design optimization of direct-coupled ironless axial flux

IEEE TRANSACTIONS ON MAGNETICS, VOL. 00, NO. 00, APRIL 2016- Under Publish

1

Design Optimization of Direct-Coupled Ironless Axial Flux

Permanent Magnet Synchronous Wind Generator with Low Cost and

High Annual Energy Yield

Ali Daghigh, Hamid Javadi, Hossein Torkaman Faculty of Electrical Engineering, Shahid Beheshti University, A.C., Tehran, Iran

In this study, an improved design of an ironless Axial Flux Permanent Magnet Synchronous Generator (AFPMSG) is presented for

direct-coupled wind turbine application considering wind speed characteristics. Partial Swarm Optimization (PSO) method is used to

perform a multi-objective design optimization of the ironless AFPMSG in order to decrease the active material cost and increase the

Annual Energy Yield (AEY) of the generator over the entire range of operating wind speed. General practical and mechanical

limitations in the design of the generator are considered as optimization constraints. For Accurate analytical design of the generator,

distribution of the flux in all parts of the machine is obtained through a modified Magnetic Equivalent Circuit (MEC) model of

AFPMSG. In this model, the magnetic saturation of the rotor back iron cores is considered using a nonlinear iterative algorithm.

Various combinations of pole and coil numbers are studied in the design of a 30 kW AFPMSG via the optimization procedure. Finally,

3-D Finite Element Model (FEM) of the generator was prepared to confirm the validity of the proposed design procedure and the

generator performance for various wind speeds.

Index Terms— Axial Flux Permanent Magnet Synchronous Generator (AFPMSG), ironless, wind speed characteristics, Annual

Energy Yield (AEY), Finite Element Analysis (FEA).

I. INTRODUCTION

ecently, wind turbines have received particular attention

globally, as a source of renewable energy. In rural and

remote areas from the network, the use of small-scale wind

systems can provide reduced connection cost while

eliminating transmission and distribution losses. Considering

the advantages of gearless systems, many manufactures use

direct-coupled generators [1] that enhance the overall

efficiency and reliability of the system. However, due to the

generator operation at low speed, large numbers of poles are

required that results in large diameter of the generator [2].

Axial Flux Permanent Magnet Synchronous Generator

(AFPMSG) with high ratio of generator diameter to generator

length is one of the appropriate choices in direct-coupled wind

turbine applications [3]. Depending on design characteristics

and according to the material used in the stator core,

AFPMSGs are classified into two types: iron-cored and

ironless generators. Compared with iron-cored generators,

ironless generators have the advantages of light weight, high

efficiency, no cogging torque, and simple construction.

Moreover, in ironless generators, considering the negligible

attraction force between the stator and the rotor, the structural

mass of the generator is of light weight and provides reliable

design for large generator diameters. However, because of the

large effective length of the air gap, which requires a larger

amount of PM material, in these generators, the active material

cost is higher than iron-cored machines. In other words, the

advantages of lower structural mass and higher generator

diameter are obtained at the cost of higher active material

price.

By increasing the application of AFPMSGs in direct-

coupled wind turbines, the analysis of their design and

optimization procedure has been an active field of recent

research. A hybrid method based on analytical and FEA has

been proposed for optimal design of a high speed ironless

stator AFPMSG in [4]. In [5], an optimal design of Radial

Flux PM (RFPM) wind generator has been presented for the

maximum annual energy production, without considering the

generator active material cost. Design optimization of an iron-

cored RFPM wind generator has been investigated in [6];

however the design procedure was not implemented properly.

Stamenkovic et al. investigated the design, analysis, and

graphical optimization of the ironless AFPMSG in [7], with a

proper parametrization study of coil dimensions.

The computer aided techniques in the design of electrical

and electromechanical machines were taken in to

consideration, in many researches [8-10]. The development of

field computation and multi-objective optimal design in

electromagnetics were presented in [11, 12]. In [13] a novel

procedure was proposed to combine the multiple objectives

into a single objective in the optimization of electromagnetic

design problems. Lowther reviewed the development and

industrial design implementation of two branches of analysis

tools, magnetic circuits and field solvers, and also investigated

the capability of combining these analytical systems in [14].

Recently, the use of heuristic algorithms has increased in the

design optimization procedure of Axial Flux Permanent

Magnet (AFPM) machines. Improved design of 30 kW iron-

cored AFPMSG using Genetic Algorithm (GA) has been

presented in [15] for minimizing the active material cost.

Some practical limitations and mechanical constraints have

been taken into account but the full analysis results were not

provided. Mahmoudi et al. evaluated an optimized AFPM

motor using GA for maximum power density and low cogging

torque [16]; however, the cost of the machine was not

considered.

In an iterative design procedure like optimal design of

R

Manuscript received Dec. 15, 2015 (date on which paper was submitted for review). Corresponding author: H. Torkaman (e-mail:

[email protected]).

Digital Object Identifier inserted by IEEE


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ironless AFPMSG, effective air gap length varies by the stator

axial length variation; thus, in this case, the Permanent Magnet

Leakage Flux (PMLF) factor and PM axial length are

changed. In the iron cored machines, stator coils are placed in

the slots and therefore, the air gap length and the leakage flux

factor value don’t change by the stator axial length variation.

