(SMC) of Permanent Magnet Synchronous Generators

International Conference on Renewable Energies and Power Quality (ICREPQ’14)
Cordoba (Spain), 8th to 10th April, 2014
Renewable Energy and Power Quality Journal (RE&PQJ)
ISSN 2172-038 X, No.12, April 2014
Sliding Mode Control (SMC) of Permanent Magnet
Synchronous Generators (PMSG)
M.S. Merzoug1, H. Benalla 2 and L. Louze1
1
Department of Electrical Engineering
L’arbi Ben M’hidi Oum el bouaghi, University
B.P. 358, Oum el Bouaghi 04000 (Algérie)
e-mail: [email protected]
2
LEC- Research Laboratory Department of Electrical Engineering Mentouri Constantine,University
Abstract. This paper presents Sliding Mode Control of
Permanent magnet synchronous generators. A comprehensive
dynamical model of the PMSG in d-q frame and its control
scheme is presented. Wind energy is a promising technology and
becomes more and more interesting player on the market of
energy production. Since the converter/capacitor model is
nonlinear, the sliding mode technique constitutes a powerful tool
to ensure the dc-bus voltage regulation. The simulation results
show the method is simple and has high precision of control, and
the system has good characteristics in steady state and during
transients with permanent magnet synchronous generator.
during the last 20 years it has shown to be a very effective
control method.
Key words
For sliding mode controller, Lyaponov stability method is
applied to keep the nonlinear system under control SMC
provides a fast and accurate dynamic response. Also
makes the system response insensitive to changes in
parameters and load disturbance.
Lyapunov, PMSG, SMC, PWM, IGBT.
1. Introduction
Permanent Magnet Synchronous Generators (PMSG) are
receiving significant attention from industries for the last
two decades. They have numerous advantages over the
machines which are conventionally used. Current research
in the design of the PMSG indicates that it has high torque
to current ratio, large power to weight ratio, high
efficiency, high power factor and robustness.
Currently, there is much interest in using brushless
electronically commutated servo machines in high
performance electromechanical systems and the
application of neodymium-iron-boron (Nd2Fe14B) and
samarium cobalt (Sm1Co5 and Sm2Co17) rare-earth
magnets results in high torque and power density,
efficiency and controllability, versatility and flexibility,
simplicity and ruggedness, reliability and cost, weight-totorque and weight-to-power ratios, better starting
capabilities [1].
Sliding mode variable structure control which sliding
mode can be designed is insensitive to system parameters
change and load disturbance, and has the advantages of
good robustness, quick response, and easy realization and
so on. Accordingly, hopeful to design a control strategy of
high quality by applying sliding mode control (SMC) to
the control of PMSM [3].
2.
Mathematical Model Of The PMSM
The dynamic model of PMSG has been built in the d-q
rotating reference frame, where the q-axis goes ahead 90
from the d-axis with respect to the direction of rotation.
The electrical model of the PMSG in the d-q synchronous
reference frame, with the voltage and torque equations are
given by : [4] [9]
d
ϕd + ωr ϕq
dt
d
Vq = − R S I q − ϕq − ωr ϕd
dt
ϕd = L d I d + ϕ
ϕq = L q I q
Vd = − R S I d −
(1)
(2)
(3)
(4)
And the electromagnetic torque Te is given by [10]:
Sliding mode control (SMC) theory was introduced for the
first time the context of the variable structure system
(VSS). It has become so popular that now it represents this
class of control systems. Even through, in its early stage
of development, the SMC theory was over-looked because
of the development in the famous linear control theory [2],
Te =
3
P[(Ld − Lq )Id Iq + Iq ϕ)]
2
(5)
ɺ t) = 0
s(x,
(10)
u n = − K sgn ( s )
(11)
Wher : K > 0
K: is the control gain
1

sgn(s) = 0
 −1

Fig. 1.a
s>0
s=0
(12 )
s<0
4. The Sliding Mode Application
Figure 2, Shows the proposed control scheme in a cascade
form in which two surfaces is required
Fig. 1.b
Fig. 1. Equivalent Circuit of a PMSG in the synchronous
reference frame (a) d-axis, (b) q-axis.
3. Description Of Sliding Mode Control
The implementation of this control method requires
mainly three stages:
A. Sliding Surfaces
J. Slotine proposes a form of general equation to
determine the sliding surface [5].
d

s(x, t) =  + λ 
dt


n −1
e(t)
(6)
With:
e(t) is the error in the output state
e(t) = x ref (t) − x(t)
Fig.2. Block diagram of DC Voltage regulation of PMSG using
SMC
(7)
The speed error is defined by:
λ is a positive coefficient
e = Vdcref − Vdc
B. Conditions of convergence
The convergence condition is defined by the equation of
Lyapunov :
S.Sɺ ≺ 0
(8)
(13)
For n=1, the direct current control manifold equation can
be obtained from equation (6) as follow
ɺ
ɺ
ɺ Ω) = V
s(
dcref − Vdc
(14)
The best way to calculate the Vdc area is to put it into the
following form:
C. Controller Design
Consequently, the structure of a controller consists of two
parts; a first concerning the exact linearization and a
second stabilizing.
u(t) = u eq (t) + u n
A. DC voltage Control [6 8]
(9)
u eq (t) Corresponds to the equivalent control suggested, It
is calculated on basis of the system behaviour along the
sliding mode described by:
V 
 dV
P* = Vdc Idc = Vdc  C dc + dc 
dt
R 

