave energy generator. Thus, this paper focuses on the optimal design of a cylindrical permanent magnet linear generator (CPMLG), which is used for the wave energy conversion system. To reduce the end effect and enhance the magnetic field performance of the CPMLG, the level-set method is applied to the design of the topology and size for the generator. In the paper, the objective air gap magnetic field is given by the mathematical analysis method and appropriate measuring points are predetermined. The measuring points can fully reflect the distribution characteristics of the air gap magnetic field. Then, topology evolution on the permanent magnet (PM) and yoke based on the level-set method are performed. The level set function corresponding to the initial shape of the PM is constructed. The algorithm is programmed and computed iteratively using the discrete time and space variables. Finally, the performances of the CPMLG with the updated PM and width of yoke are analyzed by ANSYS Maxwell. Results show that the magnetic field distortion and the unbalance of three-phase electromotive force (EMF) of the CPMLG is reduced by the optimization of the level-set method. It has also been verified that the designed CPMLG with the level-set method could be used for WEC at different wave conditions. Keywords: magnetic field; level-set method; shape design; voltage unbalance; wave energy conversion
1. Introduction Wave energy is an emerging renewable energy source, which can contribute towards sustainable development of our world [1–5]. Various wave energy generators have been used to convert wave power into useful mechanical power efficiently. The generator is one of the most important units in wave power generation system, so the design of the generator is crucial to the system efficiency. The ocean’s environment is undetermined and unstable. To achieve relatively high quality and efficiency for converting ocean wave energy into electrical energy, various ocean wave energy converters and corresponding electrical machines have been investigated. It was found that linear generators can be used in the direct drive wave energy conversion system, in which it can reduce the complexity of mechanical transmission and improve the reliability and efficiency of the system [6–11]. Several electrical generator topologies have been analyzed in the past years [12,13]. When compared with the induction generator, doubly-fed induction generator, synchronous generator, and permanent magnet synchronous generator, the permanent magnet linear generator used in wave
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energy conversion seems to have a higher reliability and efficiency because the driving force of the linear motor is in a straight-line reciprocating direction [14,15]. In addition, the primary core of the permanent magnet linear generator can be sealed by epoxy resin and other materials after being embedded, which can be applied in the marine environment. However, due to the inherent edge effect and cogging effect of the permanent magnet linear motor, the resulting positioning force will cause the electromagnetic force of the motor to fluctuate, generate noise and resonance, and hinder the stroke of the secondary motion. At the same time, the fluctuation of the positioning force will cause the air gap magnetic field to be distorted, affecting the back electromotive force waveform, and reducing the power generation quality. Therefore, it is necessary to reduce the positioning force as much as possible with the level-set method, and this is also the key target for the design of permanent magnet linear generators, which are used for wave energy conversion [16]. The study of the level-set method can be traced back to the last century. In 1987, Sethian applied the upwind numerical solution scheme to the calculation of the interface evolution problem for the first time in the study of wave propagation, and gave the definition of the weak solution of the interface motion [17], which laid a theoretical foundation for the level set method. The following year, Osher and Sethian proposed the level set method, and gave the high-precision stable numerical solution of the level set equation [18]. At that time, this method was mainly used to solve the problem of flame shape change under the thermodynamic equation. Because of the highly dynamic change of the flame shape, the change of its topological structure with time is highly complicated, so it is difficult to describe the motion process accurately with the traditional parametric equation. Therefore, the level set method based on the time and motion interface arose at the historic moment [19]. After many years of development, the level-set method has been successfully applied in many fields, such as physics, fluid mechanics, materials science, and computer graphics. The level-set method and shape design sensitivity has been used for the rotor topology optimization of switched reluctance motor (SRM) in [20], in which the shape of the salient pole rotor determines the characteristic of the reluctance torque. Various torque ripples of permanent magnet (PM) machines are reduced by using different level set methods [21,22]. In this paper, to improve the air-gap magnetic field performance and reduce the unbalance of back-EMF, the CPMLG used for a wave energy conversion system is designed by using the level-set method to increase the wave energy capturing efficiency. Additionally, its validity has been verified by its application for direct and in-direct driving wave energy conversion systems, respectively. 2. Level-Set Method 2.1. Principle of Level-Set Method Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model used in the optimization of the electrical machine is that it can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. If the solution of the new level-set function evolution equation is presented and the position of each point for shapes of the electrical machine on the zero level set are deduced, then the result of the evolution curve or surface is obtained. Suppose φ(x, y) = 0 is the implicit level set function. Then, according to the principle level-set method, we can get the partial difference equation as: ∂φ ∂C + ∇φ · = 0, ∂t ∂t where ∇φ is the gradient of φ, and ∂C ∂t represents the velocity field of the interface. The curve, C(t), obtained at pseudo time, t, is the solution of the zero level set, φ(C (x, y), t).