The accurate prediction of leakage flux coefficient is essential

for the correct calculation of average flux density in the air

gap. For the iterative design procedure of linear permanent

magnet motors, an improved MEC model was established in

[17]. Ronghai et al. observed the analysis and modeling of

PMLF for the iron-cored AFPM machine [18] in which the

iron saturation is ignored. The leakage flux consideration in

modeling of high speed ironless AFPMSG was presented in

[19]. However, the PMLF model of the machine was unclear

and the leakage flux was only considered in the inner and

outer radius of the machine. Although, the consideration of

saturation factor in the rotor discs has been considered in the

case of radial flux machines [20], but this idea has not been

implemented for the ironless AFPM machines before.

In this paper, a multi-objective design optimization of

ironless AFPMSG is considered for direct-coupled wind

turbine applications. A multi-objective function was defined

and implemented through the PSO algorithm to reduce active

material cost and increase the Annual Energy Yield (AEY) of

the generator. In order to calculate AEY of the generator over

the whole operating wind speeds, wind turbine characteristics

and the statistical distribution of the wind speed are

considered. For accurate calculation of the PMLF factor, a

modified MEC model of the ironless AFPMSG was presented

based on the quasi-3-D model of the generator. The magnetic

saturation of rotor back iron cores was considered and a

nonlinear iterative procedure was used to update the

permeability of saturable parts. In the following, wind turbine

and wind speed characteristics are given in Section II.

Analytical design procedure and modified MEC model of the

generator are presented in Section III. Section IV introduces

the design optimization procedure of AFPMSG using PSO

algorithm. In Section V, the validity of the improved design

characteristics is evaluated by 3-D FEA. Finally, conclusions

are given in Section VI.

II. WIND TURBINE AND WIND SPEED CHARACTERISTICS

The improved design of an AFPM generator for wind

turbine application needs to consider the characteristics of

wind speed, wind turbine, and control method. Currently, most

turbines have control systems to operate at maximum power

point conditions for producing more annual energy as a result

of changes in wind speed. Hence, in the design stage, beside

the other design considerations and requirements, it is so

important to consider generator performance not only at the

specific wind speed, but also over the whole range of the

operating wind speed.

A. Wind Turbine Characteristics

For the proper design of a generator used in wind turbine

applications, it is essential to consider the wind turbine

characteristics. The output power of a wind turbine (Pshaft),

which is equal to input power of the generator (Pin) in direct

drive applications can be expressed as follows [5]; 3

w

2

bpairinshaft),(0.5= vRβλCρPP (1)

where ρair is air density, Rb is radius of the turbine blade, vw is

wind speed, and Cp(λ,β) is the power coefficient, which is a

function of tip speed ratio (λ) and turbine pitch angle (β).

Different practical equations for Cp(λ,β) are proposed in [21].

The following equation has been used in this paper.

λcecβcλ

ccβλC

λ

c

643

i

2

1p

i

5

)(),(

(2)

1

035.0

08.0

113

i

ββλλ

(3)

where c1=0.5176, c2=116, c3=0.4, c4=5, c5=21, and c6=0.0068.

The variation of the output power of the turbine is shown in

Fig. 1 considering the rotational speed, and different wind

speeds. The tip speed ratio is defined as (4);

w

bm

v

Rωλ (4)

Currently, most wind energy conversion systems are

controlled via the Maximum Power Point Tracking (MPPT)

technique. Generally, in this method, below the rated speed of

the generator, Cp-opt=Cp-max is achieved for an optimum value

of λopt and β=0. However, the output power of the generator is

kept constant for higher wind speeds via pitch control (β≠0).

The correct calculation of the generator rated speed is very

important according to the average wind speed and other

characteristics of the wind turbine and can be calculated for

optimum values of Cp-opt and λopt, from (1) and (4) as follows;

out

5

wopt-p

2

optair

m2P

ηvCλρω

(5)

where η is the generator efficiency and Pout is the generator

output power (Pout= η Pshaft). The specifications of the wind

turbine are given in Table І. The selected turbine is an

adjustable pitch one, considered to control the system for wind

speeds upper the rated speed. The value of rated wind speed

(vw) depends on the average wind speed of the area, which is

considered in this paper within the range of 11-12 m/s

according to the specifications given by the wind generator

companies for 30 kW wind turbines [22-25]. Also, the specs

given in Table I are calculated based on equations given

above, and by considering the specifications of wind turbine

companies.

B. Wind Speed Characteristics

Considering the generator performance over the complete

operating wind speed range, it is required to consider the

characteristics of the wind speed. The amount of energy

generation from a wind turbine depends on the wind speed

distribution of the turbine installation area. Rayleigh

distribution which is commonly used in the wind data analysis

has considered taking into the account the characteristics of

various wind speeds in the design of the generator.


3

The probability density function P(vw) for the average wind

speed of the area vav is given by (6) and is shown in Fig. 2 for

different values of mean wind speed. 2

av

w )(4

2

av

w

w)(

2)(

v

v

ev

vvP

(6)

The use of AEY conception is one of the methods that can

be used to examine generator performance over a wide speed

range. In this case, instead of calculating the amount of output

power or generator efficiency at a certain speed, the AEY of

the generator is considered in the design procedure. The total

amount of generated energy can be calculated from (7) by

multiplying the output power and the effective operating hour

at the given wind speed [5].

