Sensitivity Analysis and Optimal Design of a Stator Coreless Axial Flux Permanent Magnet Synchronous Generator

sustainability

Article Sensitivity Analysis and Optimal Design of a Stator Coreless Axial Flux Permanent Magnet Synchronous Generator

Wenqiang Wang, Shaoqi Zhou *, Hongju Mi, Yadong Wen, Hua Liu, Guoping Zhang and Jianyong Guo

Department of Petroleum, Army Logistics University of PLA, Chongqing 401331, China; [email protected] (W.W.); [email protected] (H.M.); [email protected] (Y.W.); [email protected] (H.L.); [email protected] (G.Z.); [email protected] (J.G.) * Correspondence: [email protected]; Tel.: +86-023-8673-6189

Received: 29 December 2018; Accepted: 1 March 2019; Published: 7 March 2019

Abstract: In this paper, the modified initial design procedure and economic optimization design of a stator coreless axial flux permanent magnet synchronous generator (AFPMSG) are presented to improve the design accuracy, efficiency, and economy. Static magnetic field finite-element analysis (FEA) is applied to the magnetic equivalent circuit (MEC) method to increase the accuracy of electromagnetic parameters and reduce the iteration times. The accuracy and efficiency of the initial design is improved by the combination of MEC method and static magnetic field FEA in the design procedure. For the economic optimization, the permanent magnetic (PM) material volume model, which affects the cost of the AFPMSG the most, is derived, and the influence degree of the main structure parameters, to the performance, is distinguished and sorted by sensitivity analysis. The hybrid genetic algorithm that combines the simulated annealing and father-offspring selection method is studied and adopted to search for the best optimization solution from the different influence degree and nonlinear interaction parameters. A 1 kW AFPMSG is designed and optimized via the proposed design procedure and optimization design. Finally, 3D finite-element models of the generator are simulated and compared to confirm the validity of the proposed improved design and the generator performance.

Keywords: stator coreless; axial ﬂux permanent magnet synchronous generator (AFPMSG); hybrid genetic algorithm (GA); sensitivity analysis; economic optimization

1. Introduction In the topic of permanent magnet electrical machine construction, recent works have shown that the usage of neodymium-iron-boron (NdFeB) and types of permanent magnet motor have drastically increased over the last decades, mainly due to cheaper cost and the demand of non-pollution energy. Although there are a lot of categories of permanent magnet generators available, the axial flux permanent magnet synchronous generator (AFPMSG), with a different electromagnetic path from traditional motors, is studied here. The AFPMSG has the merits of both a permanent magnet motor and a disc type motor, which have a small size, simple and compact structure, high power density and operation efficiency, and easy processing and manufacturing. Great application prospect and development value have been exploited in fields of electric cars, large industrial equipment, wind power generation, etc. [1–4]. The AFPMSG can be single-sided or double-sided, core or coreless, surface mounted or interior permanent magnet (PM), and have a single- or multi-staged configuration [5]. Among those types

Sustainability 2019, 11, 1414; doi:10.3390/su11051414 www.mdpi.com/journal/sustainability Sustainability 2019, 11, 1414 2 of 18 of AFPMSG, double-sided AFPMSG with stator coreless has been widely researched and used in the industry. Compared with a stator core generator, the AFPMSG, without stator core, have the advantages of light weight, no cogging torque, high efficiency, and simple construction. The double-sided structure of the APFMSG has higher torque-to-volume ratio than that of other structures [6], and the greater the number of pole pairs that are designed, the higher ratio it will be. Besides, the negligible axial attraction force between the stator and rotor improves the reliable design of large diameter generators. However, because of the non-ferromagnetic stator core, the effective air gap length is larger and requires more active PM material to keep the magnetic field in the air gap, which increases the cost of the AFPMSG. Economic optimization design for the stator coreless AFPMSG becomes an important research topic. With the further research and extensive application of AFPMSGs, the analysis of their design and optimization has been a hotpot research field recently. The non-uniform distribution of magnetic field, large air gap, and the interaction of structural parameters make the stator coreless AFPMSG a multi-variable non-linear system, which brings great difficulties to its optimization design [7]. At present, the design and optimization of the AFPMSG have been extensively studied and explored. The basic theory of AFPMSG, including the design procedure, performance calculation, and application, is presented in [8,9]. The design methods and development of performance analysis tools for axial flux permanent magnet machines is presented in [10]. Kahourzade, et al. and Huang, et al. [11,12] investigated the design and analysis of the AFPMSG with the sizing equation. In [13,14], analytical methods are used for the analysis and calculation of magnetic field in the generator. Study on the flux distribution of each part of the AFPMSG via magnetic equivalent circuit (MEC) is presented in [15,16]. A new three-dimensional finite element method based on improved Maxwell equations is proposed to calculate the no-load flux of an axial permanent magnet motor in [17]. To improve the design efficiency and accuracy, hybrid methods are used in the design, analysis, and calculation of the AFPMSG in [18,19]. However, the special structure and oversimplification would lower the design accuracy for the analytical method, and the 3D finite element method takes time to do the calculation and is inconvenient for design and analysis with multi-parameters varying in a large range. The calculation time and accuracy of the MEC method are moderate, and between the analytical method and finite element method. Additionally, as computer-aid technologies blossom, new or improved design methods are still under research to make the design more realistic. Due to complex electromagnetic processes and practical application requirements, an optimal procedure is necessary to obtain better design results or to improve the performance of the motor for special needs. Recently, heuristic algorithms, especially the genetic algorithm (GA), have been used in motor optimization [20,21]. The development of field computation and multi-objective optimal design in electromagnetics were studied in [22,23]. Performance optimizations, including the maximum power density, low cogging torque, minimum material cost, etc., via the GA, are presented in [24–26]. However, in these studies, the sensitivity analysis of the generator design parameters to output performance and economy optimization is not paid enough attention, which affects the efficiency and results of the motor optimization design. Considering the special requirements for the design and application, improved design procedure and economic optimization of a stator coreless AFPMSG are proposed in the paper. The field-circuit method, combining the MEC method and static magnetic field FEA, is used in the initial design procedure to improve the design efficiency and accuracy. An active PM material volume model, which concerns the cost of the AFPMSG the most, is derived for the economical optimal design. In order to find out the influence degree of each parameter, sensitivity analysis on the main design parameters to the performance and material volume consumption of the AFPMSG is carried out. Considering the main design parameters’ non-linear interaction and different influence degree to the output performance, father–offspring selection and a simulated annealing method are applied in the GA to improve the design accuracy, globality, and fast convergence. Improved GA is used to perform the calculation of the AFPMSG economic optimization model for the best solution to reduce active Sustainability 2019, 11, 1414 3 of 18

