1

Particle Swarm Optimization of Air-cored Axial Flux Permanent

Magnet Generator for Small-Scale Wind Power Systems

B. Xia, P. C. K. Luk, W. Fei, L. Yu

Electric Power and Drive Group, School of Engineering, Cranﬁeld University, UK

Keywords: Air-cored, axial flux permanent magnet, particle

swarm optimization, wind power.

Abstract

Axial flux permanent magnet synchronous machines with air-

cored configuration is particular suitable for small scale wind

power system due to their advantages of low synchronous

reactance, cogging torque free, high efficiency and high

power factor. However, due to the number of machine

parameters, with some tightly ‘coupled’ with each other,

optimisation of the design could become extremely

challenging by conventional analytical means. Here, the

particle swarm optimization method is used in the design of

an axial flux permanent magnet generator for small-scale

wind power system. Five inter-dependent design parameters

are adjusted simultaneously to achieve an optimal solution for

the application. Three-dimensional finite element analysis is

employed to evaluate the electromagnetic performance for the

optimization. The results show the proposed optimization

method is efficient and with fast convergence.

1 Introduction

Wind power will continue to contribute an increasingly

important fraction of the future energy mix in the U.K.

Among the most mature technology in renewable energy

industry, wind power has promising prospects to achieve an

economically sustainable and environment friendly solution

to energy challenge we face [1,2]. Whilst large-scale wind

generators are now widely employed in centralized wind

farms to provide electricity to the grid [2,3], the fast

development of distributed electric generation means that the

demand for small-scale to medium-scale generators will

remain strong. In particular, for some cities and towns in

windy area, small-scale wind generators can be built in

neighbourhoods and even clustered together to power

streetlights. Moreover, in some remote areas where power

grid is not available, small-scale off-grid power systems

become more crucial as the main energy source [4,5].

With the advantages of flat shape, compact structure, high

power and torque density, axial flux permanent magnet

(AFPM) synchronous machines are widely applied in various

applications such as wind power generation, electric vehicle,

aircraft and so on [6-8]. Air-cored AFPM synchronous

machines, which have no ferromagnetic material in the stator,

could eliminate iron loss as well as cogging torque. Moreover,

such an air-cored configuration could also deliver high

efficiency, light weight and low starting torque so that it will

be ideal for small-scale off-grid wind power applications. As

the air-cored AFPM synchronous machine has relatively large

magnetic air gap, more magnets are required for the sufficient

excitation. Nevertheless, the advantage of large air gap is that

no saturation occurs in the machine, and high harmonic

components of air gap flux density distribution can be

minimized. Importantly, the large effective air gap length also

results in low synchronous reactance and hence high power

factor.

The electromagnetic performance of an AFPM generator is

affected by many parameters, and some of them have strong

interactions with one another. This could present conflicting

requirements in the optimisation stage. Many optimization

methods, such as response surface methodology (RSM),

genetic algorithm (GA) and particle swarm optimization

(PSO), have been proposed for multi-parameter and multi-

objective problems to explore the design space and achieve

optimal designs to satisfy the objectives simultaneously [9-

11]. Among these methods, PSO is a relatively simple but

efficient evolutionary computation technique originated from

the movement around searching space of intelligent swarm

societies according to a user-defined fitness function. The

instinct merits such as simple parameter adjustment, short

computational time, derivative-free, and flexibility, make

PSO easily suit to different kinds of optimization problems

[11-13]. Although it was only first developed in 1995 [14,15],

PSO has already attracted more and more interests in a variety

of research areas and applications.

PSO method has been adopted successfully in electric power

systems for economic dispatch, reactive power control and

voltage regulations [10,12,16]. In the recent years, increasing

number of researchers are applying PSO method to multi-

parameter, multi-objective optimal design of electric

machines, as many of the parameters to be optimized are

conflicting to each other during optimization. Permanent

magnet synchronous machines (PMSMs) have been

optimized using PSO to achieve high efficiency and less

weight [17,18]. Moreover, the cogging torque of a transverse

flux permanent motor (TFPM) and the force of a linear TFPM

have been optimized in [9] and [19] respectively. An PSO-

based optimal design of axial laminated synchronous

reluctance motor (SynRM) has been used for traction

applications [20]. Moreover, designs of induction machines

using PSO method have also been reported [21].