)()(=wwout

vHvPAEY (7)

where H(vw) can be computed using wind distribution as

follows;

www)( 8760=)( vvPvH (8)

III. ANALYTICAL DESIGN

Analytical design of AFPMSG based on the sizing

equations and the improved MEC model of the machine is

considered in this section. The considered AFPMSG consists

of two parallel rotors and one inner ironless stator.

Trapezoidal permanent magnets are placed on the two rotor

surfaces and the non-overlapping concentrated winding coils

in the stator are held together by the use of composite material

of epoxy resin. Structure of the ironless AFPMSG is shown in

Fig. 3.

A. Main Design Equations

Sizing equations of ironless AFPMSG have been presented

in [26]. The main sizing equation that shows the relationship

between the output power (Pout) and generator diameter (Dout)

is expressed in (9).

)cos()1()1(32

2d

2

dg

3

sw1i

3

out

ηkkABDnkα=P

out (9)

where A, Bg, ns, αp, kw1, and kd are electrical loading, flux

density of the air gap, rotational speed, magnet width to pole

pitch ratio, winding factor, and the inner to outer diameter

ratio, respectively.

The performance of the ironless AFPMSG with different

types of concentrated coil winding has been studied in [27]

and the winding specifications are calculated analytically. In

order to reduce the effective air gap length of the generator,

the thickness of the winding is decreased as much as possible.

In this case, the limited space to place the winding in the inner

radius of the generator is the most important limitation. The

minimum axial length of the winding (sd), with considering

space factor ks for sufficient mechanical strength of the stator

disc can be found as (10). Regarding the practical conditions,

the axial length of the resin stator disc (Ls) is assumed to be

greater than the axial length of the coils, which is expressed in

(11) and shown in Fig. 4. ls is the axial increment on each side

of the stator resin disc that is 0.5 mm in this study.

(a)

(b) (c)

Fig. 3. Structure of the ironless AFPMSG, a) 3-D structure b) Rotor and PMs

structure c) Stator windings

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

Wind Speed (m/s)

Pro

ba

bil

ity

4 m/s

5.5 m/s

7 m/s

8.5 m/s

10 m/s

11.5 m/s

13 m/s

Fig. 2. Rayleigh distribution for different mean wind speeds

0 50 100 150 200 250 3000

10

20

30

40

50

Rotor Speed (rpm)

Tu

rbin

e O

utp

ut

Po

wer

(kW

)

13 m/s

11.5 m/s

10 m/s

8.5 m/s

7 m/s

5.5 m/s

Fig. 1. Output power of the wind turbine versus rotational speed curves for

different wind speeds

TABLE I

THE SPECIFICATIONS OF THE WIND TURBINE

Parameter Value

Rated rotational speed (nm) 200 (rpm) Rated wind speed (vw) 11.5 (m/s)

Radius of the turbine blade (Rb) 4.8 (m)

Maximum power coefficient(Cp-max) 0.48 Air density (ρair) 1.225 (kg/m3)


4

ins

cw

d

2

Dk

Qs=s

(10)

sds2lsL (11)

where Din, Qc, and sw are, respectively, the inner diameter, the

number of the stator coils, and the cross section area of the

coil. In ironless AFPMSGs, as it can be seen in (12), the axial

length of PMs (Lpm) depends on the axial length of the stator

and PMLF factor (kpm) [26]. On the other hand, the value of

PMLF corresponds with the axial length of the stator and the

effective airgap length of the generator, so the calculation of

Lpm and kpm requires accurate MEC modeling of the generator,

which is discussed in the next part of this section.

)95.0(2

)2(

pm

g

r

spmgr

pm

k

BB

gLkBμ=L

(12)

where Br is the residual flux density of PM, g is the air gap

length and μr is the magnet permeability. kpm is considered as

the ratio of the air gap flux to the flux leaving the magnet.

In the generators with a high number of poles, in order to

calculate the rotor back iron thickness, the mechanical

constraint should be considered in addition to the maximum

permissible value of the flux density condition in the rotor

disc, considering the attraction force between PMs in the

opposite rotor discs. This problem becomes more important,

especially when the stator axial length is small and the rotor

diameter is large. Calculation of the rotor axial length, only in

relation to the saturation of the rotor discs may result in very

thin discs. In this study, the axial length of the rotor is

calculated with considering both conditions. First, it is

calculated based on the maximum permissible value of the

flux density in the rotor disc, and after that it is computed with

considering the maximum allowable deflection (ydef) of the

rotor disc in the outer radius of the machine [28]. Finally, the

maximum value of two results is selected as the rotor axial

length.

Accurate calculation of the loss value is one of the

important steps in AFPM generator design. In the ironless

AFPM machines, copper losses (Pcu) constitute the main part

of whole losses, which is calculated as follows [26];

awp

av11R12

a1cusaa

LNkI=mP

(13)

where m1 is the number of the phases, N1 is the number of

turns per phase, L1av is the average length of each turn, σ is the

electric conductivity, k1R is the skin effect factor, and sa is the

area of the armature conductor.

Other parts of the losses which are considered in this paper

are the eddy current losses in the stator winding and the

mechanical losses. The eddy current losses in the stator

winding (Pe) with round conductors can be calculated from

(14) [29].