Sustainability 2019, 11, x FOR PEER REVIEW 3 of 18

Sustainabilitymaterial volume 2019, 11, costx FOR and PEER increase REVIEW the economy on the condition of keeping the electrical performance3 of 18 unchanged.In the Thefollowin schematicg, basic overview structure of and this equations paper is showed are introduced in Figure and1. the improved initial design procedure of the AFPMSG with the field-circuit method is given in Section 2. The economic InIn the the followin following,g, basic basic structure structure and and equations equations are are introduced introduced and and the the improved improved initial initial design design optimization model, that is the active PM material volume, is derived and studied in Section 3, as procedureprocedure of of the the AFPMSG AFPMSG with with the the fieldﬁeld-circuit-circuit method is given in SectionSection2 . 2. The The economic economic well as the sensitivity analysis about the main design parameters to the performance and material optimizationoptimization model model,, that that is isthe the active active PM PM material material volume volume,, is derived is derived and and studied studied in S inection Section 3, as3, volume consumption of the AFPMSG. Section 4 introduces the improved GA for the optimal design wellas well as the as thesensitivity sensitivity analysis analysis about about the the main main design design parameters parameters to to the the performance performance and and material material of the APFMSG, which combines the simulated annealing and father-offspring selection method. In volumevolume consumption consumption of of the the AFPMSG. AFPMSG. Section Section 4 introducesintroduces the the improved improved GA GA for for the the optimal optimal design design Section 5, the validity of the improved initial design procedure and economic optimization design ofof the the APFMSG, APFMSG, which which combines combines the the simulated simulated anneali annealingng and and father father-offspring-offspring selection selection method. method. In are evaluated by comparison and 3D FEA. Finally, conclusions are drawn in Section 6. SectionIn Section 5, the5, the validity validity of ofthe the improved improved initial initial design design procedure procedure and and economic economic optimization optimization design design areare evaluated evaluated by by comparison comparison and and 3D 3D FEA. FEA. Finally, Finally, conclusions conclusions are drawn in Section 66..

Figure 1. The schematic overview of this paper. FigureFigure 1 1.. TheThe schematic schematic o overviewverview of of this this paper. paper. 2. AFPMSG and Initial Design 2. AFPMSG and Initial Design 2. AFPMSG and Initial Design 2.1. Structure and Design Analysis 2.1. Structure and Design Analysis 2.1. StructureConsidering and Design the advantages Analysis of the axial flux permanent magnetic motor and the stator coreless structure,Considering a dual the-rotor advantages AFPMSG of with the axial coreless ﬂux permanent stator was magnetic studied in motor this and paper. the The stator considered coreless Considering the advantages of the axial flux permanent magnetic motor and the stator coreless structure,AFPMSG aconsists dual-rotor of two AFPMSG outer rotor withs and coreless a coreless stator stator was clamped studied in the this middle. paper. Sector The considered permanent structure, a dual-rotor AFPMSG with coreless stator was studied in this paper. The considered AFPMSGmagnets consistsare placed of twoon the outer rotor rotors yoke and surfaces a coreless which stator formulates clamped inthe the rotor middle. structure. Sector The permanent stator is AFPMSG consists of two outer rotors and a coreless stator clamped in the middle. Sector permanent magnetsmade of are coils placed and non on the-ferromagnetic rotor yoke surfaces materials, which and formulates the non-overlapping the rotor structure. concent Therated stator coils isare made held magnets are placed on the rotor yoke surfaces which formulates the rotor structure. The stator is oftogether coils and and non-ferromagnetic supported by non materials,-ferromagnetic and the non-overlapping materials. The concentratedstructure of AFPMSGcoils are held is shown together in made of coils and non-ferromagnetic materials, and the non-overlapping concentrated coils are held andFigure supported 2. by non-ferromagnetic materials. The structure of AFPMSG is shown in Figure2. together and supported by non-ferromagnetic materials. The structure of AFPMSG is shown in Figure 2. Rin Rout Shaft R in R out  Shaft hM g  w hM g w

Permanent Magnet Coreless Stator

Figure 2. IllustrationPermanent of main Magnet structure and dimensions of axial ﬂux permanent magnet synchronous Figure 2. Illustration of main structure and dimensions of axial flux permanentCoreless Stator magnet synchronous generator (AFPMSG). generator (AFPMSG). Figure 2. Illustration of main structure and dimensions of axial flux permanent magnet synchronous generator (AFPMSG). Electromagnetic relation and sizing equations are the basics of motor design. The design of the AFPMSG follows the fundamentals is presented in [5]. The main sizing equation of the AFPMSG that Electromagnetic relation and sizing equations are the basics of motor design. The design of the shows the relationship between the output power and generator structure parameters is given by: AFPMSG follows the fundamentals is presented in [5]. The main sizing equation of the AFPMSG that shows the relationship between the output power and generator structure parameters is given by: = ( + )( − ) (1) 32 = ( + )( − ) (1) 32 Sustainability 2019, 11, 1414 4 of 18

Electromagnetic relation and sizing equations are the basics of motor design. The design of the AFPMSG follows the fundamentals is presented in [5]. The main sizing equation of the AFPMSG that shows the relationship between the output power and generator structure parameters is given by:

π3 P = n k A α B (D + D ) D2 − D2 (1) e 32 s w av i δ out in out in where, kw, ns, Aav, Bδ, Dout, and Din are winding factor, rotor speed, average electrical loading, flux density of the air gap, and outer and inner diameter of the AFPMSG, respectively; and αi is the calculated polar arc coefficient. For the sector-shaped permanent magnet structure, the calculated polar arc coefficient is equal to the polar arc coefficient αp which is the ratio of magnet width to pole pitch. When designing the AFPMSG, appropriate design values should be selected for some design quantities according to experience and design requirements, such as electromagnetic load, inner-to-outer diameter ratio, etc., and the final value should be determined after checking. Electrical load represents the number of ampere conductors per unit circumference along the armature surface, and magnetic load, namely air gap flux density, refers to the average flux density along the air gap surface when no load occurs. Electromagnetic load equations are expressed in: √ A(r) = 2mNIa/πr (2)

Bδ = ∅m/αpτLe f (3) where, m, N, and Ia are the number of the phases, coil turns per phase, and rated current of armature winding, respectively. ∅m, αp, τ, and Le f are main flux per pole, polar arc coefficient, magnetic pole pitch, and effective length of armature winding, respectively. It is obvious that the electric load is a function of radius, and the maximum electric load is at the inner diameter for the AFPMSG. In order to avoid local overheating, it should be designed according to the maximum electric load. Besides, the air gap magnetic field is the medium of energy conversion of a generator; therefore, the design of magnetic load has a significant impact on the performance of a generator. Because of the complicated flux distribution, the value is given by estimation in the initial design [27]. The diameter ratio γ is used to present the relation between inner and outer diameter, and the relation is given by: γ = Din/Dout (4) The diameter ratio γ is an important design parameter affecting the performance and economy of the AFPMSG. Based on the extreme value analysis and comprehensive consideration of copper loss, cost, etc., the value of γ is usually 0.45–0.60 [5]. However, the actual value depends on the design objectives and optimization. The energy conversion of the motor is carried out in the air gap, hence the length of the air gap affects the structure and performance of a generator. Due to the non-ferromagnetic stator structure, the air gap length of the AFPMSG is made up of the double-sided air gap and the stator, as expressed in Equation (5). Additionally, the axial length of the AFPMSG can be obtained by Equation (6).