This paper concerns the design of a 500W AFPM generator

for small-scale wind power system. Though with many

inherent merits of this air-cored AFPM synchronous machine,

Proceedings of the 7th IET International Conference on Power Electronics, Machines and Drives (PEMD 2014), Manchester, UK, 8-10 April 2014

DOI: 10.1049/cp.2014.0295

This paper is a postprint of a paper submitted to and accepted for publication in and is subject to Institution of Engineering and Technology Copyright.

The copy of record is available at IET Digital Library"

2

there is a concern about the usage of permanent magnet (PM)

materials hence the cost of the machine. Therefore, the inner

and outer diameters, pole pitch factor and thickness of PMs

should be carefully determined to obtain high power output

with minimum PM usage. Moreover, since no slot exists in

the stator, the coil configuration, which has a great influence

on the performance of the generator, needs to be optimized.

Three-dimensional (3-D) finite element analysis (FEA)

models are developed to carry out the optimization based on

PSO. Optimal design is achieved by factoring both the

performance and cost into the cost function. The

electromagnetic performance of the optimal design is

evaluated by 3-D FEA under different operational conditions.

The results have revealed that PSO is a simple yet efficient

method with fast convergence.

2 The AFPM generator

The air-cored AFPM synchronous machine with double-

outer-rotor-internal-stator configuration is illustrated in Fig. 1.

Surface-mounted PM poles have the merits of simple

structure and easy manufacture and assembly. The wind

turbine can be directly attached to the outer rotor which can

further reduce the system weight and cost. Since the gearbox

is no longer necessary, efficiency and reliability will further

improve [6]. Due to the relatively large magnetic air gap,

saturation is not considered in analysis. Therefore the main

design parameters lie on the dimensions of PM and coils.

According to its instinct flux distribution style, the sizing

equation of this AFPM synchronous machine can be obtained

2

23

(1 )(1 )

120

em p w g o

P mn K B A D

(1)

where P

em

is the electric power, α

p

is pole pitch factor, m is

the number of phases, n is rotating speed in resolution per

minute, K

w

is coil factor, B

g

is the air gap flux density, A is

the electric load, D

o

is the outer diameter, and γ is the ratio of

D

i

(inner diameter) over D

o

. Outer diameter is chosen as

250mm to meet the requirements.

The combinations of rotor poles and stator coils for such

AFPM machine have been discussed in [22]. The pole/coil

configurations of 16/12, 20/15 and 24/18 have large coil

factors and the power density of the machine with increase

with pole number. However, high pole number means strict

precision for pole location, which will increase mechanical

complexity and manufacturing cost. Therefore, a compromise

of 20/15 pole/coil configuration is adopted for the design.

The main active parts in the machine are PMs, coils and back

iron, as depicted in Fig.1. Since saturation in back iron is not

considered, the size and shape of PMs and coils are the key

parameters which have great influences on the performance of

the machine. However, the design values of these parameters,

such as pole pitch factor (α

p

), inner diameter (D

i

), thickness of

PM (H

m

), thickness of coil (H

c

) and coil-band width (W

c

),

have great interaction, sometimes conflicting influence on one

other. Consequently, they should be simultaneously optimized

to achieve an optimal solution.

3 Application of PSO

PSO is a stochastic global optimization method. Besides its

fast convergence rates, another main advantage is simplicity.

Only two vectors are associated for each particle in PSO,

position (X

i

) and velocity (v

i

) respectively. One position of a

particle is a candidate solution, and the next searching

position is determined by the velocity of the particle based on

the experience of the whole swarm. After one iteration,

position and velocity can be updated as follows [11]

( 1) ( ) ( 1)

i i i

X t X t v t

(2)

1

2

( 1) ( ) rand() ( )

rand() ( )

i i best i

best i

v t w v t c p X

c g X

(3)

where w is the initial weight, c

1

and c

2

are scaling factors,

rand() is function generating random numbers between 0 and

1, and p

best

is best solution one particle ever achieved

(personal pest), and g

best

is the best solution all particle

achieved so far (global best).