(1)
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If the arc length of the curve is expressed by s, then according to the curve evolution theorem, x y C the+ tangential = direction , of=C0is 0, i.e., it exists. it can be found that the change of = φ along (2)
s
x s
y s
s
∂φ ∂x ∂φ ∂y ∂C ∂φ = · + · = C ∇φ, ∂s ∂s ∂x ∂s ∂y ∂s ⊥ s . From Equation (2), it can be found that: ∂C From (2), it can found that: ∇φ⊥ ∂s So, theEquation normal vector of Cbecan be represented as:. So, the normal vector of C can be represented as:
=0
(2)
= − ∇φ N = −
→N
(3) (3)
|∇φ|
Thus, the level set evolution equation can be obtained as: Thus, the level set evolution equation can be obtained as:
∂φ = F t = F|∇φ| ∂t
(4) (4)
The above above level level set set evolution evolution can can be be used used for for shape shape optimization optimization as as shown shown in in Figure Figure 1. 1. The Start
Create optimization model
Create finite element model
Evaluate new mesh parameters No
Analyze with Maxwell finite elements
Adaptivity?
Error OK?
Yes
Yes
Estimate error
No
Evaluate sensitivities
Optimize B and wt
Update and re-analyze the optimization model No
Yes Stop
Figure 1. Flowchart of of the the optimization optimization algorithm Figure 1. Flowchart algorithm with with level level set set evolution. evolution.
2.2. The Solution of Level Set Function
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2.2. The Solution of Level Set Function Using the discrete grid to represent the level set function, φ(x, y, t), the distance between adjacent nodes of the discrete grid is h, the time step is ∆t, then the level set function at the (i, j) and n∆t is: φijn = φ(ih, jh, n∆t)
(5)
Therefore, the evolution Equation (4) of the level set function, φ, can be written as: φijn+1 − φijn ∆t
= Fijn ∇ij φijn
(6)
The third-order third-order weighted essentially non-oscillatory (WENO) discrete algorithm is applied to Equation (6) to obtain its solution as: 1 2 3 φx = w1 φx + w2 φx + w3 φx w1 + w2 + w3 = 1 0≤w ≤1
(7)
i
and
φ1 = x
φx2 =
φx3
=
Dφi−2 i − 7Dφ6i−1 + 11Dφ 3 6 Dφi+1 i − Dφ3i−1 + 5Dφ 6 + 3 Dφi 5Dφi+1 Dφi+2 + − 3 6 6
(8)
where wi is the weight, D is the operator, which is defined as follows: 0 Di = φi Di0+1 φ− Di0 φ D1 ∆x i+1/2 φ = 2 Di1+1/2 φ− Di1−1/2 φ 2∆x Di φ = ...
(9)
Then, discretize the corresponding time variables, we can get φ as:
φ n +1 − φ n + Fnn · ∇φn = 0 ∆t n + 2 n + φ −φ 1 + Fnn+1 · ∇φn+1 = 0 ∆t φn+1/2 = 34 φn+1 + 14 φn+2 φn+3/2 −φn+1/2 + Fnn+1/2 · ∇φn+1 ∆t φn+1 = 13 φn + 23 φn+3/2
(10)
=0
Further, the level set function has the following properties as: φ(( x, y), t) > 0 φ(( x, y), t) < 0 φ(( x, y), t) = 0
( x, y) ∈ Ω+
( x, y) ∈ Ω−
(11)
(( x, y) ∈ C )
Thus, when the level-set method is applied to the shape optimization, the surface formed by φ = 0 is used to describe the curve change of C, which can be directly applied to the shape design of electrical machines.