Lair = 2δ + δw (5)

L = Lair + 2hM + 2hr (6) where δ, δw, hM, and hr are the air gap length, thickness of stator, PM, and rotor, respectively. Lair and L are the effective air gap length and axial length of the APFMSG, respectively. Studies on the performance of the APMFSG with different types of coil winding are presented in [28] and speciﬁcations can be obtained by analytical calculation. Considering the non-ferromagnetic stator structure that increases the effective length of the air gap, the thickness of winding should be decreased as much as possible. On the contrary, too small of a space for the stator windings will lead to Sustainability 2019, 11, 1414 5 of 18 heat dissipation problems. In the AFPMSG, the inner radius is the most limited space to place the stator windings and thus the space factor is introduced from the stator core motor to calculate the thickness of the stator winding appropriately. The thickness of the armature winding coil is determined by the occupancy rate of the total cross-sectional area of armature winding coil at the inner diameter. As shown in Equation (7), the initial thickness of stator can be calculated when a proper space factor Sustainabilityis selected. 2019, 11, x FOR PEER REVIEW 5 of 18 0 2 S f = 4mNaa D /(πDinδw) (7) where , , , and are the number of turns in series for each phase winding, the number of whereparallelN , branches,a, a0, and D coilsare theper number turn wound of turns around in series the for conductor each phase, and winding, the diameter the number of ofthe parallel wire, respecbranches,tively coils. per turn wound around the conductor, and the diameter of the wire, respectively. Another important step is to calculate the loss and efﬁciencyefficiency of the AFPMSG accurately in the design procedure. Due to the absen absencece of a stator stator core, core, the loss loss of of the the AFPMSG AFPMSG consists consists of of copper copper loss loss,,

Pcu,, eddyeddy currentcurrent loss,loss, P eddy_cu_,, mechanical mechanical loss, loss,P mech , and, and rotor rotor core core loss, lossP,core . The. The calculation calculation of ofevery every aspect aspect loss loss of the of AFPMSG the AFPMSG has been has researched been researched and presented and presented in [5,29, 30 in]. [5 Thus,,29,30] the. Thus, efﬁciency the efficiencyof the AFPMSG of the isAFPMSG expressed is expressed as: as:

η = ⁄( + + _ + + ) (8) η = Pout/(Pout + Pcu + Peddy_cu + Pmech + Pcore (8)

2.2. Improved Initial Design Procedure In the the design design procedure procedure of the of AFPMSG,the AFPMSG as discussed, as discussed in the previous in the previous part, some part, design some parameters design wereparameters determined were bydetermined experience by and experience estimating. and Considering estimating. theConsidering characteristics the characteristics of the structure of andthe structuremagnetic fieldand magnetic distribution, field the distribution, analytical design the analytical and FEA design may be and either FEA inaccurate may be either for special inaccurate magnetic for specialfield distribution magnetic field or time-consuming, distribution or time due- toconsuming too many, parameters,due to too many for simulation. parameters In, for this simulation paper, the. InMEC this method, paper, the combining MEC method the static, co magneticmbining the field static FEA, magnetic was adopted field to FEA improve, was adopted the design to efficiencyimprove theand design accuracy. efficiency The way and it wasaccuracy. used inThe the way paper it was makes used full in usethe ofpaper the advantagesmakes full use of the of the static advantages magnetic offield the FEA static (accuracy magnetic and field less FEA time-consuming) (accuracy and toless both time solve-consuming the electromagnetic) to both solve parameters’ the electromagnetic inaccuracy problemparameters’ and inaccuracy reduce the problem number and of iterations reduce the in number the MEC of method.iterations The in the MEC MEC method method. in thisThe paperMEC methodconverts in the this non-uniform paper converts spatial the magnetic non-uniform field into spatial the equivalent magnetic multi-section field into the magnetic equivalent circuit, multi and- sectionapproximates magnetic that circuit, the magnetic and ap fluxproximates in each section that the is uniformly magnetic distributedflux in each along section the cross-sectionis uniformly distributedand length, andalong transforms the cross- thesection calculation and length, of magnetic and transforms field into the the calculation calculation of magnetic circuitfield into [8]. theFigure calculation3 shows theof magnetic magnetic circuit circuit [8] and. Figure calculation 3 shows is given the magnetic by: circuit and calculation is given by: H1L1 + H2L2 + ··· + Hn Ln = F (9) + + ⋯ + = (9) where Hi andand L iareare the the magnetic magnetic field field intensity intensity and and path path length length of the ofn-th the magneticn-th magnetic circuit, respectively.circuit, respectively.

Figure 3 3.. Flux paths in 3D plane for the stator coreless AFPMSG.AFPMSG.

In order toto increaseincrease the the accuracy accuracy of of design design parameters, parameters, finite finite element element analysis analysis was wa adopteds adopted in the in thedesign design procedure, procedure and, and the static the static magnetic magnetic field analysisfield analysis of the of AFPMSG the AFPMSG was mainly was mainly used to used obtain to obtain the electromagnetic parameters that were difficult and inaccurate for circuit calculation. Professional electromagnetic field simulation software, Ansys Maxwell, was used to establish the model and conduct the finite element analysis. Figure 4 shows the 2D view of the magnetic field distribution of the AFPMSG in the mean radius. The magnetic field distribution and leakage flux can be obtained through the static field analysis. Sustainability 2019, 11, 1414 6 of 18 the electromagnetic parameters that were difficult and inaccurate for circuit calculation. Professional electromagnetic field simulation software, Ansys Maxwell, was used to establish the model and conduct the finite element analysis. Figure4 shows the 2D view of the magnetic field distribution of the AFPMSG in the mean radius. The magnetic field distribution and leakage flux can be obtained SustainabilitythroughSustainability the 2019 2019 static, ,11 11, ,xfield x FOR FOR analysis.PEER PEER REVIEW REVIEW 66 ofof 1818

FigureFigure 4 4.4. .2D 2D view view of of the the magnetic magnetic field ﬁeldfield distribution distribution ofof AFPMSG AFPMSG.AFPMSG. .

BasedBased on onon the the design design design analysis, analysis, analysis, the the the improved improved improved design design design procedure procedure procedure with with with the the thefieldfield ﬁeld-circuit--circuitcircuit method method method wa wass wasproposedproposed proposed,, ,as as shown shown as shown in in Figure Figure in Figure 5 5, ,and and5, the andthe stator stator the stator coreless coreless coreless AFPMSG AFPMSG AFPMSG rated rated ratedat at 1 1 kW, kW, at 1230 230 kW, V V 230(phase), (phase), V (phase), three three-- phase,three-phase,phase, 5050 Hz,Hz, 50 andand Hz, 300300 and r/minr/min 300 r/min wawass designed.designed. was designed. TheThe classicalclassical The classical structustructure structurere modemode wawa modess adoptedadopted was adopted inin thethe designdesign in the ofdesignof thethe ofrotorrotor the rotorpole pole structurepole structure structure and and and stator stator stator winding. winding. winding. Design Design Design param param parameterseterseters werewere were chosenchosen chosen properly properly with with comprehensivecomprehensive consideration.consideration. TableTable1 1 1lists lists lists the the the detailed detailed detailed design design design results results results of of of the the the AFPMSG. AFPMSG. AFPMSG.

StartStart

ImportImport design design rated rated values values of of AFPMSG AFPMSG

StructureStructure determination determination and and Electromagnetic Electromagnetic designdesign of of AFPMSG AFPMSG

CalculationCalculation of of generator generator main main dimensions dimensions basedbased on on sizing sizing equation equation

StaticStatic magnetic magnetic field field analysis analysis for for special special parametersparameters accurately accurately

MECMEC calculation calculation for for flux flux path path

NN CheckCheck the the PM PM operating operating point point convergenceconvergence

CalculationCalculation of of generator generator output output performance performance

CheckCheck the the efficiency efficiency convergenceconvergence NN

YY OverOver

FigureFigure 5. 5. Improved ImprovedImproved design designdesign procedure procedure of of the the statorstator corelesscoreless AFPMSG.AFPMSG.

