A New Design Optimization Method for Permanent ...

energies Article

A New Design Optimization Method for Permanent Magnet Synchronous Linear Motors Juncai Song, Fei Dong, Jiwen Zhao *, Siliang Lu, Le Li and Zhenbao Pan College of Electrical Engineering and Automation, Anhui University, Hefei 230601, China; [email protected] (J.S.); [email protected] (F.D.); [email protected] (S.L.); [email protected] (L.L.); [email protected] (Z.P.) * Correspondence: [email protected]; Tel.: +86-138-5517-6449 Academic Editor: Chunhua Liu Received: 28 July 2016; Accepted: 18 November 2016; Published: 25 November 2016

Abstract: This study focused on the design optimization of permanent magnet synchronous linear motors (PMSLM) that are applied in microsecond laser cutting machines. A new design optimization method was introduced to enhance PMSLM performances in terms of motor thrust, thrust ripple, and inductive electromotive force (EMF). Based on accurate 3D finite element analysis (3D-FEA), a multiple support vector machine (multi-SVM) was proposed to build a non-parametric quick calculation model by mapping the relation between multivariate structure parameters and multivariate operation performances. The gravity center neighborhood algorithm (GCNA) was also applied to search the global optimal combination of the structure parameters by locating the gravity center of the multi-SVM model. The superiority and validity of this method are verified by experiments. Keywords: permanent magnet linear synchronous motors (PMSLM); thrust; thrust ripple; 3D finite element analysis (3D-FEA); multiple support vector machine (multi-SVM); non-parametric quick calculation model; gravity center neighborhood algorithm (GCNA)

1. Introduction Permanent magnet synchronous linear motors (PMSLM) are widely used in high-precision microsecond laser cutting machines. Compared with the conventional solution-routing motors with ball screws, PMSLM has absolute predominance in terms of high acceleration, excellent accuracy, and direct drive [1]. The quantity of manufactured products is guaranteed by the remarkable operational performances of PMSLM. High acceleration and efficiency are significant to the operating quality of PMSLM. Thrust is one of the most important indices of PMSLM performance that can influence the acceleration of the mover, which is similar to torque in rotatory motors. Thrust ripple is becoming the largest problem that can result in a decreased accuracy of the mover position [2]; it is the fluctuation of stable thrust that can lead to scratching on the surface of products, increase roughness, and produce dimension errors. The main influencing elements of the thrust ripple are the structure parameters, control method, and load; all of the elements are gathered together nonlinearly. Two main schemes for the suppression of the thrust ripple are the design optimization of PMSLM and advanced control strategies, with the former being the dominant method [3]. The design optimization of PMSLM is a multi-objective problem due to the great number of design parameters, objectives, and constraints [4]. The most important issues in the design optimization are models and algorithms. Firstly, the commonly used optimization models include the analytic model [5,6], the finite-element model (FEM), the approximate models [7–9], but it is very difficult to apply the abovementioned optimization models due to the nonlinear and high dimensionality of PMSLM Energies 2016, 9, 992; doi:10.3390/en9120992

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dimensionality of PMSLM optimization. The major shortcoming of the analytic model and approximate model is the accuracy problem: given the essential assumptions, and the veracity of the objective functions, theseshortcoming models basedofon equivalent magnetic circuit willmodel result is inthe an accuracy optimization. The major thethe analytic model and approximate decrease given [10]. The FEM model is accurate, it has low efficiency, requires a significant problem: the essential assumptions, and but the veracity of the objective which functions, these models based computing cost. Another modified method based on the FEM model are robust design optimization on the equivalent magnetic circuit will result in an accuracy decrease [10]. The FEM model is accurate, methods [11,12], whichwhich are requires generallya significant employedcomputing for high-quality designs, including product but it has low efficiency, cost. Another modified method based robustness noise factors in optimization manufacturing and operational environments. most on the FEM against model are robust design methods [11,12], which are generally The employed representative six-sigma [13,14] and Taguchi methods [15–18] which use mathematical statistics for high-qualityare designs, including product robustness against noise factors in manufacturing and and design environments. of experiments, methods that can promote the efficiency models by using an operational The most representative are six-sigma [13,14] of andFEM Taguchi methods [15–18] orthogonal array, they are widely used in industry applications’ optimization. However, the which use mathematical statistics and design of experiments, methods that can promote the efficiency of shortcoming ofusing theseanmethods is that sample space based optimization. on different FEM models by orthogonal array,only theyanalyzing are widelydiscrete used in industry applications’ combinations of motor structure parameters may in an approximate local optimum for the However, the shortcoming of these methods is thatresult only analyzing discrete sample space based on motor optimization design. Another important issue is in the design optimization algorithms. different combinations of motor structure parameters may result in an approximate local optimum for Nowadays, many modern intelligent optimization suchdesign as genetic algorithms (GA) [7] the motor optimization design. Another importantalgorithms, issue is in the optimization algorithms. and particlemany swarm optimization algorithms [19],algorithms, are becoming popular as they(GA) can handle Nowadays, modern intelligent optimization such very as genetic algorithms [7] and problems with strong nonlinearity, but the search ability and rapidity need be enhanced. particle swarm optimization algorithms [19],global are becoming very popular as they cantohandle problems of the above-mentioned methods areability inappropriate in terms and reliability in with All strong nonlinearity, but the global search and rapidity needof toaccuracy be enhanced. PMSLM design optimization. This presentare study proposes ain new design optimization method for All of the above-mentioned methods inappropriate terms of accuracy and reliability in PMSLM. First, a 3D-FEA modelThis is built to obtain theproposes accurate aoperation performances of PMSLM PMSLM design optimization. present study new design optimization methodand for the sample data space based onisdifferent structural parameters. Second, a multiple of support vector PMSLM. First, a 3D-FEA model built to obtain the accurate operation performances PMSLM and machine (multi-SVM) is introduced to map the relation between the space of the multivariate the sample data space based on different structural parameters. Second, a multiple support vector structure parameters is and the multivariate operational performance. It of is the a new machine structure learning machine (multi-SVM) introduced to map the relation between the space multivariate method for regression which have been effectively applied in function estimation [20], fault diagnosis parameters and the multivariate operational performance. It is a new machine learning method for [21], and data mining this study, the space of the discrete data based 3D-FEA[21], is translated regression which have [22]. been In effectively applied in function estimation [20], faulton diagnosis and data into a continuous space by multi-SVM regression, which can provide a non-parametric quick mining [22]. In thisdata study, the space of the discrete data based on 3D-FEA is translated into a continuous calculation model for follow-up optimization. Finally, the gravity center neighborhood algorithm data space by multi-SVM regression, which can provide a non-parametric quick calculation model for (GCNA) is presented, andthe thegravity “filling function” [24] is introduced to avoid[23] theispremature follow-up[23] optimization. Finally, center neighborhood algorithm (GCNA) presented, convergence and local optimal GCNA. The combination of the structure and the “filling function” [24] isofintroduced toglobal avoid optimal the premature convergence and localparameters optimal of is realized convergence of theofiteration. The parameters superiority is and validity of this method are GCNA. Theupon globalthe optimal combination the structure realized upon the convergence confirmed by the 3D-FEA and experimental test.of this method are confirmed by the 3D-FEA and of the iteration. The superiority and validity The paper is organized as follows: Section 2 introduces the characteristics of PMSLM; Section 3 experimental test. introduces the used Taguchi optimization method; Section 4the proposes the newof design optimization The paper is organized as follows: Section 2 introduces characteristics PMSLM; Section 3 method forthe PMSLM; Sectionoptimization 5 verifies and discusses the superiority of the introduces used Taguchi method; Section 4 proposes the and new practicability design optimization proposed method by experimental results; Section 6 draws the conclusions; and Section 7 discusses method for PMSLM; Section 5 verifies and discusses the superiority and practicability of the proposed future methodwork. by experimental results; Section 6 draws the conclusions; and Section 7 discusses future work. 2. Characteristic Presentation design model modelof ofthe thePMLSM PMLSMisisshown shownininFigure Figure1.1.The The main mechanical structure The initial design main mechanical structure of of this PMSLM was made of back-iron, coils magnets. motor three-phase, seven-pole, this PMSLM was made of back-iron, coils andand magnets. ThisThis motor has has three-phase, seven-pole, sixsix-coils X,Y, B, C, Y, C, and and a double secondary.The Themachine machinedesign designstructure structure is is the the “U”-model “U”-model coils (A, (A, X, B, and Z),Z), and a double secondary. structure. The number of coils is six, and three phases are “Y” connected. connected.