Much work has been done to achieve fast convergence on the

selection of w, c

1

and c

2

. According to [11,15], best choice for

c

1

and c

2

is 2.0 for both, and w should be between 0.4 and 0.9.

2.1 Define solution space

Generally in engineering applications, parameters are chosen

in a certain range based on practical experience and theories.

In PSO, particles are exploring the solution space for optimal

solution. Larger space means more time is required to explore

the area. Thus reasonable range should be given according to

different problem to save time for the optimization. For the

AFPM machine design, there are five key design parameters

having great influence on the machine performances:

Ratio of D

i

over D

o

(γ): Smaller D

i

is, longer the

effective length of conductors are. Yet smaller D

i

reduces the space for armatures in inner radius area.

(0.4,0.8)

Pole pitch factor (α

p

): Larger pole pitch not only can

increase the flux density in the air gap, but also

increase the amount of magnet usage and may cause

unwanted harmonics in flux distribution.

Fig. 1: Topology of air-cored AFPM machine.

3

(0.5,1)

p

Thickness of magnet (H

m

): Thicker magnet will

increase the flux density as well as cost PM material.

(2,12)

m

H

Ratio of H

c

over H

m

(β): Thicker coil means more

conductors can be used to generate higher power.

Nevertheless, thicker coils also enlarge the air gap

and lower the flux density.

(0.5,2)

Ratio of W

c

over R

i

(δ): Wider coil-band can contain

more conductors to generate higher torque, but more

conductors also can cause problems such as higher

armature resistance, higher loss, and thermal issues.

(0, /15)

Population size also necessitates special attention. Large

population can explore solution space more thoroughly, with

the cost of more time-consuming computations. However,

small population size of 10 to 20 can also provide sufficient

exploration through the solution space and effectively reduce

the optimization time [11]. Since the 3-D FEA evaluation is

quite time-consuming, 10 particles are set for this case.

2.2 Boundary conditions

As there are random functions acting on the process of

generating velocity, particles are likely to fly out of the

defined solution space. In order to address this problem,

boundary conditions are necessary to confine the particles.

Three boundary conditions are normally employed for PSO,

namely absorbing walls, reflecting walls and invisible walls.

In the first two boundary conditions, particles are strictly

restrained inside the solution space, while in invisible walls

particles are allowed to go out of defined space. However,

once a particle flies out, the position would not be calculated

or evaluated, and the particle has to fly again until it is back

within the defined region. There is more freedom of the third

boundary for particles to roam around, and thus their natural

motion would not be interfered. According to Robinson et al.

in [11], invisible walls condition is faster and more consistent,

and is thus adopted in this case.

2.3 Objective function

Each position of particles represents a possible design for the

AFPM generator. To evaluate the goodness of the designs

explored by the particles, an objective function needs to be

defined and linked to return fitness values. In order to meet

the required performance and at the same time reduce the

cost, five performance indexes, namely weight of magnet

(M

m

), weight of copper (M

c

), machine efficiency (η), volume

of the machine (V

g

) and power output (P

o

), are included in the

objective function. The weight coefficients are contained

before each performance index to indicate how much

influence of each performance index has on the overall design.

For the proposed machine, the objective function is defined as

follow

(1 )

m m c c v g po o

fitness k M k M k V k k P

(4)

where k

m

, k

c

, k

v

, k

,

and k

po

are corresponding weight

coefficients. In general, different weight coefficient choices

represent different design goals. In this case, performance and

the cost of the generator are the key criterion for optimization.

For instance, PMs are much more important and expensive

than coils, and hence the value of k

m

is much larger than k

c

.

Furthermore, small volume, high efficiency and power output

are preferred. Based on the design requirements and the range

of variation, the weight coefficients are chosen as listed in

Table 1, together with initial weight and scaling factors.

