2012 American Control Conference Fairmont Queen Elizabeth, Montreal, Canada June 27-June 29,2012
Torque Controller via Second Order Sliding Modes of WRIG Impelled by DC-Motor for Application in Wind Systems O. A. Morfin, A. G. Loukianov, R. Ruiz, E. N. Sanchez, M. I. Castellanos, F. A. Valenzuela for to do experimental testing into laboratory [2]. In addition, it has built test stations and training, so that this technology of electric power generation using wind energy is more familiar to people [3]. Therefore, we have decided built a work bench formed by a DC motor- WR induction generator group to validate experimentally the designed controllers, and promote research with this technology.
Abstract-In this paper, the authors propose three robust non-linear controllers based on a second order sliding mode technique named super-twisting method which can be applied into work bench for research related to wind system. The main control scheme is proposed to control the electromagnetic torque and stator power factor of a wound rotor induction generator connected to a local electrical network and mechanically coupled with a DC-motor whose velocity is controlled applying feedback linearization technique and supertwisting algorithm; it is known that the DC-motor can be emulate the operation of the wind turbine. Finally, a grid side converter controller is proposed to regulate the DC link bus located between to power converters connected in "back to back" standard configuration, applying linearization by block control and super-twisting algorithm. The performance of the designed controllers for the DC motor-WR induction generator group is validated through real time implementation.
I.
We propose three control schemes based on a second order sliding mode (SOSM) technique named super-twisting algorithm [4] which ensures robust tracking of the controlled output variables. The SOSM algorithms are characterized by having a bounded continuous control with discontinuities only in the control derivative [5]. These characteristics help to significantly reduce the chattering effect, i.e. dangerous high-frequency vibrations of the controlled system. In this research, the rotor side converter controller, named as torque controller of the wound rotor induction generator (WRIG) is the main; in addition, we propose regulate the angular velocity of the system by the DC-motor velocity controller, and regulate the DC-link bus into AC/DCIAC converter by the grid side converter controller; and therefore, the main control goals are to regulate the electromagnetic torque and stator power factor of the WRIG. The experimental results are obtained using a DC motor-WR induction generator group, a Dspace RTI 1104 system, and two IGBT's inverter modules, and they validate the performance of the three proposed controllers.
INTRODUCTION
T
HE renewable energy sources has been a topic of much interest in last decades, due to environmental concerns and operation cost. Today, much attention has been placed in wind power systems as an alternative energy source, due to clean production and endless power caused by wind energy. However, the main problems regarding wind power systems are related with wind irregular characteristics which are: wind speed is a random, strongly non-stationary process, and it has turbulence and extreme variations. In addition, the users demand electrical energy quality involving a good power factor, minimum current harmonic distortion and no flickers. Therefore, the wind power system performance depends strongly in the contribution of a robust automatic control system [1].
This paper is a previous work to control a wind system through a DC motor-WR induction generator group, where the operation turbine is emulated with a DC-motor and the control goals are maximize the capture of the wind energy, and regulate the stator power factor [2].
On the other hand, the trend of researches in universities about the wind systems is to build a work bench
II.
The Fig. 1 shows a scheme of a DC motor - WR induction generator group which can emulate a small-scale wind system. The WRIG may be supplied with energy at both the rotor and stator terminals. The stator winding's terminals are connected directly to electrical grid, while the rotor winding's terminals are feed by an AC/DCIAC power electronic converter via slip-rings and brushes. This converter is located between the rotor circuit and threephase feeding bus, and it is able of handling electric power in both directions [1]. The DC motor is the impeller and it
O. A. Morfin, and M. 1. Castellanos are with Universidad Autonoma de Ciudad Juarez, Chihuhua, Mexico, (phone: +52 656 688-4841; fax: +52 656688-4841; e-mail: [omorfin][mcastell]@uacj.mx). A. G. Loukianov is with CINVESTAV DEL 1.P.N., Unidad Guadalajara; on sabbatical leave at Department of Electrical Engineering, DIELEC, Universidad de Guadalajara, Jalisco, Mexico, (
[email protected]). R. Ruiz, and E. N. Sanchez are with CINVESTAV del 1.P.N., Unidad Guadalajara, Jalisco, Mexico, (phone: +52 33 3777-3600; fax: +52 33 3777-3609; e-mail: [rruiz][sanchez]@gdl.cinvestav.mx). F. A. Valenzuela is with Universidad Juarez Aut6noma de Tabasco, Cunduacan, Tabasco, Mexico, (phone: +52 99 3358-1577; fax:+52 91 4336-0940; e-mail:
[email protected]). 978-1-4577 -1096-4/12/$26.00 ©20 12 AACC
PROBLEM STATEMENT
985
works with a constant level of magnetization. This motorgenerator group has three independent controls: the DC motor velocity controller, WR induction generator torque controller, and grid side converter controller.