Figure Figure 1. 1. Permanent Permanent magnet magnet synchronous synchronous linear linear motors motors (PMSLM) (PMSLM) with with aa double double secondary. secondary.

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The layer analysis model of the PMSLM is shown in Figure 2, and there are three regions: region I are the The coils, region II aremodel the permanent magnets, and in region III2,2,isand thethere back-iron. This PMSLM design The layer analysis model thePMSLM PMSLM is shown are regions: region layer analysis ofofthe is in Figure Figure and there arethree three regions: region I are the coils, region II are the permanent magnets, and region III is the back-iron. This PMSLM I are the region II are permanent magnets, and region III is the This PMSLM implies the coils, considerations of the six main variable structural parameters: theback-iron. magnet length (τ m ), the design implies the considerations six main variable variable structural (τm(l), ), mand design implies considerations ofofsix main structural parameters: themagnet magnet length (τ ), magnet height (h),the the air gap between magnets and coils (δ), theparameters: pole pitch the (τ), the coillength length the magnet height (h), the air gap between magnets and coils (δ), the pole pitch (τ), the coil length (l), magnet (h), the airthese, gap between magnets (δ), the pole pitch (τ), the (l), thethe coil widthheight (d). Aside from the materials ofand the coils PM and back-iron should alsocoil belength considered. Energies 2016, 9,width 992 3 also of 15 be and the coil (d).Aside Asidefrom from these, these, the the materials materials of the PM and back-iron should theconsist coil width (d). theT, PM should also be Theand PMs of NdFe material with a remanence (Br ) ofof1.43 andand the back-iron stator consists of M20 steel considered. The PMs consist of NdFe material with a remanence (B r) of 1.43 T, and the stator consists considered. The PMs consist of NdFe material a remanence (B r) of 1.43 T, and the stator consists layer analysis model of thewidth, PMSLM iswith shown in Figureare 2, and there regions: silicon. TheThe magnet length, magnet and coil width fixed at are 40,three 15, and 40 region mm, and the of M20 steel silicon. Themagnet magnetlength, length,magnet magnet width, width, and coil width are fixed atat 40, 15,15, and 4040 mm, of M20 steel silicon. The and coil width are fixed 40, and mm, I are theare coils, region II are the the permanentparameters magnets, and region III is the This PMSLM turns per coil Given that are regarded asback-iron. fixed different and the turns 180. per coil are 180. Givenother that the other parameters are regarded as and fixedknown, and known, and the turns per coil are 180. Given that the other parameters are regarded as fixed and known, design implies the considerations of six main variable structural parameters: the magnet length (τ m ), combinations structure parameters result in different output performances of PMSLM. The different of combinations of structurecan parameters can result in different output performances of best the magnet height (h), of the structure air gap between magnetscan and coils (δ),in thedifferent pole pitch output (τ), the coil length (l), different combinations parameters result performances of performance are determined via the global optimization of the motor structure. PMSLM. The best performance are determined via the global optimization of the motor structure. and the (d). Asideare from these, the via materials of the PM and back-iron be PMSLM. Thecoil bestwidth performance determined the global optimization of the should motor also structure. considered. The PMs consist of NdFe material with a remanence (Br) of 1.43 T, and the stator consists of M20 steel silicon. The magnet length, magnet width, and coil width are fixed at 40, 15, and 40 mm, and the turns per coil are 180. Given that the other parameters are regarded as fixed and known, different combinations of structure parameters can result in different output performances of PMSLM. The best performance are determined via the global optimization of the motor structure. Figure 2. Layer analysis model of the PMSLM.

Figure 2. Layer analysis model of the PMSLM. Figure 2. Layer analysis model of the PMSLM. 3. Taguchi Optimization Method

3. Taguchi Optimization 3. Taguchi OptimizationMethod Method The Taguchi method is a multi-objective optimization method using the analysis of means The Taguchi method isisa3amulti-objective optimization method using theanalysis analysis means (ANOM) [10,15–18]. Figure shows the comprehensive framework of the Taguchi method: the first The Taguchi method multi-objective optimization method using the ofof means Figure 2. Layer analysis model of the PMSLM. (ANOM) Figure shows the comprehensive framework theTaguchi Taguchi method: first step [10,15–18]. is [10,15–18]. building the FEM33model PMSLM and analyzing the performances data, followed by (ANOM) Figure showsof the comprehensive framework ofofthe method: thethe first applying the Taguchi optimization method, which includes design of experiments (DOE) and step is building the FEM model of PMSLM and analyzing the performances data, followed by applying 3. Taguchi Optimization Method step is building the FEM model of PMSLM and analyzing the performances data, followed by ANOM.optimization Finally, according to the different sensitivities thedesign different factors,and select the best theapplying Taguchi method, which includes design ofofmethod experiments (DOE) ANOM. Finally, the Taguchi optimization method, which includes of the experiments (DOE) and The Taguchi method is a multi-objective optimization using analysis of means combination of motor structure parameters. All of these are shown as follows. according to the[10,15–18]. different sensitivities of comprehensive the different factors, select best combination of best motor ANOM. Finally, according to3 shows the different sensitivities of the factors, select the (ANOM) Figure the framework ofdifferent thethe Taguchi method: the first structure parameters. All of these areofshown as and follows. combination of motor structure parameters. All of these are shown as follows.data, followed by step is building the FEM model PMSLM analyzing the performances applying the Taguchi optimization method, which includes design of experiments (DOE) and ANOM. Finally, according to the different sensitivities of the different factors, select the best combination of motor structure parameters. All of these are shown as follows.