( ( x, y ) , t ) 0 x, y ) , t ) = 0 ((
( ( x, y ) ) −
(11)
( ( x, y ) C )
Thus, when the level-set method is applied to the shape optimization, the surface formed by ϕ = 0 is5 of 12 used to describe the curve change of C, which can be directly applied to the shape design of electrical machines.
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3. CPMLG Optimization with the Level-Set Method 3. CPMLG Optimization with the Level-Set Method
Figure 2 shows the electromagnetic model of the designed CPMLG, including the schematic Figure 2 shows the electromagnetic model of the designed CPMLG, including the schematic diagram, winding distribution diagram, and finite element meshing diagram. diagram, winding distribution diagram, and finite element meshing diagram. Permanent magnets
Primary winding
Shaft
Secondary part Primary part
Housing (a)
B B
b C
c c
CC
c A
a a
AA
a B
b b
(b)
(c) Figure 2. The electromagnetic modelofofthe thedesigned designed CPMLG. diagram; (b) Winding Figure 2. The electromagnetic model CPMLG.(a)(a)Schematic Schematic diagram; (b) Winding distribution diagram; (c) Finite element meshing diagram. distribution diagram; (c) Finite element meshing diagram.
1 presents main structureparameters parameters of of the the designed TableTable 1 presents thethe main structure designedCPMLG. CPMLG. Table 1. Main structure parameters of the generator.
Table 1. Main structure parameters of the generator. Structure Parameters
Structure Parameters Poles p Poles p τ (cm) Pole pitch PolePole-arc pitch τcoefficient (cm) ξ Pole-arc coefficient ξ Primary slots’ number Primary slots’ number Permanent magnet thickness d (cm) Permanent magnet thickness d (cm) Width of slot bs (cm) Width of slot bs (cm)
Value
Value8 8 8 8 0.8 0.8 9 9 2 2 4.3 4.3
Structure Parameters
Structure Primary yokeParameters height hj (cm) Primary (cm) Primaryyoke toothheight width bht j(cm) Primary tooth width b Armature effective lengtht l(cm) (cm) Armature effective Air gap widthlength δ (cm) l (cm) Air gap width δ (cm) Secondary core thickness d2 (cm) Secondary core thickness d2 (cm) Width of yoke wt (cm) Width of yoke wt (cm)
Value 3.3Value 2.37 3.3 1582.37 0.4 158
0.4 16 2.1 2.1 16
During the simulation procedure, the motion direction is set along the z-axis with the movement displacement being set to 128 cm. The solution step is set to be 0.0005 s. The optimization process is as follows: (1)
The objective air gap magnetic field is given by the mathematical analysis method. Appropriate measuring points are predetermined and the points fully reflect the distribution characteristics of the air gap magnetic field. A contour ring is chosen as the initial shape of the PM with the cross-sectional shape being a 64 mm* 10 mm rectangular. So, according to Figure 1 and Equation (11), the corresponding matrix of the zero level-set function is characterized by: The PM boundary value being 0, and the external value and the internal value are equal to 1 and −1, (0)
(0)
(0)
respectively. Then, choose another n + 1 variables Ki xi , yi along the permanent magnet edge evenly to represent the PM shape for optimization. Here, a curvature evolution scheme is
During the simulation procedure, the motion direction is set along the z-axis with the movement displacement being set to 128 cm. The solution step is set to be 0.0005 s. The optimization process is as follows: (1) The objective air gap magnetic field is given by the mathematical analysis method. Appropriate Energies 2018, 11, 3109 points are predetermined and the points fully reflect the distribution characteristics6 of 12 measuring
(2)
of the air gap magnetic field. A contour ring is chosen as the initial shape of the PM with the cross-sectional shape being a 64 mm*10 mm rectangular. So, according to Figure 1 and Equation (11), applied under the control lyapunov function (CLF)-condition of the Hamilton-Jacobi equation. the corresponding matrix of the zero level-set function is characterized by: The PM boundary Using Equation we external can get value φ andand Ki . the internal value are equal to 1 and −1, respectively. value being 0,(10), and the Then, the coordinates of K0 , K1 ... Kn are to 0 0 form the profile of the edges of the PMs in (0) used xi( ) , yi( ) along the permanent magnet edge evenly Then, choose another n + 1 variables Ki
(
)