(15)
Therefore:
2
dVdc
1  * Vdc
=
 P −
dt
CVdc 
R



(16)
P* =
2
Vdc
R
Vdn = K d sgn ( s(Id ) )
(17)
(30)
Where
When losses are not taken into
electromagnetic torque will be given by:
Te =
account,
the
K d : Positive constant
C. Quadrature Current Control
P*
Ω
(18)
The quadrature current error is defined by:
Since:
e q = I qref − I q
Te =
3
PIq ϕ
2
(19)
Then:
Iq =
2 P*
3 ωϕ
(20)
(21)
In which:
Ieq
q
Iqn
2 P*
=
3 ωr ϕ
(22)
= K Vdc sgn ( s(Vdc ) )
s(I q ) = I qref − I q
(32)
ɺ q ) = ɺIqref − ɺIq
s(I
(33)
Vqref = Vqeq + Vqn
(34)

R
L
ϕ 
Vqeq =  ɺIqref + s Iq + Pωr d Id + Pωr
 Lq


L
L
L
q
q
q


(
Vqn = K q sgn s(Iq )
)
5. Converter
(24)
For n=1, the direct current control manifold equation can
be obtained from equation (6) as follow
s(Id ) = Idref − Id
(25)
ɺ d ) = ɺI dref − ɺI d
s(I
(26)
The equivalent circuit of a voltage source converter is
presented in Fig.2.2. A sit can be seen in the figure3, a
three phase converter has 6 semiconductors (IGBTs)
displayed in three legs: a, b and c. The 6 semiconductors
are considered as ideal switches. Only one switch on the
same leg can be conducting at the same time [8].
Substituting the expression of given by equation (1) in
equation (26) we obtain
Lq
Rs
1
Id − Pωr
Iq −
Vd
Ld
Ld
Ld
(36)
Where:
K q : Positive constant
The direct current error is defined by
ɺ d ) = ɺIdref +
s(i
(35)
(23)
B. Direct Current Controller [7]:
ed = Idref − Id
For n=1, the quadrature current control manifold equation
can be obtained from equation (6) as follow
The control voltage Vqref is defined by:
The current control iq is defined by :
Vqref = Vqeq + Vqn
(31)
(27)
The control voltage Vdref is defined by:
Vdref = Vdeq + Vdn
(28)
Lq 

R
Vdeq =  ɺIdref + s Id − Pωr
Iq  Ld
Ld
Ld 

(29)
Fig. 3. Voltage Source Converter
In the figure, Sa, Sb, Sc are variables which represent the
switching status for each leg. S can only have two values:
1 for the conduction state and 0 for the block state. The
desired output voltages are obtained by programming the
duty cycles of the 6 IGBTs.
High accuracy and strong robustness of the sliding mode
control are providing by Fig.6, when resistance load (100
Ω, 10000 Ω) are applied. A rapid response is obtained and
the introduced perturbation is immediately rejected by the
control system
According to for the connection presented in Fig.3 the
applied voltages at the machine terminals, Uan, Ubn and
Ucn and the DC-link current IDC may be found as
7. Conclusion
 Uan 
 2 −1 −1 Sa 

 U DC 
 
 U bn  = 3  −1 2 −1 Sb 
 Ucn 
 −1 −1 2   Sc 
I DC = Sa
Sb
i a 
 
Sc  i b 
 ic 
(37)
In this approach the components I d and I q is regulated
(38)
C
(
(39)
)
dVdc 1
V
= Sd Vdc + Sq Vdc − dc
dt
2
RL
using (SMC) control, so that I d is zero, the controller is
designed in the total system including switching devices.
The simulation results show, Fast response without
overshoot and robust performance to parametric variation
and disturbances in all the system.
And the model of the PWM converter is :
1