Figure 3. Framework of Taguchi optimization method.

3.1. Establishment of Orthogonal Figure Array 3. Framework of Taguchi optimization method. Figure 3. Framework of Taguchi optimization method. A finite-element parametric of aofPMSLM was built.method. After the FEM mesh generation, Figure 3. model Framework Taguchi optimization 3.1.shown Establishment of4,Orthogonal Array in Figure all of the performances of the motor in the matrix experiments were analytically 3.1. Establishment of Orthogonal Array 3.1. Establishment of Orthogonal Array demonstrated in [25]. A finite-element parametric model of a PMSLM was built. After the FEM mesh generation,

A finite-element parametric model of a PMSLM wasbuilt. built. After the FEM generation, mesh generation, finite-element model ofofa the PMSLM was After experiments the FEM meshwere shown inAFigure 4, all of parametric the performances motor in the matrix analytically shown in Figure 4, all of the performances of the motor in the matrix experiments were analytically shown in Figure 4, all of the performances of the motor in the matrix experiments were analytically demonstrated in [25]. demonstrated in [25]. demonstrated in [25].

Figure 4. 3D-FEM model of a PMSLM.

Figure 3D-FEM modelofof a PMSLM. Figure4. 3D-FEM model Figure 4.4.3D-FEM model ofa aPMSLM. PMSLM.

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According to design principles of PMSLM four parameters, including height of magnets (h), air gap (δ), pole pitch (τ), and coil length (l), were selected as variable parameters. In Table 1, the value range of h is from 2.0 to 3.5 mm, δ is from 1.6 to 2.2 mm, τ is from 17 to 20 mm, and l is from 6.0 to 7.5 mm. All of the value ranges are generally satisfied with the requirements of microsecond laser cutting machines, namely, thrust should be no less than 35 N, and thrust ripple and harmonic contents should be the lowest. Table 1. Four main structural variables of the motor. No.

Parameters

Description

Value Range

1 2 3 4

h δ τ l

height of magnets air gap pole pitch coil length

2.0–3.5 mm 1.6–2.2 mm 17–20 mm 6.0–7.5 mm

The Taguchi method is known as the design of experiments using an orthogonal array to screen the experimental conditions and means [26]. In Table 2, three levels of four undetermined factors: h, δ, τ, and l were shown clearly as the level step of each factor is 0.75 mm, 0.3 mm, 1.5 mm, and 0.75 mm, respectively. As shown in Table 3, a Taguchi orthogonal array L9 (34 ) is then established for numerical experiments. In this Table, the nine-group FEA simulation experiments were conducted, all of the motor’s performances, such as F (average thrust), η (thrust ripple), and THD (harmonic distortion rate), were analyzed. Table 2. Levels of the main factor variables. Factor Variables

Level 1

Level 2

Level 3

A: h (mm) B: δ (mm) C: τ (mm) D: l (mm)

2.0 1.6 17 6.0

2.75 1.9 18.5 6.75

3.5 2.2 20 7.5

Table 3. Experimental arrays and results of finite element analysis (FEA). No. 1 2 3 4 5 6 7 8 9

Levels of Each Factor

Performances

A

B

C

D

F (N)

η (%)

THD(%)

2.0 2.0 2.0 2.75 2.75 2.75 3.5 3.5 3.5

1.6 1.9 2.2 1.6 1.9 2.2 1.6 1.9 2.2

17 18.5 20 18.5 20 17 20 17 18.5

6.0 6.75 7.5 6.75 6.0 7.5 6.75 7.5 6.0

36.084 34.143 31.102 40.797 37.514 38.168 45.109 43.083 40.315

10.804 10.796 12.273 10.542 8.342 9.973 9.135 10.161 12.897

5.214 5.517 6.425 3.333 2.243 2.155 2.067 6.031 4.153

3.2. Analysis of Mean Value The Taguchi optimization method uses the statistical mean made by orthogonal arrays and analysis results of FEA. The average values of each motor performance (F, η, and THD) are shown in Table 4: Table 4. Average values of each performance. Performances

F (N)

η (%)

THD (%)

Average Values

38.479

10.547

4.126

The average value of different performances at different levels are calculated using Equation (1):

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1 [ Px ( l1 ) + Px ( l 2 ) + Px ( l3 )] 13

G xi ( Px ) =

(1)

(1) Gxi ( Px ) = [ Px (l1 ) + Px (l2 ) + Px (l3 )] x represents the different factors: A, B,3C, D; i represents the levels of factors: 1, 2, 3; Px represents the motor performances: F (N),factors: η (%), THD i(Px) represents the average values of performances x represents the different A, B, (%); C, D;Gx i represents the levels of factors: 1, 2, 3; Px represents under x factor at i level; and Pxη(li(%), ) represents of performances under x factor at i level as the motor performances: F (N), THD (%);the Gxvalues i (Px ) represents the average values of performances shown Tableat3.i level; and P (l ) represents the values of performances under x factor at i level as under xinfactor x i Forinexample, shown Table 3. the average value of F under factor C(τ) at level 3 is calculated as Equation (2): For example, the average value of1 F under factor C(τ) at level 3 is calculated as Equation (2): GC 3 ( F ) = [ F (3) + F (5) + F (7)] 3 (2) GC3 ( F ) = 131 [ F (3) + F (5) + F (7)] (2) 1[31.102 + 37.514 + 45.109] = 37.908 = = 33 [31.102 + 37.514 + 45.109] = 37.908

All of the average values of performances under each level of factor variable index are obtained as shown in Table 5. Table 5. 5. Average Average values Table values of of performance performance under under each each level level of of factor factor variable variable index. index.

ofFactor Each Factor Variables Variables Level Level of Each 1 1 A 2 2 A 3 3 1 1 2 B 2 B 3 3 1 1 C 2 2 C 3 3 1 1 2 D 2 D 3 3

F (N) F (N) 33.776 33.776 38.826 38.826 42.836 42.836 40.663 40.663 38.247 38.247 36.528 36.528 39.112 39.112 38.418 38.418 37.908 37.908 37.971 37.971 40.016 40.016 37.451 37.451

η (%) η (%)THD (%) THD (%) 11.291 5.179 11.291 5.179 9.619 2.577 9.619 2.577 10.731 10.731 4.084 4.084 10.160 3.538 10.160 3.538 9.766 4.597 9.766 4.597 11.714 4.244 11.714 4.244 10.313 4.467 10.313 4.334 4.467 11.412 11.412 4.334 9.917 3.578 9.917 3.578 10.681 3.870 10.681 3.639 3.870 10.158 10.158 3.639 10.802 4.870 10.802 4.870