ANSYS Maxwell. The magnetic induction density, B( xm , ym ), at the detection positions, D0 , D1 , ... represent shape for optimization. Here, a curvature evolution scheme , Dmto, as well asthe thePM corresponding x and y axis components of them, B and By (under xi , yi ), are x ( xi , yiis) applied the control lyapunov the Hamilton-Jacobi equation. UsingBEquation obtained. According to function Bx ( xi , yi(CLF)-condition target magnetization values, x ), By ( xi , yi ) andoftheir ( x0 i , yi ) and can get ϕ and K i. By0 ((10), xi , ywe , we can get the objective function as: i)
(2) Then, the coordinates of K0, K1 ... Kn are used to form the profile of the edges of the PMs in ANSYS " # Maxwell. The magneticminduction density, B ( xm , ym ) , at , 2D1 , ... , Dm , 2 the detection positions, D0
( Bx ( xi , yi ) − Bx0 ( xi , yi )) + By ( xi , yi ) − By0 ( xi , yi )
1
g x, y) = ∑ (12) as well as (the corresponding x and y axis components of them, Bx (2xi , yi ) and By ( xi , yi ) , are 2 m
( Bx0 ( xi , yi )) + By0 ( xi , yi )
i =1
obtained. According to Bx ( xi , yi ) , By ( xi , yi ) and their target magnetization values, Bx 0 ( xi , yi ) and
minas: [ g( x, y)] ≤ ε By 0 ( xi , yi ) , we can get the objective function (3)
(4)
(13)
In addition, g( j) ( x, y) is denoted as the jth objective2 function. In this paper, m2 = 33. It means that 1 m ( Bx ( xi , yi ) − Bx 0 ( xi , yi ) ) + ( By ( xi , yi ) − By 0 ( xi , yi ) ) the air gap region to one pole pair is divided into 32 segments. Figure 3 shows g ( x, ycorresponding = ) (12) the 2 2 m i = 1 B x , y + B x , y ( ) ( ) ( ) ( ) detection points setting of B. x0 i i y0 i i If the Equation (13) does not hold, then keep F and t unchanged, and repeat the above steps to g ( x, y ) (13) resolve φ and the corresponding Bx ( xmin i , yi ), By ( xi , yi ), and g ( x, y ) until Equation (13) holds. In this design example, ε( j )and F are set to be 3 and 0.2, respectively. There are five cases of iterations. N is (3) In addition, g ( x, y ) is denoted as the jth objective function . In this paper, m = 33. It means the iteration time and is set to be 100, 200, 300, 400, and 500, respectively. Figure 4 shows the that the air gap region corresponding to one pole pair is divided into 32 segments. Figure 3 shows evolution results of the edge of the permanent magnets under five different conditions. the detection points setting of B.
y
B0y(xi,yi)
B0(x,y)
B0x(xi,yi) B0(xi,yi)
4
2 B0(x2,y2)
4 B0(x3,y3)
4
2
4
4
2
4
4
2
Di(xi,yi)
D1 D2 D3
4
2
Dm(xm,ym) Dm-1(xm-1,ym-1)
B0(x1,y1)
3
3
3
B0(xm,ym) B0(xm-1,ym-1)
0
3
x 1
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Figure 3. Detection points setting of the B. Figure 3. Detection points setting of the B.
N = 100
N = 200
N = 300
N = 400
N = 500
(4) If the equation (13) does not hold, then keep F and t unchanged, and repeat the above steps to resolve ϕ and the corresponding
Bx ( xi , yi ) , By ( xi , yi ) ,
and g ( x, y ) until Equation (13)
holds. In this design example, and F are set to be 3 and 0.2, respectively. There are five cases of iterations. N is the iteration time and is set to be 100, 200, 300, 400, and 500, respectively. Figure 4 shows the evolution results of the edge of the permanent magnets under five different conditions. F= 0.2
Figure 4. 4. Evolution permanentmagnets. magnets. Figure Evolutionresults results of of permanent
4. Simulation Results and Analysis
4. Simulation Results and Analysis
Figures 5 and 6 shows thethe flux density Fouriertransform transform (FFT) decomposition Figures 5 and 6 shows flux densityand and its its fast fast Fourier (FFT) decomposition of theof the designed CPLMG without optimization method,respectively, respectively, when wave energy designed CPLMG without optimizationby bythe the level-set level-set method, when the the wave energy converter drives the the generator in in constant techniques[12]. [12]. converter drives generator constantspeed speedby by special special techniques In Figure 6, it shows total harmonic distortion(THD) (THD)of offlux flux density density in In Figure 6, it shows thatthat the the total harmonic distortion in the the air airgap gapfor for the the CPLMG is about 24.43% before optimization. CPLMG is about 24.43% before optimization. The optimization results for CPLMG are shown in Figure 7 and its FFT decomposition with level-set optimization is shown in Figure 8. The related THD of the harmonic content is lower 1.25
[tesla]
0.63 0
F = 0.2 F = 0.2 Figure 4. Evolution results of permanent magnets. Figure 4. Evolution results of permanent magnets.
