Linearizing control input-output of a wind turbine permanent magnet synchronous
Riad AISSOU#1, Toufik REKIOUA#2
1,2 Laboratory of Industrial Technology and the Information « LT2I »,
Faculty of Technology, University A. Mira, Targa Ouzemour, Bejaia, Algeria
1 riadaissou@hotmail.com 2 to_reki@yahoo.fr
Abstract — In this paper, we study the control voltage at the output of a permanent magnet synchronous generator (PMSG) connected to a PWM rectifier. The input-output linearizing control is tested for PMSG. This device is intended for an application of wind energy conversion in the case of an isolated site. The results of the different simulations of the entire chain conversion performed under MATLAB / Simulink, were used to evaluate the performance of the proposed system.
Keywords— Linearizing control input-output; wind turbine;
permanent magnet synchronous generator (PMSG); PWM Rectifier;
I. INTRODUCTION
The power generation sector is the largest consumer of primary energy and two-thirds of its sources are fossil fuels. It is technically and economically capable of making significant efforts to reduce violations of human activity on climate and the environment. One possibility is to increase the rate of production of electricity from resources of non-renewable fossil type and Today, renewable generation sources, including solar and wind energy, which are the growth rate is highest.
The wind power generator, which is based on a variable speed turbine and a synchronous permanent magnet generator is connected to a DC bus through a PWM power converter. [1]
However, stand-alone operation, the rotational speed and the load is not fixed, the stator voltage can vary within wide limits. It then becomes necessary to use an appropriate control system to maintain the output voltage at a constant amplitude
and frequency.
The input-output linearizing command is a command which generalizes the vector-ensuring decoupling and linearization of the relationship between inputs and outputs. Assuming that all of the state vector is measurable, it is possible to design a nonlinear state feedback which ensures the stability of the
closed loop system [2].
This article focuses on the application of the input-output linearizing the wind energy conversion system control with a variable rate based on a permanent magnet synchronous
generator. The overall plan of studies of the system is represented by the figure 1. Results of the simulation of the dynamic behavior of the studied system are presented. From these results, we can verify the effectiveness and reliability of the applied control.
II. MODELING OF WIND GENERATOR The wind power generator, comprising a variable speed turbine coupled directly to a PMSG connected to a DC bus through a PWM power converter, is shown in Figure 1
Fig. 1. Schematic diagram of the system studied.
II.1 MODEL TURBINE
The power of the air mass that passes through the surface of the turbine S is given by [3,4]:
p ρs V (1) With
s : The effective area through which the wind, ρ : The density of the air (1.25kg / m3),
V : The wind speed,
We focus our work in the operation of a vertical axis turbine. The value of the active surface (S) was replaced by the geometric dimensions of the wing shown in Figure 2 where:
Copyright IPCO-2014 Vol.2
ISSN : 2356-5608
S W
H R
F low s λ R Ω V W be e
P T mea c
F
B
* sp C I T beh
S=2*R*H With
H : height of t R : The radius
Fig.
For describing w
speed (specific λ VR.Ω R : The radius Ω : The angula V : Wind s Wind power a
expressed in t
P Cp
The power co asures:
c λ 0.21
Figu
Fig. 3. Cu
From this pow C By replacing t
peed)
C C
II.2 MODELI The differenti havior of the tu
(2) the turbine.
s of the turbine
2. Geometric Di
g the operatin c) λ is used, w (3) s of the wind t ar speed of ro speed..
and power ext terms of the po pP oefficient Cp 121 λ 0.
ure 3 shows th
urve of the
wer, the wind t
P the value of th
R H V
ING OF THE al equation th urbine and gen
e.
mensioning Savo
ng speed of a where :
turbine blades
tation of the b tracted by the ower coefficie (4) is often deriv 0856 λ 0 he power coeff
e blade Savonius t
torque is given (6) he power by t (7 SHAFT OF T hat characteriz nerator is give
onius wing
a wind turbin
,
blades, e wind p
ent c
) ved from pra 0.2539 λ ficient Cp
type studied
n by:
the product (t 7) THE MACHIN zes the mecha en by [6].
e, the
can
actical
(5)
torque
NE
anical J wh J respe
f blade
C In with accou
C II.
Th linke
w R I V L p ω
stato
• Th Park
C II.
M ideal defin
J here :
et J : are th ectively,
et f : the co es respectively
: the sta n our applicati the generato unt), then:
J . 3 MODEL O he equations f ed to the rotor
V R V R I with :
: Resistance , I : Curren , V : Stator , L : Induct : Number of p : The pulse v
: The flux cr r windings.
• Expression he expression
P L .4 Modeling R Modeling of th ls. These sw ned by the foll
C C
he inertias of t oefficient of fr
y,
atic torque pro ion, we consi or (one the
C f Ω
OF THE SYNC for the PMSG
as follows: [7
R I L
L I
of the stator w nts in the stator
r voltages in th tions in the sta pole pairs.
voltages (rad/s reated by the
n of the power of electroma
L I I Rectifier he rectifier is witches are c lowing functio
C f f
the turbine an riction of the ovided by the der that the f wing will no Ω (9)
CHRONOUS G, can be writt
7]
I pω L pω L I
windings.
r mark Park.
he benchmark ator cyclical m
s).
permanent m
and electrom agnetic torque I (11 s made by a complementar
on [8, 9]:
f Ω (8
nd of the mach engine and of
wind.
friction associ ot be taken
MACHINE ten in a refere
I
pω (10)
k Park.
mark park.
agnet through
magnetic torque e in the reposi
)
a set of switc ry, their state
8) hine
f the
iated into
ence
)
h the
e itory
ches e is
Fig. 4. Schema of association PMSG-PWM rectifier
s 1, s I
1, s I for s=a ,b ,c
The input voltage and the output current phase can be written in terms of: Sj, and Vdc input currents ia, ib, ic.
