Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 8, N. 4
317 =
− +
+ 1
7 =
− −
− 1
+ 1
8 where i
d
is the d-axis stator currents, i
q
is the q-axis stator current, v
d
is d -axis stator voltage, v
q
is the q-axis stator voltage, Rs
is the winding resistance Ω, L
d
is winding inductance on the d-axis H, L
q
is the winding inductance on the axis q H,
ψr is permanent magnetic flux Wb, and
ω
r
is the electrical rotating speed of the PMSG rads.
The angular speed of the electric generator is dependent on the number of pole pairs P and the
angular velocity of the generator ω
g
which can be expressed by Eq. 9:
ω
r
= P ∙ ω
g
9 Electromagnetic torque produced by the PMSG in
terms of the dq model can be expressed as follows: T
e
= 1.5PL
d
- L
q
∙ i
d
t ∙ i
q
t + i
q
t ψ
r
10
III. Overview of Particle Swarm Optimization PSO
PSO is an evolutionary computing technique for optimization [21], [28]-[30]. PSO was developed by Dr.
Kennedy and Dr. Eberhart based on a social-psychology theory of the behavior of flocking animals such as ants,
termites, bees and birds. Social behavior consists of individual actions influenced by others in the group.
Fig. 1. Turbine Power Characteristics
a q- axis equivalent circuit b q- axis equivalent circuit Figs. 2. Equivalent circuit of a PMSG
Every individual, for example a bird, is expressed as a particle that has two characteristics, namely position and
velocity. The iterations of the model proceed as each particle is randomly moved through the search space,
where the movement of a particle depends on its local best-known location as well as that of the swarm.
Based on the optimum position of one particle, other particles in the group will adjust their position and
velocity [14], [15], [18]-[20]. PSO is initialized to a random solution, an iterative
search for the optimal value based on an objective function. If a particle in d- dimensional space is
expressed by X
i
= {X
i1
, X
i2
,…, X
iD
} and the best position of each particle is Pbest. The best position of the whole
swarm is called Gbest. The velocity and position of the ith particle can be defined according to the following
equations:
v
id
t = w v
id
t-1 + C
1
r
1
Pid – X
id
t-1 + + C
2
r
2
Pid – X
id
t-1 11
X
id
t = X
id
t-1 + v
id
t 12
where v
id
t is velocity of ith particle, X
id
t is position of ith particle, w is the constriction factor, C
1
and C
2
are learning factors, r
1
and r
2
are random numbers generated uniformly in the range [0 1] [21].
IV. Control Strategy
A block diagram of the wind energy conversion system is shown in Fig. 3 which consists of the PMSG,
wind turbines, control rectifiers, PWM module, current controller, and a speed controller. As each wind speed
produces a different power curve, a control system is required in order to modify the generator speed to
achieve optimum power with fluctuating wind speeds.
The PMSG speed controller consists of two stages; a current controller inner loop and a speed controller
outer loop. In this paper, the speed controller was a PSO - PI
controller while the current controller for the d-axis and q-axis was a PI controller based on pole assignment.
Fig. 3. Schematic of speed and current controller components
0.2 0.4
0.6 0.8
1 1.2
1.4 -0.2
0.2 0.4
0.6 0.8
1 1.2
1 pu Max. power at base wind speed 12 ms and beta = 0 deg
6 ms 7.2 ms
8.4 ms 9.6 ms
10.8 ms 12 ms
13.2 ms 14.4 ms
Turbine speed pu of nominal generator speed T
u rb
in e
o u
tp u
t p
o w
e r
p u
o f
n o
m in
a l
m e
c h
a n
ic a
l p
o w
e r
Turbine Power Characteristics Pitch angle beta = 0 deg
Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 8, N. 4
318 IV.1. Speed Control Using a PSO-PI Controller
The speed controller regulates the PMSG speed based on a reference value obtained through maximum power
point tracking calculations. Based on the characteristics of the WECS, the generator must rotate at an optimum
speed to achieve maximum power. Based on the method of tip speed ratio TSR, the optimum speed can be
determined by the wind speed, the length of the blade R and the optimum tip speed ratio
λ
opt
as described by Eq. 13:
= ·
13 Here we assumed an optimum tip speed ratio of 8.1. A
block diagram of the speed controller components is shown in Fig. 4. The speed controller input is the
difference between the reference speed and generator speed while the controller output is the q-axis current that
will be referenced in the q-axis current controller.