 Vsd = 2 Sd Vdc

V = 1 S V
 sq 2 q dc
A sliding mode control method has been proposed and
used for the control of a permanent magnet synchronous
generator. Simulation Results show good performances
obtained with proposed control, with a good choice of
parameters of control. The DC voltage control operates
with enough stability.
(40)
Vdc is the DC voltage
6. Simulation Results
Extensive simulations have been performed using
Matlab/Simulink software to examine control algorithm of
the Sliding Mode Control of permanent magnet
synchronous Generator.
In order to vali1date the control strategies as
discussed, digital simulation studies were made the
system described in figure 2.
Three phases PMSG parameters:
Rs=1.4 Ω, φ=0.154wb, Ld =6.6mH, Lq =5.8mH
F=0.00038N.m.s/rad, J=0.00176 kg.m2
The DC voltage regulation is obtained using the proposed
algorithm controller in spite of the presence of
disturbances such as step changing of the resistive load
and the mechanical speed (when the SEIG is driven by a
wind turbine for example)
The validity of the control is demonstrated through the
results shown in Fig. (4, 5, 6) the response for step in DC
voltage command is chow in Fig.4, the DC voltage
response is completely robust with perfect rejection of
load disturbances,
Fig.5, show the response of the components id and iq .
The stator currents which are controlled by PWM
generated by (SMC) control.
References
[1] N. Selvaganesan1and R. Saraswathy Ramya “A Simple
Fuzzy Modeling of Permanent Magnet Synchronous
Generator” ELEKTRIKA VOL. 11, NO. 1, 2009, 38-43
Faculty of Electrical Engineering Universiti Teknologi
Malaysia.
[2] O. Gjinim T. Kaneko and H. Ohaswa. “A new controller for
PMSM Servo Drive Based on the Sliding Mode Control with
Parameter Adaptation” D 123 N6, 2003 Japan
[3] S. Xin, PMSM Servo System Sliding Mode Variable
Structure Control, Master's degree thesis, essays of
professional control theory and control engineering Wuhan
University of Technology, P.R.China 2007
[4] P. Pragasan, and R. Krishnan, “Modeling of permanent
magnet motor drives”, IEEE Trans. Industrial electronics,
Vol. 35, no.4, Nov. 1988.
[5] A. Massoum, M-K. Fellah, A. Meroufe, Sliding mode
control for a permanent magnet synchronous machine fed by
three levels inverter using a singular perturbation
decoupling, Journal Of Electrical & Electronics Engineering.
Vol 5. Istanbul university – 2005
[6] L. Louze, A.L. Nemmour, A. Khezzar, M.E. Hacil and M.
Boucherma “Cascade sliding mode controller for self-excited
induction generator” Revue des Energies Renouvelables Vol.
12 N°4 (2009) 617 – 626 Ohm
[7] A. G. Assaoui, M. Abid, “Sensorless Control of Permanent
Magnet Synchronous Motors”, 2nd International Conference
on Electrical Engineering Design and Technologies
November 8-10, 2008 Hammamet, Tunisia
[8] M. O. Mora “Sensorless vector control of PMSG for wind
turbine applications” Master Thesis, Institute of Energy
Technology Aalborg University, June 2009.
[9] C. Arroyo, Modeling and simulation of permanent magnet
synchronous motor drive system, thesis Master of Science
electrical engineering University of Puerto rico Spain, 2006
[10] L. G. González, E. Figueres , G. Garcerá , O. Carranza
“Synchronization Techniques Comparison for Sensorless
Control applied to PMSG” International Conference on
Renewable Energies and Power Quality (ICREPQ’09)
Valencia (Spain), 15th to 17th April, 2009
160
250
150
200
D C V o lta g e [V ]
S p e e d [ra d /s ]
140
130
120
110
100
150
100
90
50
80
70
0
0.5
1
1.5
2
2.5
0
3
0
0.5
1
1.5
Time [s]
2
2.5
3
2
2.2
Time [s]
2
221
-2
D C V o lta g e [V ]
T o rq u e T e [N .m ]
0
-4
-6
-8
220.5
220
219.5
-10
-12
0
0.5
1
1.5
2
2.5
3
219
1
1.2
1.4
Time [s]
1.6
1.8
Time [s]
30
5
20
0
C o u ra n t Id a n d Iq [A ]
C o u ra n t Ia [A ]
Fig. 4. Response under load disturbance (variation of speed)
10
0
-10
-20
-30
-5
-10
-15
-20
-25
0
0.5
1
1.5
2
2.5
3
-30
Id
Iq
0
0.5
1
1.5
Time [s]
Time [s]
Fig. 5. Stator current under load disturbance
2
2.5
3
250
200
DC Voltage [V]
8000
6000
4000
2000
150
100
50
0
0
0.5
1
1.5
2
2.5
0
3
0
0.5
1
1.5
Time [s]
Time [s]
30
20
Courant Ia [A]
Resistance RL [Ohm]
10000
10
0
-10
-20
-30
0
0.5
1
1.5
2
2.5
3
Time [s]
Fig. 6. Response under load disturbance (Variation of resistive load)
2
2.5
3

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