ρ

ldBf=P

e4

42

g

22 (14)

where d is the conductor diameter, ρ is the specific mass

density of the conductor, l is the total active length of the

stator conductors, and f is the stator frequency.

The mechanical losses consist of friction losses (Pfr) in the

bearings and windage losses (Pwind), which are expressed as

follows [2];

sshaftrfrfr)(06.0 nmmk=P (15)

)(8

5

shaft

5

out

3

sairf

3

windDDnρc=P

(16)

where kfr, mr, and mshaft are the friction coefficient, mass of the

rotor and shaft, respectively. Dshaft is the shaft diameter and cf

is the coefficient of drag for turbulent flow, which is

calculated through the equations in [26]. Finally, the generator

efficiency can be found from (17);

)(windfrecuout

out

PPPPP

P=

(17)

B. Magnetic Equivalent Circuit Model

In the axial flux machines, because of the intrinsic 3-D

geometry of the machine, analytical analysis does not possess

adequate accuracy in the average radius of the machine. For

precise modeling of MEC of the ironless AFPMSG the quasi-

3-D computation is used in this study, similar to the iron-cored

AFPMSG [30]. The generator is considered as a certain

number of layers, and each layer is used to develop a 2-D

model. The number of computational layers is assumed equal

to 10 in this study. The magnetic circuit modeling needs to

illustrate the poles arrangement and flux path. For this

purpose, Fig. 5 shows the 2-D view of an AFPMSG in the

mean radius of the specified layer.

Based on flux paths in Fig. 5, and regarding the model

symmetry, the reduced MEC model of the ith layer for one

magnet pole is derived and shown in Fig. 6, where Rg,i, Rr,i,

and Rpm,i are in order, the reluctances of the air gap, rotor back

iron, and magnet in layer i. Rmr,i and Rmm,i are the

corresponding reluctances to the magnet-to-rotor leakage flux,

and magnet-to-magnet leakage flux, respectively. Also, Φm,i,

Φg,i, Φmr,i, Φmm,i, and Φr,i, represent the flux leaving the

magnet, air gap flux, magnet-to-rotor leakage flux, magnet-to-

magnet leakage flux, and rotor back iron flux in layer i,

respectively. Because of the large effective air gap length, the

magnetic field of the stator winding current is neglected.

Ls

lsls Sd

Stator winding

Epoxy resin

(a) (b)

Fig. 4. 2-D structure of the ironless stator, a) radial view, b) axial view.


5

As mentioned in the introduction section, in this paper, the

reluctances of rotor back iron is not neglected and the

magnetic saturation of rotor back iron is considered through a

nonlinear iterative procedure. Because of the variations of

rotor yokes flux density behind the PMs and inter polar

regions, the rotor back iron reluctances are separated into two

parts, which are defined in (18) and shown in Fig. 5. The

consideration of rotor back iron reluctances, in this

configuration, has not been implemented before, in the case of

ironless AFPM machines.

ir2,ir1,ir,2= RRR (18)

Reluctances behind the inter polar regions (Rr1,i) and behind

the PMs (Rr2,i) are obtained from (19) and (20), respectively.

ir1,if1,0

ipm,ip,

ir1,=

Aμμ

wτR

(19)

2/)(

5.0=

ir2,ir1,if2,0

ipm,

ir2,AAμμ

wR

(20)

where, μf1,i and μf2,i are the relative permeabilities of the rotor

iron parts in the corresponding layer i, which are the nonlinear

function of the flux density and determined through the B-H

curve of the rotor iron. Ar1,i and Ar2,i are the average flux

passing areas of the rotor back iron behind the inter polar and

PM regions, respectively. Fig . 7 shows the B-H curve of the

used steel in the rotor disc with grade M19-G29.

The amount of Rg,i and Rpm,i can be obtained by considering

the effective air gap length and flux passing areas. Rmr,i and

Rmm,i for each computation layer, can respectively be achieved

by computing the leakage permeance between magnet and

rotor back iron and the leakage permeance between two

adjacent magnets. In this study, similar to [18, 20], the circular

arc straight-line model has been used for computing the

leakage permeances in the air gap of an ironless AFPMSG.

Finally, after calculations of the reluctances of all flux

paths, the fluxes of MEC can be determined from (21) by

applying the KVL and KCL lows to the loops and nodes of

MEC specified in Fig. 6.

0

0

0

0

101050

0115050

22400

02020

00202pm

1

ir,imm,imr,

imm,ig,

imr,ipm,

ir,

imm,

imr,

ig,

im, F

--.