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4. Simulation Results and Analysis Figures 5 and 6 shows the flux density and its fast Fourier transform (FFT) decomposition of the Figures 5 andwithout 6 showsoptimization the flux density and its fast method, Fourier transform (FFT) decomposition of the designed CPLMG by the level-set respectively, when the wave (100) ( x, y ) = (200) (energy thanconverter 24.43%.drives It is decreased to 17.9%. In addition, there are g 8.16, g x, y) = 7.47, designed CPLMG without optimization by the level-set method, respectively, when the wave energy the generator in constant speed by special techniques [12]. (400) ( x, yin (400) ( x, y ) [12]. drives the generator constant speed by special techniques g(300)converter y)Figure = 4.18, and g = 2.84. When N = 400, g ≤ 3 satisfies the requirement, so ( x,In ) 6, it shows that the total harmonic distortion (THD) of flux density in the air gap for In Figure 6, it shows that the total harmonic distortion (THD) of flux density in the air gap for the curve evolution stops. Of course, we can set the ε to be a lower value to get a better performance of the CPLMG is about 24.43% before optimization. the CPLMG is about 24.43% before optimization. the generator.
1.25 1.25
B [tesla] B [tesla]
0.63 0.63 0 0
−0.63 −0.63 −1.25 −1.25 0
200 200
100 100
0
300 300
Distance [mm] Distance [mm]
400 400
500 500
600 600
Figure 5. Flux density without optimization.
Figure withoutoptimization. optimization. Figure5.5.Flux Flux density density without
FFT analysis FFT analysis
Fundamental (10Hz) = 1.054 , THD = 24.43% Fundamental (10Hz) = 1.054 , THD = 24.43%
120 120
Mag (% of Fundamental) Mag (% of Fundamental)
100 100 80 80 60 60
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24.43%. It is decreased to 17.9%. In addition, there are
(
0
)
5 10 = 4.18, 0 5 , and 10
(
g (100) ( x, y ) = 8.16
)
(
)
,
g (200) ( x, y ) = 7.47
,
25 30 35 40g (400)45x, y 50 60 65 70 g g x, y20 = 2.84 3 55 N =35400, the requirement, 15 20 25. When 30 45 50 satisfies 55 60 65 70 Frequency (Hz)40 Frequency (Hz) so the curve evolution course, weofcan set the to be a lower value to get a better Figurestops. 6. FFT Of decomposition flux density without optimization. (300)
x, y0
(400) 15
Figure 6. 6. FFT decomposition fluxdensity densitywithout without optimization. Figure FFT decomposition of flux optimization. performance of the generator. The1.5 optimization results for CPLMG are shown in Figure 7 and its FFT decomposition with Theoptimization optimizationisresults CPLMG are shown Figure 7 and its FFT decomposition with —— Ideal level-set shown for in Figure 8. The related in THD of the harmonic content is lower than ——isNlower = 100 than level-set optimization is shown in Figure 8. The related THD of the harmonic content —— N = 200 —— N = 300 —— N = 400
B-AirGap [T]
1 0.5 0 −0.5
−1
−1.5
−80
−60
−40
−20
0
20
40
Distance [mm]
Figure 7.7.Flux withlevel-set level-setoptimization. optimization. Figure Fluxdensity density with FFT analysis
120
tal)
100
Fundamental (10Hz) = 1.039 , THD = 17.90%
60
80
−1.5
−80
−60
−40
−20
0
20
40
80
60
Distance [mm] Energies 2018, 11, 3109
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Figure 7. Flux density with level-set optimization.