i i i 0 (12)
Input voltages between phases of the PWM rectifier can be described by:
U S S U
U S S U
U S S U (13)
The equations for the voltage phase balanced system without neutral connection can be written as:
ee
e R i
ii L i ii
U U
U (14)
with :
U . U
U . U
U . U
(15)
Finally, we deduce the equation coupling between AC and DC sides by:
c U s i s i s i i (16) The previous equations in synchronous dq coordinates are:
e Ri L ωLi U (17) e Ri L ωLi U (18) c U s i s i i (19) with :
s √ 2s s s . cos ωt √ s s . sin ωt s √ s s . cos ωt
√ 2s s s . sin ωt III. APPLICATION OF THE LINEARIZING
CONTROL INPUT OUTPUT FOR PMSG Our command is to control the stator current I and the rectified voltage V of PMSG. For this we chose as the state vector x I I V T , and as output y=V I T and control vector u V V T .
The model of PMSG, expressed in the rotor reference frame related to the form of equation of state:
X f x G x . U t
y H x (20) with :
y t y t
y t h x
h x
xx V I ;
X xx x
I I
V ; U V V ; G x
L 0
0 L
0 0
g 0
0 g
0 0
; f x
f x f x f x
a x a x
b x b x b
c c
where : a RL ; a LL ; b RL ; b LL ;
b L ;c EC; c CIL.
The linearization condition to check if a nonlinear system admits a linearization input - output is the order of the relative degree of the system.
The following notation is used for the Lie derivative of the function h x along a vector field f x f x … f x [11].
¾ L h ∑ f x X f x
¾ L h L L h (21)
¾ L L h LX G x .
¾ degree relative
The relative degree of output is the number of times that is needed to derive the output to bring up the input U.
The future output y(t+ τ) is calculated by:
• Relative degree of the rectified voltage
. (22) . (23) with :
0
0 Relative degree of is 2
•
Relative degree of the currenty t h x L h x L h x . U (24)
With :
L h g 0
The matrix defining the relationship between the physical inputs (U) and the derivatives of the outputs (y (x)) is given by the following expression :
(25)
With :
0
0
To linearize the input-output behavior of the generator in a closed loop non-linear state feedback is applied according to [24]
V
V D x A x V
V (26) The determinant of the matrix decoupling D (x) is not null
D x 0
0 (27)
(28)
Substituting (27), (28) into (26) we have:
Entries V V are calculated by imposing a static regime I I et V V and the dynamic error.
e K e 0
e K e K e 0 (29) Internal inputs V V are defined as follows
:
V K I I I
V K V V K V V V (30)
I V V 0 (31)
The coefficients (K , K , K ) are selected so that equation (30) is a polynomial HURWITZ [11]
.
0
0 (32) IV. SIMULATION RESULTS
Full operation of the device was simulated in the Matlab Simulink . In this control strategy, the reference voltage at the output of the rectifier is taken equal to Vdc−ref = 40 V and the variation of the wind speed is shown in Figure 5. In what follows, we present simulation results.
Fig. 5. Wind speed
Fig. 6. Rectified voltage
Fig. 7. Direct current
Wind speed show in Figure 8 is modeled as a sum of deterministic several harmonics [11] :
V t 10 0.2 sin 0.1047 t 2 sin 0.2665 t sin 1.2930 t 0.2sin 3.6645 t [36]
0 2 4 6 8 10 12 14 16 18 20
11 11.5 12 12.5 13 13.5 14 14.5 15
temps en sec
Vitesse du vent en (m/sec)
0 2 4 6 8 10 12 14 16 18 20
0 5 10 15 20 25 30 35 40 45
temps en sec
Vdc en (V)
Vdc Vdcref
0 2 4 6 8 10 12 14 16 18 20
-1 0 1 2 3 4 5 6
temps en sec
Id en (A)
Id Idref
Fig. 8. Wind speed
Fig. 9. Rectified voltage
Fig. 10. Direct current
The response of the voltage at the output of the rectifier is given in Figures 6, 9. We can see that the voltage is well regulated. This is also the case of the current Id and the rejection of disturbances made in this case by changes in wind speed is ensured.
V. CONCLUSION
In this article, we presented the study of voltage control system consists of a permanent magnet synchronous generator feeding a PWM rectifier. The proposed control strategy is based on the input-output linearizing control to ensure good performance.
Law control system has been detailed. The results of
different simulations were discussed and validated mathematical models of the system proposed wind.
ANNEXES
Désignation Valeur
nominal voltage Vn = 90 V
nominal current In= 4.8 A
nominal power Pn= 600 W
Number of pole pairs 2 p = 17
Winding resistance Rs = 1,137 Ω
synchronous inductance Ls = 2.7 mH
efficient flow Φeff = 0.15 Wb
Coefficient of friction f = 0,06 N.m.s/rad Inertia of the GSAP J = 0.1 N.m
Radius of the wing R = 0.5 m
Height of the wing H = 2 m
active surface S = 2 m2
Inertia of the wing J = 16 kg.m2
Density of air ρ 1.2 kg/m
REFERENCES
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0 2 4 6 8 10 12 14 16 18 20
7 8 9 10 11 12 13 14
temps en sec
Vitesse du vent en (m/sec)
0 2 4 6 8 10 12 14 16 18 20
0 5 10 15 20 25 30 35 40 45
temps en sec
Vdc en (V)
Vdc Vdcref
0 2 4 6 8 10 12 14 16 18 20
-1 0 1 2 3 4 5 6
temps en sec
Courant Id en (A)
Id Idref
battery', PhD thesis, Eole Normale Superieure de Cachan, December 2002.
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