TableTable 1. 1. Initial Initial design design results results of of the the AFPMSG. AFPMSG.

DesignDesign Parameters Parameters ValuValueses OuterOuter diameter diameter (mm) (mm) 380380 InnerInner diameter diameter (mm) (mm) 210210 ThicknessThickness of of permanent permanent magnet magnet ( (PMPM)) (mm (mm)) 1010 PolePole arc arc coefficient coefficient 0.80.8 AirAir gap gap length length (mm) (mm) 1.21.2 ThicknessThickness of of stator stator (mm) (mm) 99 ThicknessThickness of of rotor rotor yoke yoke (mm) (mm) 1010

2.3.2.3. Performance Performance Analysis Analysis Sustainability 2019, 11, 1414 7 of 18

Table 1. Initial design results of the AFPMSG.

Design Parameters Values Sustainability 2019, 11, x FOR PEEROuter REVIEW diameter (mm) 380 7 of 18 Inner diameter (mm) 210 3D finite element analysisThickness wa ofs permanent adopted magnetto simulate (PM) (mm)the output performance 10 of the AFPMSG through Ansys MaxwellPole 16.0 arcsoftware. coefficient Ansys Maxwell is the most practical 0.8 electromagnetic analysis software at present, whichAir can gap analyze length (mm) and calculate various electromagnetic 1.2 fields and multi-state Thickness of stator (mm) 9 system problems with Thicknessits advanced of rotor adaptive yoke (mm) meshing technology 10and convenient self-defined material library [31]. The main steps of the simulation analysis are shown in Figure 6 [32]. In this paper, the establishment of the model was based on the design value of the generator structure 2.3. Performance Analysis dimension.Sustainability The 2019 other, 11, x settingsFOR PEER and REVIEW conditions were set as the normal or rated value. 7 of 18 The3D finiteperformance element analysis analysis of was the adopted generator to mainly simulate focuse the outputd on the performance analysis and of research the AFPMSG of the inducedthrough3D Ansyselectromotive finite Maxwell element voltage, 16.0 analysis software. torque was, Ansys adoptedand efficiency Maxwell to simulate isunder the most theload output practicalto verify performance electromagnetic the design of results. the analysis AFPMSG The isoftwarenitialthrough simulation at Ansys present, Maxwellmodel which of can16.0 the analyzesoftware. AFPMSG and Ansys wa calculates Maxwellestablished various is the based electromagnetic most on practical the initial electromagnetic fields design and multi-state structure analysis dimensionsystemsoftware problems values, at present, with as shown its which advanced in can Figure analyze adaptive 7a. The and meshing initial calculate operation technology various condition andelectromagnetic convenient was under self-defined fields a rated and constant materialmulti-state speedlibrarysystem of [31 300 ]. problems Ther/min main with with steps a itsthree of advanced the-phase simulation symmetrical adaptive analysis meshing-rated are resistiv showntechnologye inload. Figure and The 6 convenient[output32]. Inperformance this self paper,-defined curvethematerial establishment, including library the of [31] the induced. The model main wasvoltage steps based andof on the torque the simulation design of the value analysis AFPMSG of the are generator, cshownould bein structure Figureobtained dimension.6 [32]through. In this Thesimulationpaper, other settingsthe, as establishmentshown and in conditions Figure of 8.the were model set aswa thes based normal on or the rated design value. value of the generator structure dimension. The other settings and conditions were set as the normal or rated value. The performance analysis of the generator mainly focused on the analysis and research of the Establishing Material and Boundary and Initial Operation Solution and induced electromotiveModel voltage,Excitation torqueSetting , andMesh efficiency Generation underCondition load Setting to verifyPost -theprocessing design results. The initial simulation model of the AFPMSG was established based on the initial design structure dimension values, as shown in Figure 7a. The initial operation condition was under a rated constant Figure 6. The s simulationimulation and design flow flow chart of Ansys Maxwell. speed of 300 r/min with a three-phase symmetrical-rated resistive load. The output performance curveThe, performanceincluding the analysis induced of the voltage generator and torque mainly of focused the AFPMSG on the analysis, could andbe obtained research of through the inducedsimulation electromotive, as shown voltage, in Figure torque, 8. and efficiency under load to verify the design results. The initial simulation model of the AFPMSG was established based on the initial design structure dimension values, as shown in Figure7a. The initial operation condition was under a rated constant speed Establishing Material and Boundary and Initial Operation Solution and of 300 r/min withModel a three-phaseExcitation symmetrical-rated Setting Mesh Generation resistive load.Condition The Setting outputPost performance-processing curve, including the induced voltage and torque of the AFPMSG, could be obtained through simulation, as shown in Figure8. Figure 6. The simulation and design flow chart of Ansys Maxwell.

(a) (b)

Figure 7. 3D model and air gap magnetic density of the AFPMSG. (a) 3D model of the AFPMSG; (b) Magnetic flux density (Mag B) distribution in the air gap.

InducedVoltage vs Time Load

375.00 Curve Info InducedVoltage(WindingA) Setup1 : Transient InducedVoltage(WindingB) Setup1 : Transient 250.00 InducedVoltage(WindingC) Setup1 : Transient

125.00 (a) (b)

0.00 FigureFigure 7. 3D7. 3D model model and and air air gap gap magnetic magnetic density density of of thethe AFPMSG.AFPMSG. ((aa)) 3D3D model of of the the AFPMSG AFPMSG;; (b)

InducedVoltage [V] InducedVoltage (b)Mag Magneticnetic flux ﬂux density density (Mag B B)) distribution in the air gap. -125.00

InducedVoltage vs Time Load 375.00 -250.00 Curve Info InducedVoltage(WindingA) Setup1 : Transient InducedVoltage(WindingB) Setup1 : Transient 250.00 InducedVoltage(WindingC) -375.00 Setup1 : Transient 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Time [ms] 125.00 (a) (b)

Figure 0.00 8. Output performance of the AFPMSG. (a) Output-induced voltage waveform; (b) torque against the rotor angle. InducedVoltage [V] -125.00

-250.00

-375.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Time [ms] (a) (b)

Figure 8. Output performance of the AFPMSG. (a) Output-induced voltage waveform; (b) torque against the rotor angle. Sustainability 2019, 11, x FOR PEER REVIEW 7 of 18

3D finite element analysis was adopted to simulate the output performance of the AFPMSG through Ansys Maxwell 16.0 software. Ansys Maxwell is the most practical electromagnetic analysis software at present, which can analyze and calculate various electromagnetic fields and multi-state system problems with its advanced adaptive meshing technology and convenient self-defined material library [31]. The main steps of the simulation analysis are shown in Figure 6 [32]. In this paper, the establishment of the model was based on the design value of the generator structure dimension. The other settings and conditions were set as the normal or rated value. The performance analysis of the generator mainly focused on the analysis and research of the induced electromotive voltage, torque, and efficiency under load to verify the design results. The initial simulation model of the AFPMSG was established based on the initial design structure dimension values, as shown in Figure 7a. The initial operation condition was under a rated constant speed of 300 r/min with a three-phase symmetrical-rated resistive load. The output performance curve, including the induced voltage and torque of the AFPMSG, could be obtained through simulation, as shown in Figure 8.