FFT analysis
Fundamental (10Hz) = 1.039 , THD = 17.90%
120
Mag (% of Fundamental)
100 80 60 40 20 0 0
5
10
15
20
25
30 35 40 Frequency (Hz)
45
50
55
60
65
70
Figure 8. FFT decomposition of of flux density with level-set optimization. Figure 8. FFT decomposition flux density with level-set optimization.
Table 2 gives the detection values of B with the level-set optimization at m = 33. Figure 9 presents Table 2 gives the detection values of B with the level-set optimization at m = 33. Figure 9 presents the optimized permanent magnets model of the generator. the optimized permanent magnets model of the generator. Table 2. Detection values of Bx (xi ,yi ) and By (xi ,yi ) with F = 0.2 and N = 400. i
Di
Bx (xi ,yi )/T
By (xi ,yi )/T
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D20 D21 D22 D23 D24 D25 D26 D27 D28 D29 D30 D31 D32 D33
0 0.0042 0.0061 0.0089 0.0106 0.0276 0.0160 0.0279 0.0626 0.4018 0.0529 0.2064 0.0417 0.0101 0.0062 0.0034 −0.0011 −0.0799 −0.0084 −0.0075 −0.0074 −0.1429 −0.2201 −0.5251 −0.2306 −0.3441 −0.4879 −0.7347 −0.0795 −0.0100 −0.0050 −0.0034 −0.0006
0 0.1613 0.4108 0.8153 0.8039 0.6216 0.6447 1.0606 0.9632 0.8331 0.9581 0.9991 0.9743 0.7552 0.3950 0.1381 −0.0375 −0.1977 −0.2380 −0.4606 −0.8741 −0.9562 −0.9101 −0.7700 −0.9326 −0.9018 −0.8190 −0.6013 −0.9223 −0.7101 −0.3483 −0.0928 −0.0074
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19 D19 −0.0084 −0.2380 20 D20 −0.0075 −0.4606 21 D21 −0.0074 −0.8741 22 D22 −0.1429 −0.9562 23 D23 −0.2201 −0.9101 24 D24 −0.5251 −0.7700 25 D25 −0.2306 −0.9326 with 26 Figure Dmodel 26 −0.3441 −0.9018 Figure 9. 9. PM PM model optimized optimized with the the level-set level-set method. method. 27 D27 −0.4879 −0.8190 Figure 10 shows the three-phase back-EMFs of the CPMLG. In this condition, Figure 10 shows the CPMLG. In this−0.6013 condition, the the width width of of the the 28 the three-phase D28 back-EMFs of −0.7347 yoke soso that thethe unbalance of three-phase back-EMFs is reduced compared with yoke isisoptimized optimizedas29 as1.9 1.9cm cm that unbalance of three-phase back-EMFs is reduced compared D29 −0.0795 −0.9223 the corresponding resultsresults in [23]. with the corresponding in [23]. 30 D30 −0.0100 −0.7101 Further, energy converter with aa mass-adjustable float is used Further, the the wave wave energy converter with mass-adjustable float is used to to drive drive the the CPMLG CPMLG 3111 presents the D31amplitude response −0.0050 curve of the float −0.3483 directly [24]. Figure in the heaving motion directly [24]. Figure 11 presents the amplitude response curve of the float in the heaving motion 32shows the related D32 induced voltages −0.0034 with the wave −0.0928 direction. Figure 12 frequency being equal to direction. Figure 12 shows the related induced voltages with the wave frequency being equal to 0.2 Hz 33 D 33 −0.0006 −0.0074 0.2 Hz, respectively. Figure it can seen that similarmaximum maximumpeak peak power power and andHz 0.3and Hz,0.3 respectively. FromFrom Figure 12, it12,can be be seen that similar and captured energy per period in the steady state are achieved at different wave frequencies. captured energy per period in the steady state are achieved at different wave frequencies.
3
Table 2. Detection values of Bx(xi,yi) and By(xi,yi) with F = 0.2 and N = 400.