Now, a sliding surface s is formulated as a linear combination of the tracking error £1 and its derivative £2:
To enforce a 8M motion in the system we design the controller using the super-twisting algorithm [4],
~Qs
PS~
ua
= ~ IslYz sign s +U
j ,
. {Va' U
Vabs los lor Ibs Dr Ics Icr
DC motor -
if cl > 0 then the tracking error
=~ (K i - B OJ J mae m e
&1
(t) asymptotically tends
to zero. The control input (5) is implemented in real time using PWM which is the feeding of the bridge IGBT's gates to establish the adequate voltage for armature winding.
The mathematical model of a DC motor with a constant level of magnetization is governed by [6]
dt
= ~signs, lual:o:Va,
(6)
WR induction generator group control.
I. Dc MOTOR VELOCITY CONTROLLER
d OJm
(5)
Iual > Va
where Va is the nominal value of the DC-motor's armature voltage. In [4] and [12] it was shown that there exist }q>O and A2>O such that the state vector of the closed-loop system (4) - (5) converges to the surface s == 0 in finite time (see subsection VII). The a 8M motion is described by
vea. Fig. 1
j
T ) e
di 1 _ a ==--(R i +K OJ -u ) dt L aa m m a'
II. WOUND ROTOR INDUCTION GENERATOR MODEL
(1)
The WRIG is provided with three-phase windings on both the stator and rotor. These windings have a symmetric physical structure and they are electrically balanced. It is a common practice applying a similitude transformation [8] to refer all the electrical variables from abc three-phase system to a new base named dq orthogonal coordinate frame whose axes rotate agree with the electric network frequency [6].
a
where OJm and i a are the shaft angular velocity and armature current; R; and La are the armature winding resistance and inductance, respectively; K; is a motor constant, T; is the electromagnetic torque produced by the WR induction generator; J m and B m are the inertia moment and frictional coefficient of shafts' train of both machines, and u; is the armature feeding voltage which is the control input for the plant. To design a 8M controller, a tracking error is defined
Thus, the WR induction generator model in dq frame has the following form:
d:
di
(2) The dynamics of this variable can be obtained using the plant model (1), and defining a new error variable £2 which is the derivative of £1, we have
==
Allis +A l2ir +A l3vs +Blv r
(7)
where the voltage and current vectors for the winding stator and rotor are defined, respectively, as T
vs
= [vds
v J' qs'
iT s
= [ids
i J' qs'
T
vr
= [vdr
v J' qr'
iT r
= [idr
i J' qr'
and the matrices parameters of (7) are shown in [8]. The plant mathematical model (7) is fourth order, has two known inputs which are the grid voltages, Vds and vqs ; and two control inputs by the rotor circuit, Vdr and vqr .
Having this transformation, the plant model (1) is represented in the new variables £1 and £2 in reduced to the following canonical form:
III. WR INDUCTION GENERATOR TORQUE CONTROLLER
(4)
The control objectives for WR induction generator controller are: 1) Achieve a set point of the electromagnetic torque Te , 986
2) Keep the stator power factor in a value desired pf,
s* =G(is,ir)+v r
Therefore, the vector of selected control output variables consist of two components: the electromagnetic torque Te , and stator reactive power Qs. The mathematical expression for these outputs is given by [8]
[J:
Qs ]T s == [i™T i r
0
'i4 PL
vsMQi s
where M
T
=
3 --PL
4
=
MQ
d; = A 2lis +A 22 i r +A 23vs +B 2vr,
where the current rotor vector i, is considered as the internal (residual) dynamics variable. To induce sliding mode on s* == 0 (14) and s* == 0 (15), we apply the super-twisting algorithm (4)
(8)
3
0
m
0
m
r· -
2
3
l-. > IV I Iv I< IV I,
2
a: r
where V dc == [Vdc
=
[I:ret
p
f
Substituting (8) and (9) into (10) yields
( s
2
T
-b20~
s = [GT
(9)
ere!