According to Table 5, the influence of each factor on F, η, and THD are as shown in Figure 5. It According to Table the influence(A3, of each factor onD2) F, η,contribute and THDtoare asmaximization shown in Figure 5. is noted in Figure 5 that,5,combination B1, C1, and the of F, It is noted in Figure that, combination (A3,toB1, and D2) contribute to the maximization F, combinations (A2, B2,5 C3, and D2) contribute theC1, minimization of η, combination (A2, B1, C3, of and combinations (A2, B2, C3, and D2) contribute to the minimization of η, combination (A2, B1, C3, and D2) contribute to the minimization of THD. However, no elements are obviously selected to D2) contribute to the minimization of THD. design However, no elementsF,are obviously selected constitute the combination of the optimum for maximum minimum η, and THDtoatconstitute the same the combination of the optimum design for maximum F, minimum η, and THD at the same time. time.

Figure Figure 5. 5. Main Main factor factor effects effects on on motor motor performances: performances: (a) (a) F F (N); (N); (b) (b) ηη (%); (%); and and (c) (c) THD THD (%). (%).

3.3. Proportions of Influences Produced by Each Factor’s Different Levels on Motor Performances 3.3. Proportions of Influences Produced by Each Factor’s Different Levels on Motor Performances The sum of error squares (SS) is a measure of the deviation of the experimental data from the The sum of error squares (SS) is a measure of the deviation of the experimental data from the average value of data, SS generated by various factors and different levels can be calculated as average value of data, SS generated by various factors and different levels can be calculated as Equation Equation (3). All of the results of SS and proportion are reported in Table 6. (3). All of the results of SS and proportion are reported in Table 6.

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3

SS = 3 ∑

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3

∑ [ Mxi ( Pxi )− M( Pj )]2

(3)

i =13 j=31

SS = 3 [ M xi ( Pxi ) −M ( Pj )]2 (3) =1 i =1 i jrepresents x represents the different factors: A, B, C, D; the levels of factors: 1, 2, 3; P represents the motor performances: F (N),factors: η (%), A, THD (%); performances i (Pxi ) represents x represents the different B, C, D; Mx i represents the levelsthe of average factors: 1,values 2, 3; P of represents the under x factor at i level as shown in Table 5; and M(P ) represents the average values of performance j motor performances: F (N), η (%), THD (%); Mxi(Pxi) represents the average values of performances as under shownxin Table factor at 4. i level as shown in Table 5; and M(Pj) represents the average values of performance as shown Table in Table 4. 6. Proportion of influences produced by various factors on motor performance. Table 6. Proportion of influences produced by various factors on motor performance. Various Factors

Various Factors A BA CB DC Total

D Total

F

SS

SS 123.67 123.67 25.89 2.19 25.89 11.03 2.19 162.78

11.03 162.78

F Proportion (%) Proportion (%) 76.0 76.0 15.9 1.3 15.9 6.8 1.3 100 6.8 100

THD

η

SS

SS 4.35 4.35 6.36 3.60 6.36 0.70 3.60 15.01

0.70 15.01

η Proportion (%) Proportion 29.0 (%) 29.0 42.3 24.0 42.3 4.7 24.0 100 4.7 100

THD SS Proportion (%) SS 10.53 Proportion 64.9 (%) 10.53 64.9 1.74 10.7 1.38 8.6 1.74 10.7 2.57 15.8 1.38 8.6 16.22 100 2.57 15.8 16.22 100

As shown in Table 6: (1) F is most sensitive to A, as B has a relatively weak effect on F; (2) η As shown 6: (1) is most to factor A, as Bas has relatively weakaeffect on F; (2) ηon is η; is sensitive to A, in B,Table and C, B isF the mostsensitive sensitive A,a C have nearly similar effect to A,effected B, and C, is as theDmost factor as A, have nearly a similar effect on η; (3) (3)sensitive THD is most byBA, has asensitive moderate effect. To C achieve this multiple-optimization goal, THD is most effected by A, as D has a moderate effect. To achieve this multiple-optimization goal, according to Figure 5 and Table 6, the best combination of factors is selected to be (A2, B2, C3, and D2). tothe Figure 5 andperformances Table 6, the best combination of factors is selected be (A2,N, B2,η C3, and In according conclusion, PMSLM based on the Taguchi method are: F to is 40.241 is 6.342%, D2). In conclusion, the PMSLM performances based on the Taguchi method are: F is 40.241 N, η is and the THD is 2.66%. 6.342%, and the THD is 2.66%. 4. Proposed New Optimization Method 4. Proposed New Optimization Method In Section 3, the Taguchi method was conducted, which had some effect on PMSLM optimization. In Section 3, the Taguchi method was conducted, which had some effect on PMSLM The optimization results are F is 40.241 N, η is 6.342%, and the THD is 2.66%. However, it had one optimization. The optimization results are F is 40.241 N, η is 6.342%, and the THD is 2.66%. However, relatively serious shortcoming: in the optimization process, the four parameters and three levels it had one relatively serious shortcoming: in the optimization process, the four parameters and three can result in 34 = 81 combinations. This method based on FEA analysis, in which only nine groups levels can result in 34 = 81 combinations. This method based on FEA analysis, in which only nine of parameter combinations are analyzed, can promote the efficiency by using ANOM. However, groups of parameter combinations are analyzed, can promote the efficiency by using ANOM. given the one-sidedness produced by the discrete sample space of FEA, this method can only obtain However, given the one-sidedness produced by the discrete sample space of FEA, this method can theonly bestobtain optimization results fromresults 81 discrete not from global optimum of the the best optimization from 81combinations, discrete combinations, notthe from the global optimum parameter continuous value range. This method results in aninapproximate local optimum of the parameter continuous value range. This method results an approximate local optimumfor forthe motor optimization design. the motor optimization design. This study method to toovercome overcomethe theshortcomings shortcomings This studyproposes proposesa anew newdesign designoptimization optimization method of of thethe Taguchi method. AA flowchart 6. Taguchi method. flowchartofofthis thismethod methodisisshown shown in Figure 6.

Figure6.6.Flowchart Flowchart of of the the proposed proposed optimization Figure optimizationmethod. method.