..-

RRR

R-R

RR

φ

φ

φ

φ

φ

(21)

The amount of relative permeability of iron parts as shown

in Fig. 8 is updated in a nonlinear iterative procedure using the

B-H curve of the rotor iron. First, the reluctance matrix of the

MEC model is developed and solved by assigning an initial

value to the relative permeability of the iron parts. Then, the

flux densities in the saturable parts are computed from (22)

and (23), and the new permeabilities are updated using B-H

curve and equations (24) and (25) [20].

ir1,

ir,

ir1,=

A

φB (22)

2ir2,ir1,

ir,

ir2,)/A(A

φ=B

(23)

1)-(k

ir,0

1)-(k

ir,(k)

if,=ˆ

Hμ

Bμ (24)

)1(1)(k

if,

(k)

if,

(k)

if,ˆ=

dd

μμμ

(25)

0 2 4 6 8 100

0.5

1

1.5

2

H (kA/m)

B (

T)

Fig. 7. B-H curve of the used steel in the rotor disc

Fig. 8. The nonlinear algorithm of flux and leakage flux calculation

Fig. 6. The reduced form of MEC model (one half of a pole pair)

Fig. 5. 2-D model of the ironless AFPMSG for the specified layer


6

where, k is the number of iterations and d is the damping

factor, which is set to 0.1 [20, 31]. The iteration continues

until the convergence condition is satisfied for all

permeabilities as follows;

01.0)-(

1)(k

if,

1)(k

if,

(k)

if,

μ

μμ (26)

As it can be seen, the flux distribution inside the generator

is achieved based on the MEC model of the generator and the

reluctance of the flux paths. After the calculation of the flux

distribution, PMLF factor can be obtained as the ratio of the

air gap flux to the flux leaving the magnet, as shown in (27).

N

1i

im,

1

ig,

m

g

pm=

φ

φ

φ

φk

N

i (27)

IV. DESIGN OPTIMIZATION ALGORITHM

This section presents a PSO-based design optimization

algorithm including design requirements, constraints, and the

optimized dimensions of the generator.

A. Particle Swarm Optimization (PSO)

PSO is a metaheuristic global optimization method based on

the behavior of a swarm of insects. Each particle in a swarm

moves through the search space using its own best found

solution (Pbest,j) and the best found solution of all the swarm

particles (Gbest). Particle j has two characteristics; its position

Xj, and its velocity Vj. Particle positions and velocities are

initialized randomly in the design space. Subsequently, the

algorithm operates iteratively, and the particle velocities and

positions are updated on each iteration, as follows;

))]()((

))()(()([)1(

jbest22

jjbest,11jj

tXGtrc

tXPtrctVχtV

(28)

)1()()1(jjj tVtXtX (29)

where c1 and c2 are the individual and global learning rates,

respectively. r1 and r2 are uniformly distributed random

numbers in [0,1], which are redrawn in every iteration. t is the

iteration number and 1 is the constriction factor to reduce

the velocity of particles. The specifications of the used PSO

algorithm are described in Table II.

During the optimization process, particle positions may be

placed out of the bands. Different methods of bound handling

techniques are considered in [32]. In this study, the Reflect

method is used to reflect the infeasible solution to feasible

space. The new position of the violated variables is calculated

as follows;

))1(()1( then )1( ifmaxjmaxjmaxj

XtXXtXXtX (30)

))1(()1( then )1( ifjminminjminj tXXXtXXtX (31)

B. Design Requirements and Constraints

Accurate definition of the objective function and constraints

in terms of optimization variables is a key step in the PSO

based design procedure of the ironless AFPMSG. Four

optimization variables are selected containing the inner to

outer diameter ratio (kd), magnet width to pole pitch ratio (αp),

flux density of the air gap (Bg), and electrical loading (A)

which are restricted between upper and lower band limits. kd is

one of the important parameters in AFPMSG design, which

has a great influence on the performance of the generator [3].

αp is an influential parameter in the shape of the airgap flux

density and the amount of PMLF factor. Considering the large

effective airgap length in the ironless AFPM machines, the

value of Bg is lower in comparison to the one of the iron cored.

The value of Bg is considered in the range of 0.35-0.7 (T).

In the axial flux machines considering the large ratio of

machine diameter to length, the cooling process of the

machine is better than the radial flux ones. Also, the increment

in machine diameter improves the cooling conditions, so the

value of A can be increased in relation to D0.5 [33]. The

winding cooling is taken into account with restricting the

maximum values of A and current density (J). In this paper,

the value of J is considered equal to 4 A/mm2 and the value of

A is assumed in the range of 10-50 kA/m, for an air cooled

AFPM generator based on the values demonstrated in [2, 33,

34].

The sensitivity of each design variable is investigated in

Section IV-C, after describing the objective function and

design procedure. Another important constraint is the

maximum ratio of outer diameter to axial length of the

generator. Similarly, some mechanical limitations such as

minimum axial length of rotor and stator discs are considered

in the base of analytical design equations. All the design

constraints and requirements are listed in Table ІІI.

The design of a 30 kW ironless AFPMSG applied to wind

turbine application is investigated in this study. In order to

design a generator with high AEY and reduced material cost,

TABLE II

SPECIFICATIONS OF THE PSO ALGORITHM

Parameter Value

Individual learning rate (c1) 2.05

Global learning rate (c2) 2.05

Constriction factor (χ) 0.72984 Population size 20

Uniformly distributed random numbers (r1,r2) [0,1]

Bound handling technique Reflect method PSO neighbor topology Ring topology

TABLE III

DESIGN CONSTRAINTS AND REQUIREMENTS OF THE IRONLESS AFPMSG

Parameter Value

Rated power (kW) Pout =30

Rated speed (rpm) nm =200

Rated phase voltage (V) 220 Physical air gap length (mm) g =1.5

Ratio of inner to outer diameter 0.4< kd <0.75

Magnet width to pole pitch ratio 0.5< αp <0.9 Airgap flux density (T) 0.35< Bg <0.7