2.4 Implement of PSO

The flow chart of the PSO program is illustrated in Fig. 2.

After initialization, the process of each iteration can be

divided into 4 steps. Firstly, simulate and evaluate each

position (solution) of the iteration. And then update p

best

and

g

best

. Thirdly, the velocities and the positions of next iteration

are generated. At last, a judgement is carried out to make sure

all particles are searching within solution space for the next

iteration.

In this study, the fitness of objective function has been

successfully converged after 30 iterations. The average fitness

of particle positions, average fitness of personal best, and

global best of each iteration is depicted in Fig. 3. It is shown

that from 20

th

iteration, the personal bests are already

Initialization

Evaluations

Update p

best

and g

best

Generate and

update velocities

and positions

Within solution

space?

Record the

positions for the

next iteration

Yes

No

Iteration +1

Fig. 2: Flow chart of PSO program.

w

c

1

c

2

k

m

k

c

k

v

k

η

k

po

0.65

2.0

2.0

30.0

2.0

1000

150

0.15

Table 1: Values of factors and coefficients for PSO

Fig. 3: Fitness of objective function against iteration.

4

localized around their global best position in solution space.

After 30 iterations good convergence is achieved, and the best

solution found in iteration 26, is taken as the optimal design.

As shown in Fig. 4(a), after the first several iterations’

fluctuation, the design parameters of H

m

, α

p

, and W

c

find their

corresponding optimal values, while D

i

has a great impact on

the value of H

c

with large fluctuations, as shown in Fig. 4(b).

It should be noted that all the design parameters stabilise after

the 26

th

iteration, with optimal solution obtained.

The trends of performance indexes are derived and depicted

in Fig.5. As expected, usage of PM has higher influence to the

evaluation. After an attempt to increase PM material, the

usage is reduced gradually from 7

th

iteration, while the usage

of copper is increased to maintain the output. Though this

brings out the issue of low efficiency and big in volume, the

benefit of smaller amount of PM thus lower in cost and high

power output makes the design a more practical choice.

4 3-D FEA verification

To validate the performance of the final optimised design

from PSO algorithm, simulation should be carried out by

using the FEA method. Because of the instinct structure of

AFPM machines, 3-D FEA model is a used to obtain more

accurate results. Since only back iron, PM poles and coils

matter in the electromagnetic simulation, housing and

connection parts of the machine are not included in the 3-D

FEA model. The machine in 3-D model is illustrated in Fig.1,

and the main design parameters are listed in Table 2.

4.1 No load characteristics

Firstly, the no-load characteristics of the machine are

calculated. The flux distributions of the back iron and the air

gap are shown in Fig. 6. As is seen from Fig. 6(a), the

magnitude of flux density in the air gap is between 0.5-0.6 T,

which is relatively low compared to machines with slotted

cores. For the back iron, its flux density should be at the knee

point of its B-H curve, which is about 2.0 T, to make full use

of the material as well as avoiding saturation. The back EMF

is another most important parameter for the machine design.

For a three phase generator, sinusoidal wave form is preferred

to avoid extra loss because of harmonic components. As is

shown in Fig. 7(a), the back EMF wave form of the machine

illustrates that the machine can achieve the output voltage

requirement. Moreover, from Fig. 7(b), it can be seen that the

output contains few harmonics. More specifically, the third

harmonic is less than 2%, which can be eliminated by using

the star armature connection. For other higher harmonics, the

magnitude of phase EMF is negligible. Therefore, such EMF

is perfect for three-phase generator.

4.2 On load characteristics

To simulated performances of the generator at different

operating points, FEA models at different loads are also

constructed. For small-scale off-grid generator, loads can be

classified into two categories, which are AC load for common

household appliances and DC load for battery charging. To

simulate the different conditions of these two loads, AC load

and DC load are connected respectively to assure the output

characteristics.

(a) (b)

Fig. 4: Trends of best design parameters against iteration.

(a) (b)

Fig. 5: Trends of performance indexes against iteration.