In this method, the 3D-FEA model is developed to obtain the accurate motor output In this method, the 3D-FEA model is developed to obtain the accurate motor output performances performances after the definition of the design variables, constraints, and objectives. A total of 256 after the definition of structure the design variables,are constraints, and objectives. A total 256 combinations combinations of the parameters analyzed by 3D-FEA to prepare forofthe following multi- of

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the structure parameters are analyzed by 3D-FEA to prepare for the following multi-SVM modeling. Multi-SVM is then introduced to build a quick non-parametric calculation model based on 3D-FEA analysis data. After regression, the multi-SVM motor model translates the discrete motor data space into continuous data space, which can avoid local one-sided optimization. Finally, the gravity center neighborhood algorithm (GCNA) is presented to optimize the multi-SVM motor model by locating the gravity center of the objective function. The global optimal combination of the structure parameters is obtained upon convergence of the iteration. 4.1. 3D-FEA for Modeling Data Gaining The value range of four factors are determined by performance requirements of microsecond laser cutting machines as shown in Section 3. Table 7 shows the value ranges and the incremental step for these parameters: h is from 2.0 to 3.5 mm, δ is from 1.6 to 2.2 mm, τ is from 17 to 20 mm, and l is from 6.0 to 7.5 mm as the incremental level step of each factor is 0.5 mm, 0.2 mm, 1.0 mm, and 0.5 mm, respectively. Table 7. Levels of the structural parameters. Variables

Level 1

Level 2

Level 3

Level 4

h δ τ l

2.0 mm 1.6 mm 17 mm 6.0 mm

2.5 mm 1.8 mm 18 mm 6.5 mm

3.0 mm 2.0 mm 19 mm 7.0 mm

3.5 mm 2.2 mm 20 mm 7.5 mm

According to Table 7, four different factors and four levels of each factor result in 44 = 256 different combinations. All of the 256 groups’ data of the performances of PMSLM after the calculation of 3D-FEA is shown in Table 8. The data shown in Table 8 conceal the relationship between the different structural parameters and motor output performance. These data are prepared for following multi-SVM, exploring the relation between input and output variables. Table 8. Sample data space based on 3D-FEA. Structure Parameter Variables

Output Performances

No.

h

δ

τ

l

F (N)

η (%)

THD (%)

1 2 3 4 5 ... ... 254 255 256

2.0 2.0 2.0 2.0 2.0 ... ... 3.5 3.5 3.5

1.6 1.6 1.6 1.6 1.6 ... ... 2.2 2.2 2.2

17 17 17 17 18 ... ... 20 20 20

6.0 6.5 7.0 7.5 6.0 ... ... 6.5 7.0 7.5

36.084 35.862 36.591 36.289 36.335 ... ... 37.432 39.484 39.118

10.804 10.817 11.207 11.159 10.791 ... ... 9.381 9.385 9.316

5.214 5.127 6.517 6.425 6.333 ... ... 4.89 5.12 5.67

4.2. Multiple SVM for PMSLM Exploring the objective function—F(x), which can calculate any output value y for any input x, is the essence of SVM [27–31]. The multi-SVM is the modified SVM algorithm, which contains the multidimensional vector output y. All of the structural parameters can be seen as the input parameters compared to all of the performances, which can be seen as the output parameters, by training the input parameters and output performances shown in Table 7. The relations between the multidimensional inputs and outputs are obtained, and the output performances of any other input is subjected to regression calculation using the non-parametric relation between the inputs and outputs of the training data. The core steps of multi-SVM are as follows:

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Step 1: Single SVM modeling. By training the input parameters and calculations for the output function F(x), different SVM models (i.e., thrust SVM and THD SVM) are established. The kernel q d function—K(x, z) ofKthe (4) ( x, single z) = ρSVM x, zmodel − ρis selected x − y +as1,follows: ρ + ρ = 1, q = 1, d = 2 1

(

)

2

1

2

K ( x, z)of=dimension, ρ1 ( x, z)d − qρ2isk xconstant − ykq + number, 1, ρ1 + ρand q = 1, d = 2 (4) 2 =ߩ1, where d is the number ଵ and ߩଶ are the proportions of different kernels. where d is2:the number dimension, q is constant number, and ρ1 of and ρ2 are the of Step Fusion of theofsingle SVM models. Following the principle structure riskproportions minimum, the different kernels. regression problem is translated into a constrained optimization problem after the best W and B in Step 2: of the single SVM models. Following the principle of structure risk minimum, the Equation (5)Fusion are searched with the fusion algorithm: regression problem is translated into a constrained optimization problem after the best W and B in 2 L Equation (5) are searched with 1 N the fusion algorithm: (5) wi + C Q(hi ), hi = ei , ei = yi − (W , φ ( xi )) + B min R(W ) =



2 Ni =1

 i =1

2

L 1 k w k + C h i = k ei k, C ei is = the yi −penalty (W, φ( xiparameter, )) + B (5) minR ( W ) = i of output i ), ∑ Q(hperformances, where N is the dimension2 ∑ number hi is the i =1 i =1 empirical error and φ(xi) is the nonlinear mapping function. where N is the dimension of output C is the parameter, hi is the Aimed at the function number fitting problem of performances, N dimension inputs andpenalty M dimension outputs, N M ∈ ∈ empirical error and φ(x ) is the nonlinear mapping function. training sample is set asi {(xi, yi), i = 1, 2, 3…, L, xi R , yi R }. The multi-SVM objective function Aimed at theasfunction inputs and outputs, F(x) is expressed follows,fitting whereproblem K (xi, x)ofis N thedimension kernel function, bi is M thedimension constant, and wj is the the N , y ∈ RM }. The multi-SVM objective function training sample is set as {(x , y ), i = 1, 2, 3 . . . , L, x ∈ R regression coefficient: i i i i F(x) is expressed as follows, where K (xi , x) is lthe kernel function, bi is the constant, and wj is the regression coefficient: y = f ( x) = (∂*i −∂)K ( xi , x) + b (6) l i =1

y = f (x) =

∑ (∂i∗ − ∂)K(xi , x) + b

i =1

(6)

l

* K ( xi , x) = [φ ( xi ) ∗ φ ( x)], w j =  l (∂ i − ∂ ) ∗ φ ( xi ) i =1 j K ( xi , x ) = [φ( xi ) ∗ φ( x )], w = ∑ (∂i∗ − ∂) ∗ φ( xi )

i =1

 + b1 1+ (w2 ∗φ2 ( x))1 +b2   w ( x)  1 (w1 ∗φ1 ( x))   f1 1 , w2  1 (w f1 (x)   w  11 , w21  "φ1 ( x)  # 1 ∗φ1 ( x )) + b1 + ( w2 ∗ φ2 ( x )) + b2   F (.. x) =    =   =  ..  *  φ1 ( x) + B = W *φ ( X ) + B, .. F(x) =  + B = W ∗ φ( X ) + B, = =  ∗ φ ( x) . . . ) n n n n n n   2φ2 ( x     ( ) f x n n n n n n ∗ + + ∗ + ( φ ( )) ( φ ( )) , w x b w x b w w n  1 ∗φ1 (1x)) + (w f n (x)  w 1 b1 + ( w 1 2 ∗ φ22( x ))2+ b2 2  1 ,1w2 2  1



1

1

1

1

(7) (7)

1

B = [b1 , b2 ..., bn ]

B = [b1 , b2 . . . , bn ]

(8) (8)

After 100 groups of input parameters data are imported to the multi-SVM model, 100 groups of output parameter multi-SVM areare compared with the original output parameter data. parameterdata datapredicted predictedbyby multi-SVM compared with the original output parameter Figure 7 shows that the of theofmulti-SVM motor model can reach 95.7% or even moremore and data. Figure 7 shows thataccuracy the accuracy the multi-SVM motor model can reach 95.7% or even that the error rate can be limited to 1.2%. After the regression fitting of multi-SVM. The discrete data and that the error rate can be limited to 1.2%. After the regression fitting of multi-SVM. The discrete space by 3D-FEA is translated into a continuous data space, which canwhich provide a quick calculation data space by 3D-FEA is translated into a continuous data space, can provide a quick model for the succeeding GCNA optimization. calculation model for the succeeding GCNA optimization.