The system's model of the grid side converter in a wind system application is constituted by a bridge three phase inverter with six IGBT's (insulated gate bipolar transistor), a capacitor bank, three coupling coils, and the electric network [9]; see Fig. 1.
vsMQi s
-or: r·
This system is modeled agree to the equilibrium voltages at coil terminals in the three phases, matching the electric power between the input and output of the bridge inverter. Afterward, the system is represented in dq coordinate frame applying the transformation T dq [8] whereupon the model takes a simple form, like a DC system. Therefore, the system's model of the grid side converter takes the following form [10]:
(11)
r
G = a, Q
(12)
dv dc 1 3 . -----v 1 dt - C 2 ds ds b v dc
d .
[i~MTBI + i~MTB2 ] T
d .
-1 d t qs .
[vdr vqrJ, we apply the following (14)
=
G(is,i r)+ Yr'
1
c
c
=
R
•
•
1
c --1 -lVs 1ds - v L qs L ss
c
(18)
>
c
The system (18) is uncoupled respect to the two control inputs Vdg and V qg; therefore, two outputs are selected to be controlled which are:
Thus
=B~l (is,ir)s+B~1 (is,ir)[f(is,ir)+BIVr ]
1
where ids, i qs, Vds and vqs are the grid current and voltages on dq coordinate frame, lVs is electric frequency velocity, R; and L; are the resistance and inductance of the coupling coils, respectively; and Cb is the capacitor's capacitance connected at the link DC-bus.
transformation:
s*
.
c
vsMQB I Therefore, to decouple the system (13) with respect to the =
R; .
-1 +-V d t ds =--1 L ds +lV1 s qs L ds - L vdg
However, the system (13) is coupled with respect to the control input V n with the matrix:
control inputs v~
is the DC-link voltage of the
IV. GRID SIDE CONVERTER CONTROLLER
(13)
(Is' r, ) -
dc
control errors (10) converges to zero in finite time (see subsection, VII), and the control goals are achieved.
with the following dynamics [8]
BI
J, V
A and a have components values can be chosen such that the
]T
The dynamics of both output variables (11) have the relative degree one; then, the sliding manifold is defined as
_
Vdc
AC/DCIAC converter, see Fig. 1. The the diagonal matrices
o
Define the tracking error vector as
. .
dc
r
where Terej is a set point of the electromagnetic torque, lVsy n is the synchronous velocity in rad/s, and pf, is the desired stator power factor.
E=[isMTi r
(17)
dc
0
--
Meanwhile, the reference vector for the two output variables (8) is defined as [8]
[Teret
(16)
di
1) The DC link voltage between two converters
(15)
Vdc,
2) The reactive power delivered to the electric network by the circuit rotor, Qc.
Using (14) and (15), the WRIG system (7) is represented in the new variables of the following form: 987
To control the tracking DC-voltage error, the minus sign is used in both the control law gains Ai and ai, while to control the tracking reactive power error, the plus sign is used.
The main control goal of the grid side converter controller is to keep the DC link voltage constant at desired value regardless of the direction of rotor power flow [1OJ. Therefore, define the tracking error for the DC bus voltage & I = V dcref - V de , ( 19) where
vdcrejis
v.
STABILITY ANALYSIS
Substituting the control law of super-twisting algorithm per component (27) into (26) yields
the set point of DC voltage.
Applying a feedback linearization technique named block control [11], the dynamics of £1, involving a desired stable dynamics -kl &1 with k; > 0 , takes the form
. -_ -Ai'] ISn lli SIgn · (Sn ) + Si2 sn Si2
=
+ gi
(28)
i = d, q.
-aisign (Sn)
where
(20) From (20), the stator current reference idsrej, which establishes the first order dynamic of £1, is solved as .