(a) air gap

(b) back iron

Fig. 6: Flux distribution density in the machine

Parameters

Value

Parameters

Value

P

o

(W)

500

D

o

(mm)

250

n (rpm)

300

D

i

(mm)

168

U

o

(V)

220

H

m

(mm)

7

m

3

α

p

0.75

p

20

H

c

(mm)

17.4

q

15

W

c

(mm)

16.8

Table 2: Parameters of final design from PSO.

5

Under the AC full load condition, both the induced voltage

and armature current are quite sinusoidal, and the power

output is over 520W with a smooth torque, as depicted in Fig.

8. In fact, the loss is mainly caused by copper loss, because

eddy current loss and core loss are eliminated due to the

absence of the stator core. Also, because of the large airgap

and thus low armature reaction, the iron loss in stator back

iron is negligible. Since more copper conductors are used,

107W is consumed by armature resistance. The resulting

efficiency is 82.9%.

Under DC load condition, a non-controllable three-phase

rectifier is connected to obtain DC output. And because of the

influence of the electronic switches during commutation,

distortion can be observed in the voltage wave form, and thus

output DC voltage is fluctuating six times in one electrical

period in terms of the six switches of the three-phase full-

wave rectifiers, as is shown in Fig. 9(a) and 9(b). In addition,

from Fast Fourier transform (FFT) analysis of the DC voltage

depicted in Fig. 9(c), it can be seen that there are only the 6th

harmonic and the harmonics which are multiple of six except

for DC component. Besides, greatly influenced by the

fluctuation from voltage, the output power and magnetic

torque also suffer from variations as shown in Fig. 9(d). The

fluctuation is up to 20% of the average value. With the

fluctuating currents, the copper loss is 9W higher than that

with DC load. Thus the efficiency at DC load is 81.2%, which

is lower than that under AC load. However, it should be noted

that no filter or control algorithm is used in the simulation. In

practical applications, power electronic circuit with controller

and filters should be applied to smooth the output voltage and

obtain a stable DC voltage. Due to the instability of wind

speed, the wind turbine should be designed to adapt all these

operating conditions at a wide wind-speed range. Thus,

simulations are necessary to carried out at various load

conditions to predict the output characteristics of the wind

generator. Generator voltage regulation and efficiency at

different load conditions with AC load and that with DC load

are illustrated in Fig. 10. Generally speaking, the trends of

these two figures are quite similar. In particular, it can be seen

that the operating efficiency is above 81%, and the lowest

efficiency appears at full load, which is because higher

current leads to higher resistance loss. In addition, the output

voltage and efficiency is linear against output power, which

means that the generator can be treated as a linear system.

Last but not least, for the small wind generator, the turbine is

fixed and cannot be adjusted in terms of different wind speeds,

which means that the energy captured by generator is

changing all the time. In other words, it is impossible for the

machine to operate at full load condition at all times.

Therefore, the average operating point is probably at around

400W, and the average efficiency should be about 87%.

In summary, according to PSO algorithm, a design with more

copper and less PM is chosen as the optimal design. 3D FEA

model is built and simulated under different load

circumstances. Simulation result reveals that the final design

has good performance with acceptable efficiency, and

achieves the design requirements for small wind power

systems. Furthermore, the usage of PM material for the

machine is reduced with approximately same output features,

which would minimize the cost of the machine.

5 Conclusion

In this paper, an AFPM synchronous machine with rated

power of 500W at 300rpm is proposed for small-scale wind

power applications. A sophisticated multi-parameter and

multi-objective PSO algorithm based on 3-D FEA is

developed and employed to carry out optimization with a user

defined objective function. The performance of the final

optimal machine has also been comprehensively evaluated by

(a) back EMF waveform (b) back EMF spectrum

Fig. 7: Phase back EMF waveform and spectrum.

(a) phase voltage and current (b) loss, torque and power

Fig. 8: Output characteristics at AC load

(a) phase voltage and current (b) DC voltage and current

(c) DC voltage spectrum (d) loss, torque and power

Fig. 9: Output characteristics at DC load power output.

(a) AC load (b) DC load

Fig. 10: Generator voltage regulation and efficiency at varied

loads