Figure 7. Error rate of multi-SVM: (a) error rate of thrust (%); and (b) error rate of THD (%). Figure 7. Error rate of multi-SVM: (a) error rate of thrust (%); and (b) error rate of THD (%).

4.3. Gravity Center Neighborhood Algorithm 4.3. Gravity Center Neighborhood Algorithm Gravity center can be seen as the force balance point that satisfies the “level principle” [14]. The Gravity center can be seen as the force balance point that satisfies the “level principle” [14]. thin plank model-f(x) is shown in Figure 8, where the points B, C, and D are the top of three bulges. The thin plank model-f (x) is shown in Figure 8, where the points B, C, and D are the top of three bulges. The gravity center of the thin plank is determined by the object-hanging method. The gravitational The gravity center of the thin plank is determined by the object-hanging method. The gravitational lines la and le are obtained by choosing points A and E to hang up the plank, and the intersection O is the gravity center of this plank.

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lines la and le are obtained by choosing points A and E to hang up the plank, and the intersection O is the gravity center Energies 2016, 9, 992 of this plank. 9 of 15 Energies 2016, 9, 992

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Figure object-hanging. Figure 8. 8. Determination Determination of of the the gravity gravity center center by by object-hanging. Figure 8. Determination of the gravity center by object-hanging.

Evidently, points B and D are the extrema of f(x), given that point C is the global maximum of points B and D are the extrema of f (x), given that point C is the global maximum of Evidently, f(x). An interesting phenomenon is that the gravity f(x), center O is very close to the highest point C along f (x). An Aninteresting interesting phenomenon phenomenon is that the the gravity gravity center O is very close to the highest point C along f(x). the X-axis. This phenomenon canisbethat explained by the knowledge of the moment equilibrium because the X-axis. This This phenomenon can be explained explained by the the knowledge knowledge of the the moment moment equilibrium because the moments onphenomenon both sides of the gravitational lines are equal. C must lie on theequilibrium left by la and on the moments on on bothsides sides ofthe the gravitationallines linesare areequal. equal. C must on left la and on the moments lie lie onclose thethe left by lby a and onofthe right by le, whichboth can resultof in thegravitational location of the intersection OCatmust the very neighborhood C the right by l , which can result in the location of the intersection O at the very close neighborhood e right which Assuming can result in thethe location the intersection O at the very close neighborhood C alongby thele,X-axis. that plank of model f(x) can be expressed by the equations as f(x),ofthe of C along the X-axis. Assuming that the model plank model can be expressed by the equations as along the X-axis. Assuming plank f(x) canf (x) be by the equations aslines f(x), the optimization problem is tothat findthethe global maximum of expressed f(x). The two gravitational are f (x), the optimization problem is tothe findglobal the global maximum of f (x). The twogravitational gravitationallines lines are optimization is tooffind maximum of f(x). The two determined byproblem the location the gravity center. CF and CG are the shortest distances from C to la determined by the location of the gravity center. CF and CG are the shortest distances from C toCla to and by the location of the gravity center. CF and CG are the shortest distances from l and le. The neighborhood area of the global maximum can be expressed as [min (xf, xg), max (xf, xg)],a le . The neighborhood areaarea of the maximum can be expressed as [min (xf , xg(x ),f,max (xf , xg(x)], and and e. The neighborhood ofglobal the global maximum can be expressed as [min xg), max f, x g)], and lthe neighborhood area decreases gradually by several times of the iterative computations. The the neighborhood area decreases gradually by several times times of the of iterative computations. The global and the neighborhood area decreases gradually by several the iterative computations. The global maximum is then obtained easily using the gravity center location method. maximum is then is obtained easily using the gravity center location method. global then obtained easilyand using the gravity center Tomaximum avoid premature convergence a local optimum, the location GCNA ismethod. introduced. Its operating avoid premature convergence and a local optimum, optimum, the GCNA is introduced. introduced. Its operating steps,Toas shown in Figure 9, are as follows. steps, as shown in Figure 9, are as follows.

Figure 9. Flowchart of the GCNA. Figure Figure 9. 9. Flowchart Flowchart of of the the GCNA. GCNA.

Step 1—Basic Conditions Definition Step 1—Basic Conditions Definition Step 1—Basic Definition ConfirmConditions the objective function and optimize the variables and the range of parameters. Confirm the objective function and optimize the variables and the range of parameters. Confirm the objective function and optimize the variables and the range of parameters. Step 2—Pretreament using a Filling Function Step 2—Pretreament using a Filling Function Select the “filling function” [32,33] shown in Equation (8) to promote the accuracy of the gravity Select the “filling function” [32,33] shown in Equation (8) to promote the accuracy of the gravity center location and to avoid the local optimum: center location and to avoid the local optimum:

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Step 2—Pretreament using a Filling Function Select the “filling function” [32,33] shown in Equation (8) to promote the accuracy of the gravity center location and to avoid the local optimum: p( x, x ∗ ) = −φ[ f ( x ) − f ( x ∗ ) + α] × k x − x ∗ k2

(9)

where x* is the current local extreme point, α is the parameter named “filling factor”, which influences the filling effect largely, and φ(t) satisfies the following conditions: (1) (2)

φ(0) > 0; φ(t) is derivable and t 6= 0, φ’(t) > 0

Step 3—Initialization of Population Scale and Search Scope Initialize the search scope Ω(0), which is determined by the variable ranges; the initialized population scale P(0) is determined by considering convergence rate and optimizing accuracy. Step 4—Gravity Center Definition The gravity center location of the objective function is determined via the following search strategy: P(k)

F ( xb ) xb + ∑ F ( xi (k )) xi (k) i =2 P(k)

xo (k ) =

(10)

F ( xb ) + ∑ F ( xi (k )) i =2

where F(xb ) is optimal of the before k − 1 times iterations. This strategy can lead to movement of the “gravity center” along the direction of the global optimum effectively. Step 5—Determine the “Neighborhood Scope” of Global Optimum The neighborhood scope of the gravity center can be expressed by the follow equation: Ω0 = Ω(0)η d /N

(11)

where η (η < 1) is the scope compression rate, d is the dimension of the optimizing scope, and N is the search times Step 6—Narrow the Search Scope and Population Scale (1)

The kth search scope can be expressed as follows: Ω (k ) /Ω (k − 1) = η d

(2)

(12)

The kth population scope is expressed as follows: P ( k ) = P ( k − 1) e γ ( η γ = ρη d /d

d −1) η d / ( k − N −1)

,k ≤ N

(13)

where ρ is an adjustable parameter. Step 7—Condition of Convergence Ω ( k ) = Ω0

(14)

When Equation (13) is set up, the search scope is equal to the neighborhood scope of the gravity center. The current gravity center can be seen as the global optimum. Otherwise, return to Step 4.