InduceVoltage [kV]
2 1 0 −1 −2
−3
0
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Di Bx(xi,yi)/T By(xi,yi)/T D1 0 0 D2 0.0042 0.1613 D3 0.0061 0.4108 D4 0.0089 0.8153 D5 0.0106 0.8039 D6 0.0276 0.6216 D7 0.0160 0.6447 D8 0.0279 1.0606 D9 0.0626 0.9632 D10 0.4018 300 0.8331 100 200 400 D11 0.0529 0.9581 Time [ms] D12 0.2064 0.9991 Figure 10. Back-EMF waveform of the CPMLG. D 13 0.0417 0.9743 Figure 10. Back-EMF waveform of the CPMLG. D14 0.0101 0.7552 D15 0.0062 0.3950 D16 0.0034 0.1381 D17 −0.0011 −0.0375 D18 −0.0799 −0.1977
500
−3
0
100
200
300
400
500
Time [ms] Energies 2018, 11, 3109
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Figure 10. Back-EMF waveform of the CPMLG.
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Figure 11. Amplitude of the float motion in the heaving heaving direction. direction.
3
—— Phase A —— Phase B —— Phase C
InduceVoltage [kV]
2 1 0 −1 −2 −3
0
1
2
3
4
5
Time [s]
(a)
3
—— Phase A —— Phase B —— Phase C
InduceVoltage [kV]
2 1 0 −1 −2
−3
0
1
2
3
4
5
Time [s]
(b) Figure Figure 12. Induced Induced voltages voltages of of CPMLG CPMLG directly directly driven by WEC. (a) 0.2Hz; (b) 0.3Hz.
5. Discussions Discussions 5. The results thethe level-set method applied to theto topology and sizeand design the generator The resultsshow showthat that level-set method applied the topology sizeofdesign of the is highly reliable in terms of the improvement of the magnetic field distortion. In the electromagnetic generator is highly reliable in terms of the improvement of the magnetic field distortion. In the electromagnetic model of the designed CPMLG, the unbalance of three-phase back-EMFs and the distortion of air-gap flux density will affect the efficiency and stability of the CPMLG. After setting the optimal air-gap magnetic field and selecting the appropriate detection point, topological evolution of the permanent magnet shape and yoke size are conducted based on the level set equation.
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model of the designed CPMLG, the unbalance of three-phase back-EMFs and the distortion of air-gap flux density will affect the efficiency and stability of the CPMLG. After setting the optimal air-gap magnetic field and selecting the appropriate detection point, topological evolution of the permanent magnet shape and yoke size are conducted based on the level set equation. In the paper, the optimization results for CPMLG are compared and discussed. Results show that the related THD of the flux density is decreased after evolution. When the number of iterations reaches 400, it satisfies the optimal condition, and the curve evolution stops. Taking all of these into consideration, it has been proven that the optimal CPMLG electromagnetic model with the set-level method is successful, reliable, and widely applicable. 6. Conclusions In the paper, to improve the efficiency and reliability of the CPMLG used in a wave energy conversion system, the level set curve evolution method was applied to the design process of the topology evolution of the PM and yoke. The process of topological evolution was analyzed and discussed in detail. The objective air gap magnetic field was set and an appropriate detection point was given. Based on the specific size of the linear generator, the ideal magnetic waveform was given by the mathematical analysis method. At the same time, air gap magnetic field detection points were selected in the appropriate place in the air gap region. The points fully reflect the distribution characteristics of the air gap magnetic field. Then, topological evolution of the permanent magnet shape was made based on the level set equation. The algorithm was programmed and computed iteratively. Additionally, the shape evolution computation for the PM and yoke could be obtained by using the discrete time and space variables. Finite element analysis of CPMLG was carried out to obtain the distribution of the air-gap magnetic field and the optimum yoke. The results show that the designed CPMLG can reduce harmonics of the flux density and the unbalance of three-phase back-EMFs. Thus, the adaptability of the generator in wave energy conversion was improved. It has been found that the designed CPMLG can be used for WEC at different wave conditions. Author Contributions: H.F. designed level set method and wrote the original manuscript; R.S. performed the numerical simulation and analyzed the numerical results; Z.X. visualized the results, revised the manuscript and polished the English. Acknowledgments: The work was supported by a grant from National Natural Science Foundation of China (No. 51577124, No.51877148). Conflicts of Interest: The authors declare no conflict of interest.
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