Electrical loading (A/m) 10000< A <50000

Maximum ratio of outer diameter to axial length (Dout/Ltot)>10 Maximum deflection of the rotor disc in the

outer radius (mm) [28] ydef =g/10

Space factor in the inner radius of the stator disc ks =0.8

Axial increment on each side of stator disc (mm) ls =0.5

NdFeB remanent magnetic flux density (T) Br=1.2


7

the multi-objective function is defined as a minus per unit

profit function (Ptot,pu) that is expressed as (32);

)()(),,,(pum,2pue,1putot,pdmg

CwPw=PαkABf (32)

where Pe,pu and Cm,pu respectively are the per unit values of the

annual energy profit and the total material cost which can be

calculated from (33) and (34). w1 and w2 are the corresponding

weight coefficients that are chosen based on the design

requirements and goals.

basee,enpue,/)( PAEYc=P (33)

basem,PMPMFeFecucupum,/)( Cmcmcmc=C (34)

where mcu, ccu, mFe, cFe, mPM, and cPM are respectively the mass

and price of the copper, steel, and PM material. According to

average electricity price in the world [35] the energy price cen

is considered equal to 0.15 €/kWh. The base values of annual

energy profit Pe,base and active material cost Cm,base are selected

as the maximum available values according to the calculation

results. The mass of each material is computed based on the

material volume and the corresponding mass density. In the

calculations, the material prices of copper, steel, and PM are

considered equal to 15, 3, and 80 €/kg, respectively [2, 34].

C. Improved Design Procedure

The improved design procedure for various combinations of

the pole and the coil numbers is presented in this section. The

rated speed of the generator is calculated according to the

turbine characteristics and the rated wind speed. In selection

of the pole and coil numbers, the following considerations

have been taken into account:

● The number of poles is chosen to obtain a valid operating

frequency based on the rated speed of the generator and the

variation of the wind speed. The typical range of the operating

frequency for small wind generators is reported as 10–70 Hz

[6]. Therefore, the number of poles is selected in the range of

20–32, in this study.

● The number of coils is chosen in relation to the number of

poles to ensure a three phase output of the generator. Various

layouts exist for concentrated-coil windings. The procedure of

determining the valid layouts is given in [27].

According to the given algorithm in [27], For the pole

numbers in the range of 20-32, the obtained combinations of

the pole and coil numbers with one coil in a coil phase group

are (20, 15), (24, 18), (28, 21), (30, 18), and (32, 24). The

optimization algorithm is executed for each combination.

The flowchart of the PSO algorithm is shown in Fig. 9. The

value of the objective function in each iteration is calculated

through the improved analytical design procedure. The best

values of the objective function and design variables are saved

in each iteration and the iterative process is continued until the

convergence condition of PSO algorithm is established.

In Fig.10, the flowchart of the analytical design procedure

of AFPMSG is shown. At first, the values of design variables

are imported from PSO algorithm. A computer aided (CA)

program was evaluated based on the sizing equations and

nonlinear MEC model of the generator. In the proposed CA

program, the value of the generator efficiency is the output of

the analytical design procedure, which is calculated based on

the design variables and is used to find the AEY of the

generator. However, at first stage, an initial value is

considered for the efficiency, and the generator dimensions are

calculated based on the sizing and the electromagnetic

equations.

According to (12) PM axial length (Lpm) depends on the

PMLF factor (kpm) and the value of the airgap flux density.

Also, the value of PMLF factor is in relation to the PM axial

length, magnet width to pole pitch ratio, effective airgap

length, and some other design parameters. So, the accurate

prediction of PMLF factor is essential for the correct

calculation of PM dimensions and cost. The PMLF factor and

axial length of PMs are computed accurately according to the

MEC model of the generator and PM leakage flux model,

given in Section III.

In the following, the practical and mechanical constraints

are checked and the required design modification is carried

out. Furthermore, the loss calculation is evaluated and the new

value of the efficiency is obtained, which is compared with its

previous value. If the required accuracy in the calculation of

the efficiency is not reached, the CA program is re-run with

Fig. 10. Improved analytical design procedure of ironless AFPMSG

Fig. 9. Flowchart of the constrained PSO algorithm


8

the new value of the efficiency to update the value of

generator parameters. Normally, the value of the efficiency is

converged in 2 to 4 iterations with an acceptable accuracy.

The required accuracy is considered equal to 0.2% for the

efficiency value in this study. After calculating the design

parameters, the generator output power and AEY is computed

for a wide range of the wind speed due to the statistical wind

speed distribution and turbine characteristics. Since the value

of the efficiency is an effective parameter in AEY of the

generator, so the efficiency value is increased in relation to

objective function of the optimization in PSO algorithm. The

value of the objective function is obtained based on AEY and

active material cost (as shown in (32)), which is exported to

PSO algorithm subsequently.

Considering the design constraints and requirements which

are presented in Table ІІI, Table ІV lists various design

parameters of AFPMSG corresponding to various

combinations of the pole and coil numbers optimized through

the PSO algorithm. Taken from the results in Table ІV, the

obtained values of the profit function in the rows 1, 2, 3, and

also 5 of Table ІV are approximately the same with little

difference in the third decimal place, which is of low

importance. In other words, the above combinations

(Q=3(2p)/4) causes proper results, in this case.