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This study Energies 2016, 9, 992 adopts

the GCNA to optimize the multi-SVM motor model. The parameters 11 ofofthe 15 This adopts the 9, GCNA optimize the multi-SVM motor of dthe GCNA arestudy shown in Table wheretoP(0) is initialized population scale,model. Ω(0) isThe theparameters search scope, is GCNA are shown in Table 9, where P(0) is initialized population scale, Ω(0) is the search scope, d This study adopts the GCNA to optimize the multi-SVM motor model. The parameters of the the dimension of the optimizing scope, η is the scope compression rate, N is the search times, α is is the dimension of the scope, η is scope compression rate,evolutionary N is the search times, α is dthe GCNA arenamed shown in optimizing Table 9, where initialized population scale, Ω(0) is the search scope, is parameter “filling factor”, andP(0) ρ is anthe adjustable parameter. The process diagrams parameter named “filling factor”, ρ the is an adjustable parameter. The the dimension of the optimizing scope, scope compression rate, N is theevolutionary search times, process α is the of the GCNA are shown in Figure 10.andη is diagrams ofnamed the GCNA are shown Figure parameter “filling factor”,inand ρ is10.an adjustable parameter. The evolutionary process Table 9. Parameters of GCNA. diagrams of the GCNA are shown in Figure 10. Ω(0) P(0) 4 3768 P(0)

P(0) 768

768

Table 9. Parameters of GCNA. d 9. Parameters η N Table of GCNA. Ω(0) d η N α 4 0.45 250 4 3 250 0.0001 Ω(0) d4 0.45 η N α 34

4

0.45

250

0.0001

α ρ 0.0001 0.02 ρ

ρ 0.02

0.02

Figure 10. 10.Evolutionary Evolutionaryprocess process diagrams: (a) Evolution progress of thrust; (b) Evolution Figure diagrams: (a) Evolution progress of thrust; and (b) and Evolution progress progress of THD. Figure 10. Evolutionary process diagrams: (a) Evolution progress of thrust; and (b) Evolution of THD. progress of THD.

The global optimal combination of the PMSLM structure parameters is obtained after the The global optimal combination of the PMSLM structure parameters is obtained after the iteration iteration of GCNA. Table 10 the comparison the structural parameters of the The calculations global optimal combination of shows the PMSLM structure ofparameters is obtained after calculations of GCNA. Table 10 shows the comparison of the structural parameters of the Taguchi Taguchi method, newof method, experimental method clearly. of the structural parameters of the iteration calculations GCNA.and Table 10 shows the comparison method, new method, and experimental method clearly. Taguchi method, new method, and experimental method clearly. Table 10. Comparison of structural parameters. Table 10. Comparison of structural parameters. Table 10. Comparison of structural parameters. Parameters Taguchi New Experimental Parameters Taguchi New Experimental h 2.75 mm 3.05 mm 3.05 mm Parameters Taguchi New Experimental h δ τ l

hδ δτ τl l

1.90mm mm 2.75 2.75 mm 20.00 mm 1.90 mm 1.90 mm 20.00 mm 6.75 mm 20.00 mm 6.75 6.75mm mm

2.093.05 mmmm 3.05 mm 19.12 mm 2.09 mm 2.09 mm 19.12 6.57 mm 19.12 mmmm 6.576.57 mmmm

2.09 mm mm3.05 mm 3.05 19.12 mm 2.09 mm 2.09 mm 19.12 mm 6.57 mm 19.12 mm 6.57 mm 6.57 mm

5. Experimental Verification Verification 5. Experimental Verification To validate the performances of optimized PMSLM, a prototype of PMSLM is manufactured and performances of optimized PMSLM, the test platformthe is established as shown in Figure 11. a prototype of PMSLM is manufactured and To validate performances the test platform is established as shown in Figure 11. 11.

Figure 11. Test platform for the manufactured PMSLM: (a) Prototype of PMSLM; and (b) Test platform. Figure Prototype of of PMSLM; PMSLM;and and(b) (b)Test Testplatform. platform. Figure11. 11.Test Testplatform platformfor forthe themanufactured manufactured PMSLM: PMSLM: (a) (a) Prototype

In Figure 12, the no-load inductive electromotive force (EMF) lines and harmonic analysis of the Taguchi method, method, and experimental method respectively In Figure 12, new the no-load inductive electromotive forceare (EMF) lines andshown. harmonic analysis of the Taguchi method, new method, and experimental method are respectively shown.

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In Figure 12, the no-load inductive electromotive force (EMF) lines and harmonic analysis of the Taguchi method, new method, and experimental method are respectively shown. Energies 2016, 9, 992 12 of 15

Figure 12. Inductive EMF and harmonic analysis by three methods: (a) Taguchi; (b) New; and (c) Experimental.

Figure 12. Inductive EMF and harmonic analysis by three methods: (a) Taguchi; (b) New; and (c) Experimental. The sinusoidal characteristic of the new method is evidently promoted relative to the Taguchi method. The comparative results of the harmonic amplitude after spectrum analysis [34,35] are The sinusoidal characteristic thetonew method is evidently to theThe Taguchi shown in Figure 12 and Table 11.ofDue the star connection of coils,promoted there is no relative 3rd harmonic, experimental results showresults that theof harmonics of the 5th, 7th, 9th, and are restrained by 0.76%, method. The comparative the harmonic amplitude after11th spectrum analysis [34,35] are 0.33%, 0.32%, 12 andand 0.18%, respectively, which indicates that theofharmonic content significantly shown in Figure Table 11. Due to the star connection coils, there is no 3rd harmonic. decreases. The THD of the Taguchi method, new method, and experimental results are 2.66%, 0.80%, by The experimental results show that the harmonics of the 5th, 7th, 9th, and 11th are restrained and 0.91%. The THD of experimental results are remarkably reduced 69.92% compared with the 0.76%, 0.33%, 0.32%, and 0.18%, respectively, which indicates that the harmonic content significantly Taguchi method. Given the decrease in harmonic content and improvement in THD, the stability of decreases. The THD of the Taguchi method, new method, and experimental results are 2.66%, 0.80%, PMSLM can be enhanced significantly.