Establishing Material and Boundary and Initial Operation Solution and Model Excitation Setting Mesh Generation Condition Setting Post-processing

Figure 6. The simulation and design flow chart of Ansys Maxwell.

(a) (b)

SustainabilityFigure2019 7. 3D, 11 model, 1414 and air gap magnetic density of the AFPMSG. (a) 3D model of the AFPMSG; (b)8 of 18 Magnetic flux density (Mag B) distribution in the air gap.

InducedVoltage vs Time Load 375.00 Curve Info InducedVoltage(WindingA) Setup1 : Transient InducedVoltage(WindingB) Setup1 : Transient 250.00 InducedVoltage(WindingC) Setup1 : Transient

125.00

0.00 InducedVoltage [V] -125.00

-250.00

-375.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Time [ms] (a) (b)

Figure 8. OutOutputput performance of the AFPMSG.AFPMSG. (a) Output-inducedOutput-induced voltage waveform;waveform; ( b) torquetorque against the rotor angle.angle.

In Figure7b, the flux density on the middle plane of the effective air gap is illustrated. The air gap magnetic density decreased from the center of the magnetic pole to both sides. The phase-induced voltage and torque curves are presented in Figure8a,b. It was evident that the AFPMSG had a sinusoidal-induced voltage and relatively stable electromagnetic torque output. The output-induced voltage was stable with low harmonics. The electromagnetic torque ripple caused by eddy current and error was small, and overall was balanced in a cycle, which can be neglected. The maximum amplitude of the torque fluctuation was about 1.5 Nm. At rated speed, the torque was about 35.2 N.m, the output power was 1.105 kW, the phase voltage was about 322.54 V, and the efficiency was 90.3%. The results showed good agreement with the basic design requirements. However, as parts of the design parameters were designed with empirical value, the design results could still be further optimized for actual practical needs.

3. Optimization Model and Sensitivity Analysis

3.1. Optimization Model of the AFPMSG The optimization of a motor includes the performance, economy, and other special needs. Structure parameters are the main optimization objects of a motor, and optimization is to adjust the structural parameters to get the best results so that some performance or indicators of the motor can be optimized. Under the condition of keeping the shape and structure of the AFPMSG unchanged, the main structural parameters affecting the performance and economy were as follows: inner-to-outer diameter ratio, γ, effective air gap length, δ, the thickness of stator, δw, the thickness of PM, hM, rotor yoke thickness, hr, and polar arc coefficient, αp. When γ increased, the effective material consumption and performance of the AFPMSG decreased, while the economy improved. In order to maintain the flux density of the air gap, the longer the effective air gap was, the more PM materials were needed, which would make the generator larger. Additionally, the thickness of the permanent magnet and rotor yoke was mainly to meet the needs of the air gap magnetic field of the AFPMSG, to ensure that the magnetic circuit was set at the saturation critical point and the yoke structure was not deformed. The first priority of optimization is determining the optimization objective. According to Section2, the electrical performance of the initially designed AFPMSG could basically meet the design requirements, but the design of the structural parameters were rough and the economy of the AFPMSG could be improved. The total material cost of the AFPMSG could obtained from the structure and material used in the generator, as expressed by:

Cm,total = Ccumcu + CFemFe + CPMmPM (10) Sustainability 2019, 11, 1414 9 of 18

where mcu, mFe, mPM, Ccu, CFe, and CPM are, respectively, the mass and price of the copper, steel, and PM material. The mass of each material was computed based on the material volume and the corresponding mass density. In the calculations, the material price of copper, steel, and PM were considered equal to 15, 3, and 80 Ero/kg, respectively [29,33]. In view of the material price, the PM material cost was the main part of the production cost for the stator coreless AFPMSG. Meanwhile, the processing and manufacturing of PM was much more difficult, and it was easy to waste PM material, which lowered the economy of the AFPMSG. Therefore, the objective function of PM volume consumption was established for economic optimization in this paper, as given below. π min : f (X) = α D2 1 − γ2 h (11) 4 p out M The optimization of the AFPMSG was carried out on the premise of ensuring that the electrical performance and external structure remained unchanged. Therefore, the outer diameter and axial length was taken as the structure constraints, while the electrical performance parameters, such as flux density of air gap and rotor yoke, efficiency, etc., were taken as the electrical requirements. The constraints function could be obtained as follows:  g (X) = η0 − η ≤ 0  1  g (X) = B − B ≤ 0  2 δ0 δ  g (X) = B − B ≤ 0 3 r0 ro0 (12) g (X) = D − D ≤ 0  4 max 0   g5(X) = hmax − h0 ≤ 0  g6(X) = P0 − P ≤ 0 where η0, Bδ0, Br0, D0, h0, and P0 are the initial design value of efficiency, flux density of air gap and rotor yoke, size of generator diameter, PM thickness, and output power, respectively. Therefore, the multi-parameter AFPMSG economic optimization problem can be transformed into a mathematical model, with the volume consumption of permanent magnets as the objective function.

 ( ) = π 2 − 2  min : f X 4 αpDout 1 γ hM  T  X = D , h , α , γ, h , δ out r p M (13)  s.t.Bδ > Bδ0; hmax ≤ h0; Dmax ≤ D0   η ≥ η0; Br0 ≤ Bro0; P ≥ P0

3.2. Sensitive Analysis for Design Parameters The optimization design of the AFPMSG was a process of repeatedly modifying parameters. In order to eliminate the factors that had little inﬂuence on the objective function and reduce the time spent in modeling and optimization, it was necessary to examine the correlation between design parameters and objective functions by sensitivity analysis before optimization began. The optimum design of the AFPMSG was performed based on ensuring the demand of electrical performance. Material consumption and economy of the AFPMSG are also affected by the relationship between electrical performance and structural parameters. Therefore, the sensitivity of the AFPMSG structure parameters to the performance were mainly analyzed in this paper. Considering the complexity and inaccuracy of the AFPMSG design calculation, this paper combined analytic calculation with ANSYS simulation. Assume the performance function is y = f (X), the AFPMSG parameter is 0 00 X, their range of variation is xi, xi , and the corresponding performance dependent variable is y, 0 00 the range of variation y is yi, yi . The relative variation of the dependent variable and independent Sustainability 2019, 11, 1414 10 of 18 variable are ∆X/X and ∆y/y, respectively. The sensitivity of the AFPMSG performance to the parameters is expressed as follows:

∂y ∂X X ∂y SY = / = (14) X y X y ∂X

For linear y = f (X), the partial derivative ∂ f /∂xi is constant, and the range of performance independent variables varies: n 0 0 0 y = f u = a0 + ∑ aiui (15) i=1 n 00 00 00 y = f (u ) = a0 + ∑ aiui (16) i=1 where in the formula ai is a linear coefficient and ui is an extreme point. The corresponding range 0 00 of variation to xi is ui, ui . The performance function of the AFPMSG structural parameters is y a multi-variable non-linear function. In this paper, the differential sensitivity s is obtained by Xi taking linear approximation at the design value X0, and the variation range of the dependent variables y is obtained. n y y0 ≈ y + S u0 − x (17) 0 ∑ Xi i i0 i=1 n y 00 y00 ≈ y + S u − x (18) 0 ∑ Xi i i0 i=1 where y0 = f (X). The influence of different parameters on the AFPMSG performance can be obtained from the above formula. In order to analyze the sensitivity of the structural parameters to the performance of the dual rotor coreless AFPMSG, analytical calculation and finite element simulation were used, due to the complicated flux distribution and the interaction of multi parameters. Figure9 shows the sensitivity of the main structural parameters to the electrical performance, including inner-to-outer diameter ratio, γ, air gap length, δ, thickness of PM, hM and rotor yoke, hr, and the pole arc coefficient, αp. It can be seen that some structural parameters, like diameter ratio, γ, and PM thickness, hM, improved the output performance with the increasing of parameter value, while parameters such as air gap length, δ, decreased the performance as the parameter value increased. As shown in Figure9a,e, there was a critical vale of the diameter ratio, γ, and rotor thickness, hr, before which the relation between them and the output performance of the AFPMSG was nearly linear. Form Figure9b, increasing the air gap length reduced the rated output of the AFPMSG and the utilization ratio of the permanent magnet, but the appropriate air gap length was conducive to reduce the torque fluctuation and make the generator run more smoothly. The increase of pole arc coefficient, αp, added the area of PM and magnetic flux of each pole, but at the same time also increased the leakage flux between each magnetic pole. The influence of pole arc coefficient, αp, to the output performance of the AFPMSG is shown in Figure9c. It is obvious that the relationship between PM thickness and generator performance was monotonically increasing in Figure9d. Moreover, the influence degree of the AFPMSG structural parameters on the output performance and PM material consumption could be obtained and arranged in sequence as follows: air gap length, permanent magnet thickness, pole arc coefficient, internal and external diameter ratio, and rotor yoke thickness. The sensitivity analysis of the structural parameters obtained by simulation was consistent with the theoretical analysis, which is helpful to understand and optimize the AFPMSG. Sustainability 2019, 11, x FOR PEER REVIEW 10 of 18

≈ + ( − ) (18) where = (). The influence of different parameters on the AFPMSG performance can be obtained from the above formula. In order to analyze the sensitivity of the structural parameters to the performance of the dual rotor coreless AFPMSG, analytical calculation and finite element simulation were used, due to the complicated flux distribution and the interaction of multi parameters. Figure 9 shows the sensitivity ofSustainability the main2019 structural, 11, 1414 parameters to the electrical performance, including inner-to-outer diameter11 of 18 ratio, , air gap length, , thickness of PM, ℎ and rotor yoke, ℎ, and the pole arc coefficient, .

Amplitude Voltage Average Torque Amplitude Voltage 350 Average Torque -32 -16 340 345 -18 330 -20 320 -22 -34 310 -24 330 300 -26

290 -28 -30 280 315 -36 -32 270 -34 260 Amplitude Voltage (V) Amplitude Voltage (V)

-36 Average Torque (N·m) Average Torque (N·m) 250 -38 300 -38 240 -40 230 -42

1.4 1.6 1.8 2.0 2.2 0.6 0.8 1.0 1.2 1.4 Ratio of outer to inner diameter airgap length (mm)

(a) (b)

Amplitude Voltage Amplitude Voltage 325 Average Torque -25 375 Average Torque -22

-24 360 320 -26 345 -28 315 -30 330 -30 310 -32 315 -34 305 300 -36 -35 -38 300 285 Amplitude Voltage (V) Amplitude Voltage (V) Average Torque (N·m) -40 Average Torque (N·m) 270 295 -42

255 -44 290 -40 0.65 0.70 0.75 0.80 0.85 0.90 0.95 6 8 10 12 14 16 pole arc coefficients PM thickness (mm)

0.70 Average gap mag_B 4 Max yoke mag_B

0.68

0.66

0.64 Max yoke mag_B (T) Max Average gap mag_BgapAverage (T) 0.62

0.60 0 4 6 8 10 12 14 Rotor yoke thickness (mm)

(e)

FigureFigure 9. 9. TheThe sensitivity sensitivity of of the the main main structural structural parameters parameters to to the the electrical electrical performance. performance. (a (a) ) The The sensitivitsensitivityy of of diameter diameter ratio ratio,, γ,, to ( b) thethe sensitivity ofof airair gapgap length,length,δ , to, to the the output output performance performance of ofthe the AFPMSG, AFPMSG (c, )( thec) the sensitivity sensitivity of poleof pole arc arc coefﬁcient, coefficientαp,, to (d, )to the (d) sensitivity the sensitivity of PM of thickness, PM thicknesshM, to, ℎthe, outputto the output performance performance of the AFPMSG; of the AFPMSG (e) the sensitivity; (e) the sensitivity of rotor yoke, of rotorhr, to yoke the output, ℎ, to performance the output performanceof the AFPMSG. of the AFPMSG.

4. Improved Genetic Algorithm It can be seen that some structural parameters, like diameter ratio, , and PM thickness, ℎ , improveMulti-parametersd the output performance and the interaction with the increasing between of parameters parameter makevalue, thewhile design parameters and optimization such as air gapof the length AFPMSG, , decrease mored complicated. the performance It is difficultas the parameter to optimize value the increase designd and. As improveshown in the Figure economy 9a,e, on the basis of ensuring the output electrical performance. As discussed previously, the economic optimization of the AFPMSG is a multi-parameter and non-linear optimization process. The selection of the optimization algorithm has a significant impact on the optimization results and efficiency. Multi-parameter non-linear optimization algorithms are mainly the direct search method and random search method. Modern heuristic motor optimization methods, such as genetic algorithm (GA) and simulated annealing (SA), overcome the shortcomings of initial conditions, local optimization, and difficult multi-dimensional considerations, and have been rapidly developed and applied. In this paper, a hybrid genetic algorithm, which combines the simulated annealing method and father–offspring hybrid selection method, is proposed based on a traditional GA. GA is a random search algorithm Sustainability 2019, 11, 1414 12 of 18 based on the natural selection and natural genetic mechanism to search the optimal solution of the problem [34,35]. It has the advantages of simple process, high degree of parallelism, strong global search ability, and scalability. SA comes from the simulation of cooling in solid annealing. Its basic idea is to use a thermodynamic system to represent the optimization problem of demand solution, and to simulate the optimization process by gradually cooling the system to the lowest energy state. In the process of optimization, it has the ability of a probability jump, which can avoid falling into the local optimum solution, and makes up for the deficiency of GA in local optimization. In this paper, exponential annealing was used to reduce the temperature, as shown in Equation (19), and the Metropolis criterion shown in Equation (20) was used to judge whether to accept new solutions or not. The iteration process of “generating new solutions, judging, accepting or abandoning” was realized to find the optimal solution at this temperature. i−1 Ti = T0k (19) ( 1, f (Y0) ≥ f (Y) p = n −[ ( )− ( 0)] (20) exp f Y f Y , f (Y0) < f (Y) Ti where Ti is the temperature of the current annealing process, k is the attenuation coefficient, T0 the initial stability, and Y and Y0 are the current solution and the new solution, respectively. p is used to determine whether the new solution is acceptable or not. The father–offspring selection method improves the algorithm and adds stochastic factor control parameters so that the best individuals of the previous generation can be inherited, unconditionally, to the next generation. Therefore, the fitness sequence must be monotonic, that is, the evolutionary algebra is monotonic. The algorithm is the execution strategy of the genetic algorithm to ensure convergence, and it solves the problem that the traditional genetic algorithm may miss the global optimum, etc. The steps to implement the method were as follows: Step 1: (Initialization) Random generation of the initial population of M pairs, X(0), t = 0; Step 2: Population evolution:

(1) Generate intermediate population Y(t) by crossing M pairs in X(t) independently; (2) Execute mutation independently for each intermediate individual in Y(t) to generate progeny population Z(t); (3) Select M pairs’ mother population as the new generation population X(t + 1) from the union of parent population X(t) and offspring population Z(t);

Step 3: End and output the optimal individual in X(t + 1) after termination criterion is satisfied. In this paper, improved GA was proposed for the problems that the algorithm itself has such as slow convergence and premature convergence. SA was applied in the fitness function to improve convergence speed and avoid premature convergence, while the father–offspring hybrid selection method was used to ensure that the best individuals of the previous generation could be inherited to the next generation, to realize elite legacy and improve global accuracy. The improved GA calculation flowchart is shown in Figure 10. The main steps of the improved hybrid GA were as follows: Step 1: Set up the initial parameters, such as generator variables, random control parameters, population size, N, maximum evolutionary algebra, Mmax, initial and minimum annealing temperatures, T0 and Tmin, and temperature attenuation coefficient, k. Initial population is generated randomly. Step 2: Carry out fitness calculation and evaluation, and annealing operation is adopted to produce new solutions. Using Metropolis criterion to judge whether to accept or not, and according to the maximum number of iterations and the minimum annealing temperature to select the calculation program, if not satisfied, go to step 3; when satisfied, stop iteration, jump to step 4. Sustainability 2019, 11, x FOR PEER REVIEW 12 of 18

convergence, and it solves the problem that the traditional genetic algorithm may miss the global optimum, etc. The steps to implement the method were as follows: Step 1: (Initialization) Random generation of the initial population of pairs, (0), = 0; Step 2: Population evolution: (1) Generate intermediate population () by crossing pairs in () independently; (2) Execute mutation independently for each intermediate individual in () to generate progeny population (); (3) Select M pairs’ mother population as the new generation population ( + 1) from the union of parent population () and offspring population (); Sustainability 2019, 11, 1414 13 of 18 Step 3: End and output the optimal individual in ( + 1) after termination criterion is satisfied. In this paper, improved GA was proposed for the problems that the algorithm itself has such as Step 3: Renew the annealing temperature, and obtain the intermediate generation population slow convergence and premature convergence. SA was applied in the fitness function to improve through genetic, crossover, mutation, and other operations. Combine the parent population with the convergence speed and avoid premature convergence, while the father–offspring hybrid selection intermediate population; then select and arrange individuals according to ﬁtness and form of the next method was used to ensure that the best individuals of the previous generation could be inherited to generation population with individuals generated randomly. Go to step 2. the next generation, to realize elite legacy and improve global accuracy. The improved GA calculation Step 4: Take the ﬁrst individual in the current population as the optimal solution of the problem, flowchart is shown in Figure 10. and the calculation ends.

Start

Parameter analysis of motor Consideration of practical economy optimization and performance restrictions

Determining Coding Rules, Variable Object Functions and Constraints

Generating Initial Population X(t)

Fitness calculation and Maternal population X (t+1); evaluation Optimal Individual Retention

Simulated annealing Individuals operation based on fitness Progeny generated population Z(t) randomly

Paternal population Judgment of Acceptance of Selection, X(t); optimal New Solutions Based on crossover, individual preservation Metropolisz Criterion mutation

Renewal annealing Gen

Obtain the best fitness Export results Over

Figure 10. Flowchart of hybrid genetic algorithm (GA). Figure 10. Flowchart of hybrid genetic algorithm (GA). 5. Optimal Design and Finite Element Analysis The main steps of the improved hybrid GA were as follows: BasedStep 1: onSet the up initialthe initial design parameters scheme, andsuch optimization as generator modelvariables, of therandom AFPMSG, control the parameters, structural parameters of the generator were optimized by using the hybrid genetic algorithm programmed by population size, , maximum evolutionary algebra, , initial and minimum annealing Matlab. The optimized dimension parameters were obtained and compared as shown in Table2. temperatures, and , and temperature attenuation coefficient, . Initial population is generated randomly. Table 2. Improved dimensions of AFPMSG.

Values Structural Parameters of AFPMSG Initial Design Optimized Design Outer diameter (mm) 380 380 Inner diameter (mm) 210 227.5 Thickness of PM (mm) 10 9.3 Pole arc coefﬁcient 0.8 0.77 Air gap length (mm) 1.2 1 Thickness of stator (mm) 9 9 Rotor yoke thickness (mm) 10 8.6 Sustainability 2019, 11, 1414 14 of 18

Finite element analysis (FEA) was proceeded on the AFPMSG to evaluate the improved design presented in the previous sections. 3D models of AFPMSG were developed and simulated by Ansys Maxwell 16.0 software based on the improved dimensions of the AFPMSG in Table2, as shown in Figure7. The models before and after optimization were set and simulated with the same conditions. The electrical performance was the first concern, and the consumption of the main material was the target for optimization. Figure 11 shows the distribution of the magnetic flux density along the circumference and comparison of the magnetic flux density (Mag B) curve under one pole arc. It can be seen that the distribution of Mag B presented a flat top wave under one pole arc and was periodically distributed along the circumference direction. The average value of Mag B changed little between the initial design and optimal design. As mentioned in Section4, the decrease of the rotor yoke thickness Sustainabilityincreased the2019 value, 11, x FOR of thePEER max REVIEW flux density in the rotor yoke, while the air gap flux density14 almost of 18 remains unchanged under the integrated influence of the air gap length and PM thickness decreasing. Sustainability 2019, 11, x FOR PEER REVIEW 800 14 of 18

700 800 600 700

500600

400500 Mag B for optimizaiton design 300400 Mag BB for for optimizaiton initial design design 300 Mag B(mT) Mag g B(mT) 200 Mag B for initial design

Ma 200 100 100 0 0 -100 -100420 430 440 450 460 470 480 420 430 440 450 460 470 480 Distance(mm) Distance(mm) (a) (a) (b()b )

FigureFigu 11.11.re The 11. simulation The simulation results results results of of ofMag Mag B. B. ( a ()a )The The distributiondistribution ofof Mag Mag B Balong along the the circumference circumference directiondirection;direction;; ( b) the (b ) comparison the comparison of Magof Mag B curveB curve under under one one polepole arc.

ThreeThree-phaseThree-phase-phase voltage voltage and and torque torque of ofthe the generator generator areare important aspects aspects of of output output performance, performance, whichwhich are are discussed discussed are discussed in in the in the thepaper. paper. paper. The The induced induced induced voltage voltage voltage andand andtorque torque of of the the ofoptimal optimal the optimal design design under design under rated underrated load was achieved and compared with the initial design, as shown in Figure 12. Obviously, the output loadrated wa loads achieved was achieved and compared and compared with the with initial the design, initial as design, shown asin Fi showngure 1 in2. FigureObviously, 12. Obviously,the output phase voltage of the optimal design still had a good sinusoidal curve and the torque performed phasethe output voltage phase of the voltage optimal of the design optimal still design had a good still had sinusoidal a good curve sinusoidal and the curve torque and performthe torqueed smoothly. The comparison of the induced voltage and torque curves showed that the output smoothly.performedperformance The smoothly. comparison of the The AFPMSG comparison of the after induced optimization of the voltage induced was almost and voltage torque the andsame curves torquewith the show curves initialed design, showedthat the which that output the performanceoutputcan performance satisfy of the the AFPMSG requirements of the AFPMSG after of optimizat the after design optimizationi on purpose. was almost Besides, was the almost the same results the with same were the with consistent initial the design, initial with design,thewhich whichcan satisfycomparison can satisfy the requirements of the Mag requirements B. of the of design the design purpose. purpose. Besides, Besides, the the results results were were consistent consistent with with the comparison of Mag B.