In this study, the combination of 2p=24 and Q=18 is

selected by considering the value of the active material cost

(Cm,pu), and also the better frequency range compared to the

other combinations (rated frequency=40 Hz). By increasing

the number of poles, the active material cost increases due to

an increment in axial length of PMs and outer diameter of the

generator. With taken into account the high cost of the magnet

in recent years, the amount of magnets plays a major role in

the total active material cost of the PM machines. Final design

parameters and characteristics are given in Table V.

For further investigation, the optimization procedure is

implemented for various pairs of weight coefficients (w1, w2)

in (32), and the approximation of the Pareto optimal front of

the problem is shown in Fig. 11. Obviously, the obtained

values of (-Pe,pu) and (Cm,pu) (in Table IV, row II) are placed

on the Pareto optimal front of the problem.

The sensitivity of the profit function is estimated and shown

in Table VI with respect to the variation of each design

variable. The values of the constant variables are considered

equal to average value of the corresponding band limit in

Table III. The variation of kd and Bg is more effective in

comparison to the other variables.

TABLE VI SENSITIVITY OF THE PROFIT FUNCTION WITH RESPECT TO THE VARIATION OF

THE DESIGN VARIABLES

kd Ptot,pu αp Ptot,pu Bg (T) Ptot,pu A

(kA/m) Ptot,pu

0.4 0.214 0.5 0.460 0.35 0.479 10 0.444 0.47 0.303 0.58 0.449 0.42 0.460 18 0.469

0.54 0.373 0.66 0.433 0.49 0.429 26 0.449

0.61 0.430 0.74 0.410 0.56 0.375 34 0.390 0.68 0.470 0.82 0.373 0.63 0.306 42 0.347

0.75 0.500 0.9 0.317 0.7 0.214 50 0.297 kd: Ratio of inner to outer diameter αp: Magnet width to pole pitch ratio

Bg: Airgap flux density A: Electrical loading Ptot,pu: Profit function)

-0.832 -0.828 -0.824 -0.82 -0.816 -0.8120.7

0.8

0.9

1

1.1

1.2

1.3

Annual energy profit (-Pe,pu)

To

tal acti

ve m

ate

rial co

st

(Cm

,pu

)

Fig. 11. Approximation of the Pareto optimal front of the problem

TABLE V

FINAL DESIGN PARAMETERS OF THE IRONLESS AFPMSG

Parameter Value

Rated power (Pout) 30 (kW) Rated speed (nm) 200 (rpm)

Number of phases (m) 3

Number of pole pairs (p) 12 Number of stator coils (Qc) 18

Physical air gap length (g) 2*1.5 (mm)

Outer diameter (Dout) 1030 (mm) Ratio of inner to outer diameter (kd) 0.75

Magnet width to pole pitch ratio (αp) 0.63

Airgap flux density (Bg) 0.39 (T) Electrical loading (A) 39300 (A/m)

Axial length of PM (Lpm) 13.4 (mm)

PM leakage flux factor (kpm) 0.815 Axial length of stator (Ls) 30 (mm)

Number of turns per phase (N1) 372

Copper losses (Pcu) 2410 (W) Eddy current losses in the stator winding (Pe) 202 (W)

Mechanical losses (Pfr+Pwind) 138 (W)

Efficiency at rated speed(η) 91.5% Annual energy yield (AEY) 164.6 (MWh)

Weight of copper (mcu) 61.8 (kg)

Weight of PM (mpm) 44.9 (kg) Weight of steel material in rotor discs (mFe) 134.8 (kg)

Total active material cost 4900 (€)

TABLE IV

IMPROVED DESIGN RESULTS OF THE IRONLESS AFPMSG FOR VARIOUS COMBINATIONS OF POLE AND COIL NUMBERS

No. of Poles

and coils

(p,Q)

Parameters

Dout (mm)

Lpm (mm) Ls

(mm) N1 Lr (mm) kd αp Bg (T)

A (A/m)

Cm,pu Pe,pu Ptot,pu

(20, 15) 1015 12.8 29 355 23.4 0.75 0.68 0.39 38564 0.765 0.822 0.504

(24, 18) 1032 13.3 29.7 372 23.5 0.75 0.64 0.39 39303 0.767 0.823 0.505 (28, 21) 1050 13.8 30.1 378 23.7 0.75 0.60 0.39 40003 0.772 0.823 0.504

(30, 18) 1121 14 32 432 26.2 0.75 0.63 0.36 42695 0.952 0.816 0.463

(32, 24) 1073 14.5 30.7 400 24 0.75 0.56 0.39 40760 0.782 0.823 0.502 (Dout: Outer diameter Lpm: Axial length of PM Ls: Axial length of stator N1: Number of turns per phase

Lr: Axial length of rotor disc kd: Ratio of inner to outer diameter αp: Magnet width to pole pitch ratio Bg: Airgap flux density

A: Electrical loading Cm,pu: Total material cost, Pe,pu: Annual energy profit Ptot,pu: Profit function)


9

The energy yield of the generator with respect to wind

speed is achieved through the analytical results and shown in

Fig. 12. As it is clear, the energy yield has the peak value at

11.5 (m/s) which is equal to rated wind speed.