and 0.91%. The THD of experimental results are remarkably reduced 69.92% compared with the Taguchi method. Given the decrease and improvement in THD, the stability of Table in 11. harmonic Comparisoncontent of harmonic content. PMSLM can be enhanced significantly. Harmonic Order THD% 1st 5th 7th 9th 11th Table 11. Comparison of harmonic content. Taguchi 100% 2.34% 1.16% 0.31% 0.40% 2.66% New 100% 0.68% 0.31% 0.25% 0.16% 0.80% Harmonic Experimental 100% 0.76% 0.33%Order 0.32% 0.18% 0.91% Amplitude% 1st 5th 7th 9th 11th Amplitude%

THD%

As shown in Table 12 100% and Figure2.34% 13, the thrust results that the motor Taguchi 1.16% experimental 0.31% 0.40%indicate2.66% average thrustNew reaches 43.713 N and the thrust ripple to 1.969%, represents an 100% 0.68% 0.31%is suppressed 0.25% 0.16% which 0.80% Experimental 0.33% 0.32% 0.18% 0.91% enhancement of the Taguchi100% method. 0.76% Table 12. Comparison of thrust. As shown in Table 12 and Figure 13, the thrust experimental results indicate that the motor Taguchi average thrust reaches 43.713 N and the thrustNew ripple Experimental is suppressed to 1.969%, which represents F (N) 40.241 43.713 43.279 an enhancement of the Taguchi method. η (%)

6.342%

1.937%

1.969%

Table 12. Comparison of thrust.

F (N) η (%)

Taguchi

New

Experimental

40.241 6.342%

43.713 1.937%

43.279 1.969%

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Figure Figure 13. 13. Comparison Comparison of of motor motor thrust. thrust.

All the performances of PMSLM are influenced by the motor structure parameters. From All the performances of PMSLM are influenced by the motor structure parameters. From Taguchi Taguchi to the new method, the volume of the magnets increases by 10.9%, owing to the increase in to the new method, the volume of the magnets increases by 10.9%, owing to the increase in magnet magnet height which can enhance the magnetic field intensity. The thrust curve of the new method height which can enhance the magnetic field intensity. The thrust curve of the new method is more is more stable because of the lowest thrust ripple, which is mainly affected by the air gap and pole stable because of the lowest thrust ripple, which is mainly affected by the air gap and pole pitch pitch when the motor mover passes by different adjacent magnets. The combination of a 2.09 mm air when the motor mover passes by different adjacent magnets. The combination of a 2.09 mm air gap gap and 19.12 mm pole pitch is the most suitable for magnetic distribution. A 1.937% thrust ripple and 19.12 mm pole pitch is the most suitable for magnetic distribution. A 1.937% thrust ripple can can guarantee the placid operation of the mover. All of these findings indicate the validity of the guarantee the placid operation of the mover. All of these findings indicate the validity of the proposed proposed new method. new method. 6. Conclusions 6. Conclusions The PMSLM applied in microsecond laser cutting machines with no cutting force is presented The PMSLM applied in microsecond laser cutting machines with no cutting force is presented in in this paper. A new optimization method is introduced to the motor design optimization. Through this paper. A new optimization method is introduced to the motor design optimization. Through the the 3D-FEM analysis of PMSLM performance, multi-SVM is proposed to develop the mapping 3D-FEM analysis of PMSLM performance, multi-SVM is proposed to develop the mapping relation relation from multivariate structure parameters to multivariate operational performances, which can from multivariate structure parameters to multivariate operational performances, which can result in result in the formation of a non-parametric quick calculation model. After the optimization iteration the formation of a non-parametric quick calculation model. After the optimization iteration calculations calculations of the GCNA, the global optimal combination of motor structure parameters, such as the of the GCNA, the global optimal combination of motor structure parameters, such as the height of height of magnet, air gap, pole pitch, and coil length, are obtained by the gravity center location of magnet, air gap, pole pitch, and coil length, are obtained by the gravity center location of the objective the objective function. Through the analysis of harmonic content, the THD %, thrust, thrust ripple, function. Through the analysis of harmonic content, the THD %, thrust, thrust ripple, and operational and operational performances of PMSLM are enhanced remarkably. This method is verified by an performances of PMSLM are enhanced remarkably. This method is verified by an experimental test, experimental test, and the design goals and optimization results have a great consistency. and the design goals and optimization results have a great consistency. 7. 7. Discussion Discussion In In the the future future work, work, the the authors authors will will continue continue to to research research the the following following directions directions to to enhance enhance this this optimization method: optimization method: (1) Enlarge the value range of the design parameters. In the larger value ranges, attempt to find (1) Enlarge the value range of the design parameters. In the larger value ranges, attempt to find more more suitable combinations of design parameters which can satisfy the requirements of suitable combinations of design parameters which can satisfy the requirements of microsecond microsecond laser cutting machines. Namely, thrust should be no less than 35 N, and thrust laser cutting machines. Namely, thrust should be no less than 35 N, and thrust ripple and ripple and harmonic contents should be the lowest. harmonic contents should be the lowest. (2) Research more design parameters (such as 10–15 parameters) to certify this method. On the basis (2) of Research more design parameters as 10–15 parameters) certify this method. to Onverify the basis four parameters shown in Table(such 1, more design parameterstowill be investigated the of four parameters shown in Table 1, more design parameters will be investigated to verify the effectiveness of the proposed new optimization method. The additional parameters include the effectiveness of thethe proposed new optimization method.ofThe additional the width of magnets, length of magnets, the thickness back-iron, theparameters structure ofinclude coils, and width of magnets, the length of magnets, the thickness of back-iron, the structure of coils, and so on. so on. (3) Promote the computational efficiency. The authors will reduce the unnecessary finite-element meshing in FEM software, ANSOFT (ANSYS, Pittsburgh, PA, USA), and to study new software,

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Promote the computational efficiency. The authors will reduce the unnecessary finite-element meshing in FEM software, ANSOFT (ANSYS, Pittsburgh, PA, USA), and to study new software, INFOLYTICA (INFOLYTICA, Montreal, Quebec, Canada) which can reduce the computing time by at least about 35%.

Acknowledgments: This work is supported by the National Natural Science Foundation of China under Grant Nos. 51277002, 51577001 and 51637001. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. Author Contributions: All authors contributed to this work by cooperation. Juncai Song, Fei Dong, Le Li and Zhenbao Pan are the main authors of this manuscript and this work was conducted under the advisement of Jiwen Zhao and Siliang Lu. Conflicts of Interest: The authors declare no conflict of interest.

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5. 6. 7.

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