I
2CbktVdeGt
dsref
==----
3V
(21)
ds
A second error variable can be defined as (22)
Applying the following transformation [12]:
and its dynamics are • &
didsref
C;~ = [ISill~ sign (Sil ),
dids
(23)
=----
dt
2
dt·
Si2
J,
I~ill = ISill~ ,
(29)
the system (28) is represented of the following form:
Finally, the third error variable, involving the reactive power, is defined as
(30)
(24) and its dynamics, with a set point for the power reactive reference, take the form . . . 3 diqS &3 = Qeref - Qe = VdS --;;[. (25)
To analyze the SM stability conditions for the system (30), we apply the following Lyapunov function [12]
"2
Joining (20), (23), and (25), the system (18) represented in block control form, as follows
&1
=
-kl &1
n,
s 2 -OJ& &. = - -k, && --& -2 V 1 2 L 2 3v 3 de e ds
e k, +R+( V L
de
e
~
e
3
3v
OJ
= ~&
2
2
R ----.£& L 3 e
(31)
t;il
is
where
J.
e
1
dsref
Taking its derivative along the trajectories of (30) yields
2 OJs 1 1 +--Q - - v +-v 3v cref L ds L dg .
v: (c;i ) = _11I c;~Pic;i ,
(26)
e
3v
OJ
-~i
2
dsref
R +---.£Q L cref e
3v 2L
(32)
-~V
qg.
e
The system (26) is the 3th order, minimum phase, and it is decoupled with respect to control inputs Vdg and V qg. From (26), we choose the sliding variables as
Assuming that the perturbation term is bounded by
Igil = s, ISill~,
s, ~ 0,
(33)
and using (33) into (32) it can be shown that [12] We use again the super-twisting algorithm:
(34) (27)
i=d,q.
988
power factor constant fp, = 1, thereafter, at 10.0s the torque is reduced again to 0.4 N.m. For the super-twisting controller, we use the gain values Ad = 147, ad = 97, Aq = 147, and a q = 97.
where
-(.It -~8.) I
2
I
For Qi be definite positive, the controller gains must complain the following relationships:
A. > 28., and a. I
I
I
As result, the variable zero in finite time I [12]
£2
Ai
1 Ai 2 8. . 8 Ai _ 28 i
= -
(23) and
2~ ~2 (O)Sil
(35)
I
£3
and
Ui
(36)
(25), converge to
(0)
(37)
where Fig. 3. Tracking for output variables, T; varies from 0.4 to 1.0 N.
VI.
The electromagnetic torque and stator reactive power responses achieve a good tracking performance as is shown in Fig. 3a and 3b, respectively. Meanwhile, the rotor voltages Vdr and v qn as control inputs of the system are bounded, see Fig. 4.
EXPERIMENTAL RESULTS
The prototype consist on a DC motor - WR induction generator group whose capacity and their parameters are presented in Appendix A, two modules of IGBT's inverters from the company Semikron, two DS1104 data acquisition boards with RTI interface software to show the controller performance, and TTL-CMOS and measurement interfaces, for signal conditioning, Fig. 2. The DC-motor velocity controller establishes the operation velocity at 184 rad/s, and the grid side converter controller regulates the DC link voltage at 100 V, whereas the rotor side converter controller regulates the electromagnetic torque and stator's unity power factor. The results for validating the three proposed controllers are reported in real time.
Fig. 4. Rotor voltages on d and q axes as control inputs.
The sliding manifold on d an q axes are in zero value in steady state and their low oscillations are shown, when the reference function for the electromagnetic torque have changed, see Fig. 5.
Fig. 5. Sliding manifold on d andq axes.
Fig. 2. DC motor- WR induction generator group test prototype.
The stator and rotor current, both in phase A, are shown in Fig. 6; the stator current increase when the set point of the electromagnetic torque is changed from 0.4 to 1.0 N.m.
We have set up the following experimental event: at 2 s, the electromagnetic torque reference Terej is increased from 0.4 to 1.0 N.m (nominal torque) by keeping the stator unity 989
Whereas, the rotor current keep its low frequency due to the slip velocity.
ApPENDIX A Table 1. Induction generator and DC motor parameters
WR Induction Generator Power Stator voltage Rotor voltage Rater stator current Rater rotor current Stator resistance, R, Rotor resistance, R, Stator inductance, L, Magnetizing ind., L m Rotor inductance, L, Inertia moment, J 2
Fig. 6. Stator and rotor currents in phase -A.
Stator-rotor turn ratio
Frictional coeff., B 2
The controller performance of the grid side converter controller is shown at Fig. 7, the controlled output Vdc is kept at the set point of 100 V DC-bus regardless of the torque changes, and the control inputs v dg and v qg are bounded as shown in the same figure. Moreover, the angular velocity is regulated at 184 rad/s by the DC-motor velocity controller, it keeps unchanged in all experimentation, see Fig. 8.