V. FINITE ELEMENT ANALYSIS

FE analysis is performed on the generator to evaluate the

validity of the improved design procedure presented in

previous sections. A 3-D FEM of the ironless AFPMSG is

simulated using Ansys Maxwell 16.0 software.

The FE model of the AFPMSG was developed based on the

final design parameters in Table V. The magnetic flux density

distribution for the four-pole pitch and three stator coils of the

generator is shown in Fig. 13. Asymmetry of the flux

distribution in the rotor disc is due to the flux density of

armature reaction, which is considered for the phase currents

equal to Ic=Imax , Ia=Ib=-Imax/2.

As mentioned in section III-A, in the design procedure of

the generator, the consideration of the mechanical constraints

increases the axial length of the rotor, which results in lower

values of the maximum flux density in the rotor disc. In Fig.

14, the flux density variation on the middle plane of the

effective air gap is illustrated for a four-pole pitch of the

generator.

The three phase voltages of the generator and their

harmonic contents are achieved and presented in Fig. 15 under

the rated load and power factor=0.9. It is evident from the

results that the generator has a sinusoidal output voltage and

the harmonic content of the voltage is insignificant. The

voltage transient results of 3-D FEM and it’s fundamental

component are also shown in Fig. 16, and compared with the

fundamental component of analytical results. The results show

good agreement between the FEM and analytical results. This

also confirm that only fundamental of the voltage is necessary

to be considered in the analytical calculations for ironless

AFPM machines, as demonstrated in [27].

10 15 20 25 30 35

-300

-200

-100

0

100

200

300

Time (ms)

Ph

ase V

olt

ag

e (

V)

FEM

FEM Fundamental

Analytic Fundamental

Fig. 16. Comparison between the voltage transient results of 3-D FEM and fundamental component of the analytical results at 200 rpm and rated load

condition with power factor=0.9

2 4 6 8 10 12 14 16 18 20 22 240

0.2

0.4

0.6

0.8

1

1.2

Wind Speed (m/s)

En

erg

y Y

ield

(p

u)

Fig. 12. Energy yield of the generator with respect to wind speed

(a)

(b)

Fig. 15. a) 3 phase voltages and, b) harmonic analysis of the AFPMSG under

rated load and power factor=0.9

Fig. 14. Magnetic flux density distribution on the middle plane of the

effective air gap (four-pole pitch and three coils)

Fig. 13. Magnetic flux density distribution in AFPMSG (four-pole pitch and

three coils)


10

The performance of the generator with resistive-inductive

load model is predicted for different values of load currents

and rotor speeds. Predicted analytical results of the generator

are evaluated and compared with 3-D FEM results. Fig. 17

shows the variation of the generator phase voltage with respect

to the load current (0 to In) under the different constant speed

values. Phase voltage has linear relationship with the load

current.

Output voltage versus rotor speed characteristics are shown

in Fig. 18 for constant resistive-inductive load. It is observed

that there is a linear relationship between voltage and speed of

the rotor, even at low speed range, which makes it suitable for

direct drive applications such as low speed wind turbines. It is

observed that the analytical results match well with the FEM

in all voltage-current and voltage-speed characteristics.

Additional comparisons between the analytical design

results and the 3-D FEM results are given in Table VII. The

air gap leakage flux factor is calculated according to the

proposed PMLF model in Section III-B and FEA. It is clear

that the 3-D FEM results also confirm the analytical results.

VI. CONCLUSION

In this paper, a multi-objective optimization procedure was

studied to design an ironless AFPMSG for a direct-coupled

wind turbine using PSO algorithm. For proper design of the

wind generator, wind speed and wind turbine characteristics

were considered through calculating the AEY of the generator.

The objective function of the optimization was considered as a

combination of the active material cost and proceeds from the

AEY, in which the generator performance can be enhanced

over the entire range of the operating wind speed, despite

maintaining a low cost material.

Employing a modified MEC model, based on a nonlinear

iterative algorithm to consider flux distribution in the rotor

back irons, the accurate analytical design of the AFPMSG has

been resulted. A 30 kW 200 rpm AFPMSG with two rotors

and one ironless stator was designed using the proposed

procedure. The performance of the optimized generator was

predicted for different values of the wind speeds and load

currents. Results show that the output voltage of the generator

is very close to sinusoidal waveform and has almost linear

relationship with the variation of the load current. Comparison

between 3-D FEM results and the results of the proposed

design procedure illustrate the accuracy of the proposed

model.

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50 100 150 20050

100

150

200

250

Speed (rpm)

Ph

as

e V

olt

ag

e (

V)

Analytical

FEM

Fig. 18. Variation of phase voltage with respect to rotational speed under

rated load and power factor=0.9

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50

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Load Current (A)

Ph

as

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V)

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FEM (200 rpm)

Analytical (125 rpm)

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Analytical (50 rpm)

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Parameter Analytical

results

3-D FEM

results

Output power at 200 rpm (kW) 30 30.57

Induced RMS no load phase voltage at 200 rpm (V)

286.2 292.5

Maximum air gap flux density (T) 0.55 0.546

PM leakage flux factor 0.815 0.836 Maximum flux density in the rotor disc

core (T) 1.28 1.24


11

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design optimization of direct-coupled ironless axial flux

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