(a) (b)

Figure 12. The comparison of the AFPMSG output performance. (a) Output-induced voltage waveform; (b) output(a) torque waveform. (b)

Figure 12. 12.The The comparison comparison of the of AFPMSGthe AFPMSG output output performance. performance. (a) Output-induced (a) Output voltage-induced waveform; voltage waveform(b) output; torque(b) output waveform. torque waveform.

Sustainability 2019, 11, 1414 15 of 18

The economy of AFPMSG was improved through the consumption optimization of effective material, especially the PM, and the comparison of the electrical performance and the main material consumption are shown in Figure 13. The efficiency and output power of the AFPMSG also changed little, while the volume of the main material reduced a lot, as the calculation results shown in Table3. It was observed that the whole volume of the rotor yoke and PM for the AFPMSG was reduced obviously after optimization, 19% and 15.8%, respectively. The results show that economic optimization of the AFPMSG, through the main material volume cost model and improved GA method, can be achieved on the condition of keeping the electrical performance unchanged. It was clear that the 3D FEA results confirmed the economic optimization of the AFPMSG. The validity of both the improved initial design procedure with MEC, combining the static magnetic field FEA, and the economic optimization design through main material consumption model and improved GA, were confirmed by the results and analysis in the paper, which can be applied or used as a reference for better design. SustainabilityIn addition, 2019 the, 11 research, x FOR PEER of this REVIEW paper helps a lot for the further understanding of the AFPMSG.15 of 18

Figure 13. TheThe comparison of electrical performance and main material consumption consumption..

Table 3. Comparison of output performance and main material consumption. The economy of AFPMSG was improved through the consumption optimization of effective materialGenerator, especially Parameters the PM, and the Before comparison Optimization of the electrical After Optimization performance and Change the main Rate material consumptionInduced are voltageshown (V)in Figure 13. The 322.54 efficiency and output power 316.06 of the AFPMSG−2% also changed little, whileAverage the volume torque (N of·m) the main material 35.2 reduced a lot, as the 34.4 calculation results− 2.27%shown in Table 3. It was observedOutput power that (kW) the whole volume 1.105 of the rotor yoke and 1.08PM for the AFPMSG−2.26% was reduced obviously afterEfficiency optimization, 19% and 91.3% 15.8%, respectively. 92% The results show 0.76% that economic 3 228.6 228.9 0.13% optimizationCopper of the (mm AFPMSG) , through the main material volume cost model and improved GA Fe (mm3) 1543 1249 −19% method, can bePM achieved (mm3) on the condition1235 of keeping the electrical 1040 performance unchanged.−15.8% It was clear that the 3D FEA results confirmed the economic optimization of the AFPMSG. The validity of both the improved initial design procedure with MEC, combining the static magnetic field FEA, and 6. Conclusions the economic optimization design through main material consumption model and improved GA, wereIn confirmed this paper, by modified the results initial and designanalysis procedure in the paper, and which economic can optimizationbe applied or design used as were a referenc studiede forto improve better design. the design In addition, efficiency the and research economy of this of an paper AFPMSG. helps Toa lot improve for the the further accuracy understand and efficiencying of theof initial AFPMSG. design, the MEC method and static magnetic field FEA were combined in the initial design procedure, which solved the problems of inaccuracy of electromagnetic parameters and too many iteration timesTable in the 3. MEC Comparison method, of output by employing performance the and advantages main material of the consumption. static magnetic field FEA (i.e., accuracy and less time-consumption). For the economic optimization design, the PM material Generator Parameters Before Optimization After Optimization Change Rate volume model, which concerned the cost of the AFPMSG the most, was derived and taken as the Induced voltage (V) 322.54 316.06 −2% optimizationAverage torque object. (N·m) The sensitivity analysis35.2 was introduced and34.4 used to distinguish and−2.27% sort the performanceOutput power influence (kW) degree of the main1.105 structure parameters in the1.08 model for further optimization−2.26% calculation.Efficiency Considering the different91.3% influence degrees and multi-parameters’92% interactions,0.76% the improvedCopper GA, which(mm3) combined the simulated228.6 annealing method and228.9 father-offspring hybrid0.13% selection method, wasFe (mm studied3) and implemented to1543 search for the best solution1249 of economic optimization.−19% PM (mm3) 1235 1040 −15.8%

6. Conclusions In this paper, modified initial design procedure and economic optimization design were studied to improve the design efficiency and economy of an AFPMSG. To improve the accuracy and efficiency of initial design, the MEC method and static magnetic field FEA were combined in the initial design procedure, which solved the problems of inaccuracy of electromagnetic parameters and too many iteration times in the MEC method, by employing the advantages of the static magnetic field FEA (i.e., accuracy and less time-consumption). For the economic optimization design, the PM material volume model, which concerned the cost of the AFPMSG the most, was derived and taken as the optimization object. The sensitivity analysis was introduced and used to distinguish and sort the performance influence degree of the main structure parameters in the model for further optimization calculation. Considering the different influence degrees and multi-parameters’ interactions, the improved GA, which combined the simulated annealing method and father- Sustainability 2019, 11, 1414 16 of 18

Employing the initial design procedure and economic optimization design, a 1 kW 300 r/min AFPMSG with two outer rotors and an inner coreless stator was designed and optimized on the condition of keeping the electrical performance meeting the requirements. The results show that the output voltage of the AFPMSG was very close to the sinusoidal waveform, and the torque ripple was small, which were in accordance with the structure characteristic of the AFPMSG. Moreover, the effective material cost was reduced, especially the rotor yoke and PM, while the electrical performance of the AFPMSG changed little. The 3D FEA of the generator designed by the procedure proposed in the paper, and comparison before and after optimization, illustrates the validity of the study in the paper. The improved design and analysis proposed in this paper are beneﬁcial to the improvement and further understanding of the AFPSMG design. In the future, thermal behavior, control, other optimizations for application conditions, a speciﬁc detailed design for prototype manufacture, and a prototype experiment will be considered in the subsequent research process.

Author Contributions: W.W. and Y.W. designed the main parts of the research, including objective function establishment and optimal methodology design. S.Z. was responsible for guidance, a number of key suggestions, and manuscript editing. J.G. and H.M. mainly contributed to the writing of the paper and were also responsible for project administration. H.L. and G.Z. made contributions to simulations in Ansys Maxwell and gave the final approval of the version to be published. Funding: This research received no external funding. Acknowledgments: The authors would like to acknowledge the technological and financial support from “The National Science and Technology Support Program (supported by the Ministry of Science and Technology of P.R.C. No. 2014BAC01B05)”, “Key projects of PLA (BY113C001)”, “Army Funding for military graduate students (2016JY486)”, and “2017 Military Logistics Open Research and Research Projects (AY117J004)”. The authors are grateful that comments and suggestions provided by the anonymous reviewers and editor helped to improve the quality of the paper. Conflicts of Interest: The authors declare no conflicts of interest.

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