[4]
Fig. 7. Tracking for output variables, fp, changes from 1.0 to 0.7. 200..-------------,------------,---------------..-,.
[5]
180
[6]
g,
160
'J
140
[7]
~ 120
[8]
100L..-------------'------------'----------------..-J. 10 15 o time (5)
3.1mH 0.360 2128 JlF
Rc Cb
1. Munteanu, A. Bractu, and E. Ceanga, Optimal Control of Wind Energy Systems, Springer, 2008. B. Rabelo, and W. Hofmann, "Power flow optimization and grid integration of wind turbines with the doubly-fed induction generator", Power Electronics Specialists Conference, PESC'06, pp. 2930-2936, Recife, Brazil 2005. J. R. Arribas, C. Veganzones, F. Blazquez, C. A. Platero, D. Ramirez, S. Martinez, J. A. Sanchez, and N. Herrero, "Computer-based simulation and scaled laboratory bench system for the teaching and training of engineers on the control of doubly fed induction wind generators", IEEE Trans. on power systems, vol. 26, no.3, pp. 15341543, 2011. L. Fridman, and A Levant, "Higher order sliding modes as a natural phenomenon in control theory", Lectures Notes in Control and Information Science, Springer-Verlag, vol. 217, pp. 107-133,1995. A. Levant, "Sliding order and sliding accuracy in sliding mode control", Int. J. Control, vol. 58, no. 6, pp. 1247-1263, 1993. P. Krause, O. Wasynczuk, and S. Sudhoff, Analysis of Electric Machinery and Drive Systems, 2° edition, IEEE Press Power Engineering Series, 2002. V. 1. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electromechanical Systems, Taylor & Francis, 1999. O. Morfin, 1. Canedo, and A. Loukianov, "Direct Electromagnetic Torque Controller of a wound rotor induction generator via second order sliding modes", 6 th International Conference on Electrical
Engineering, Computing Science and Automatic Control, ICEEE 2009, Toluca, Mexico, 2009.
Fig. 8. Angular velocity.
VII.
t,
REFERENCES
[3]
!
3AHP 100 V 7.5 A 0.6 A 1750 rpm 1.270 0.821 0.0022 0.0015
This work was supported in part by the UACJ and PROMEP under grant PROMEP/103.5/11/901 Folio UACJ/EXB/155 and CONACYT under grant 129591.
[2]
:5o
DC Motor Power Field Voltage, Vf Armat. current, I, Field current, If Rotor velocity, W m Armature resist. R, Armature indue. La Inertia moment, Jm Friction coeff., Bm
ACKNOWLEDGMENT
[1]
~
180W 208/120 V 104/60V 0.86/1.5 A 1.15/2 A 12.60 16.70 0.376 H 0.352 H 0.38 H 0.0016 Nms2 516/264 0.00094 Nms
[9]
O. Barambones, P. Alkorta, and M. De La Sen, "Wind turbine output power maximization based on sliding mode control strategy", IEEE International Symposium on Industrial Electronics, ISlE 2010, pp. 364-369, Bari, Italy, 2010. [10] O. A. Morfin, A. G. Loukianov, R. Ruiz. E. N. Sanche, F. Valenzuela, M. 1. Castellanos, "Grid Side Converter Controller Applied in Wind Systems via Second Order Sliding Modes", 8 th International
CONCLUSION
A control scheme based on second order SM supertwisting algorithm, as main controller, has been proposed to regulate the electromagnetic torque Te , and stator power factor Pfs of a WR induction generator. Meanwhile, the DCmotor controller regulates the angular velocity and the grid side converter controller regulates the DC-link voltages into AC/DCIAC converter applying super-twisting algorithm, too. The experimental results show a robust performance of the three controllers achieving the control goals of each when the set point for the electromagnetic torque is changed.
Conference on Electrical Engineering, Computing Science and Automatic Control, ICEEE 2011, Merida, Mexico, 2011. [11] A. Loukianov, "Robust block decomposition sliding mode control design", Mathematical Problems in Engineering, vol. 8, pp. 349-365, 2002. [12] 1. Moreno, M. Osorio, "A Lyapunov approach to second-order sliding mode controllers and observers", IEEE Conference on Decision and Control, CDC 2008, pp. 2856-2861, Cancun, Mexico, 2008.
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