Fig. 4. Speed control using the PSO-PI controller
The PI controller used a PSO algorithm to tune the parameters Kp and Ki to get a suitable output transient
response. Every particle in PSO method has two dimensions consisting of Kp and Ki. The initial value for
each particle is determined randomly and the speed and position are updated based on equations 11 and 12.
The objective function at PSO is based on criteria performance of controller.
The most common performance criteria are the integrated absolute error IAE, the integrated square
error of time ITSE and the integrated square error ISE, which were evaluated in the frequency domain.
These performance criteria have some disadvantages where minimization can result in a response with a small
overshoot. Settling time becomes greater because the ISE performance criterion weighs all errors equally
independent of time. The ITSE performance criteria can overcome these shortcomings but the derivative
processing of the ITSE formula becomes more complex and time-consuming [21].
Therefore the performance criteria used in this paper to evaluate the PI controller are based on the time
domain. Performance criteria in the time domain include maximum overshoot M
p
, steady state error E
ss
, rise time t
r
and settling time t
s
. These can be used in the PSO algorithm as follows [21]:
WK = 1 – e
- α
·M
p
+E
ss
+ e
- α
·t
s
-t
r
14
Fig 5. Flowchart of the PSO-PI controller method
In this study, to reduce overshoot and steady state error, the weight value
α can be set to a value greater than 3. For reducing the rise time and settling time, the
α value can be set to less than 3.
The value of the fitness function using the PSO method is determined based on the minimum value
achieved from performance criteria. A flowchart of the PSO-PI control used in this study is
shown in Fig. 5. The PI controller tuned by PSO had the following parameters; number of iterations = 20, size of
the swarm = 20, C
1
= C
2
= 1.2, and a constriction factor w = 0.45.
IV.2. Current Controller Current controller functions to regulate the d-axis I
d
and q-axis I
q
current and to generate a controller output in the form of a d-axis V
d
and q-axis V
q
voltage. In this study, I
dref
is zero and I
qref
is the speed controller output. The current controller used a proportional
integrator PI based on pole assignment. A block diagram of the PI control for the d-axis and q-
axis is shown in Figs. 6. Based on the dynamic model generator and PI controller outputs, V
d
to control the d- axis current can be expressed by Eq. 15:
= − +
+ −
− 15
Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 8, N. 4
319 where Kp
d
is a proportional constant for the d-axis current controller and t
i d
is the constant of integration time for the d-axis current controller.
The output of the q-axis current controller V
q
can be expressed by:
= − +
+ −
+ +
+ 16
where Kp
q
is the proportional constant for the q-axis current controller and t
i q
is the constant of integration timefor the q-axis current controller.
a d-axis current controller
Iq +
- PI
Controller +
Iqref
r d d
r r
Vqt +
b q-axis current controller Figs. 6. PI control of a d-axis and b q-axis currents
The PI controller design is determined by pole assignment then the proportional gain and the integration
time constant for d-axis and q-axis current are calculated as follows:
Kp
d
= 2 ζL
d
– Rs 17
= 2
− 18
Kp
q
= 2 ζWnL
q
– Rs 19
= 2
− 20
where ζ is the damping coefficient selected to be 0.707 and Wn is the natural frequency which was selected
based on the desired closed loop settling time. The larger the value of Wn, the shorter the closed loop settling time.
V. Results and Discussion
