4 are considerably smaller when magnetic saturation is included in the control algorithm, observer, and load torque estimator than in the case when magnetic saturation is neglected.. In
Trang 1II-10 Impact of Magnetic Saturation of Induction Motor 253 Taking into account equations (18), the position controller and the rotor flux linkage controller are given by (19)
vα = k α0 e1+ k α1 ˙e1+ k α2 ¨e1+ y···∗
1
v β = k β1 e2+ k β2 ˙e2+ ¨y∗
After inserting (19) in the linearized system (15), the tracking error dynamics of the closed loop system is given by (20)
e
···1+ k α2 ¨e1+ k α1 ˙e1+ k α0 e1= 0
with k(·)being positive constants The desired dynamics of the tracking errors e1and e2is assured by selecting the corresponding eigenvaluesλ( ·)of the characteristic equations (21).
λ3+ k α2 λ2+ k α1 λ + k α0= 0
Observer design
The state variables of the selected IM model are necessary to realize control, based on the described input-output linearization The corresponding observer, similar to the one presented in [15], is given by (22) It is based on the electromagnetic subsystem of the
two-phase is , Ψrstate-space IM model (3) in theαβ reference frame The coefficients k( ·)
are determined in the literature [14]
d
dt
⎡
⎢
⎢
⎣
ˆi s α
ˆi s β
ˆ
ψr α
ˆ
ψr β
⎤
⎥
⎥
⎦= (C + ω rW)
⎡
⎢
⎢
⎣
ˆi s α
ˆi s β
ˆ
ψr α
ˆ
ψr β
⎤
⎥
⎥
⎦+ D
u s α
us β
+
⎡
⎢
⎢
⎣
k1 −ω r k2
−ω r k2 k1
k3 −ω r k4
−ω r k4 k3
⎤
⎥
⎥
⎦
ˆi s α
ˆi s β
−
is α
is β
(22) The symbol (ˆ·) denotes the observed values
Experimental results
The experiments have been performed to test the proposed input-output linearizing tracking control The elements of the experimental system are the three-phase Semikron IGBT inverter, the three-phase 3 kW IM Sever with wound rotor, whose parameters are given in Appendix C, and the DC motor Mavilor Mo2000 with an Infranor DC power converter, as the dynamic load The control algorithm was executed on the dSPACE DS1103 microcontroller board A block diagram of the proposed IM drive’s tracking control that includes magnetic saturation is presented in Fig 2
Experiments were done using the reference value2 ∗
r = 1.6 (V s)2
The smooth refer-ence trajectories for the positionθr and the speedωr were generated from the kinematic
model and are shown in Figs 3(a) and 4(a) The step changes of the load torque t lvs time are shown in Fig 3(d) The results of the input-output linearizing tracking control with the included saturation were compared with the results obtained with the same type of control
Trang 2Encoder
E (x)−1
D(x)
Observer
Inductances calculation (saturation)
Load torque estimator
T(x)
(y 1 , y 1 ,y 1 , y 1)*
(y 2 , y 2 ,y 2)*
u s β
u 2
2
3
3
s α
v s α
ω r
i a
i b
i c
−
Figure 2 Block diagram of the IM’s input-output linearizing tracking control.
Figure 3 Reference and measured rotor position trajectory ∗
Trang 3II-10 Impact of Magnetic Saturation of Induction Motor 255
Figure 4 Reference and measured rotor speed trajectoryω∗
without included saturation [11] The settings of the controllers were equal in both cases:
k α0 = 750,000, k α1 = 25,000, k α2 = 275, k β1 = 90,000, and k β2= 600
An analysis of the results showed that the position error r in Fig 3 and the rotor speed error rin Fig 4 are considerably smaller when magnetic saturation is included in the control algorithm, observer, and load torque estimator than in the case when magnetic saturation is neglected
It is obvious from the results in Fig 5 that tracking control with the included magnetic
saturation performed the position task with a slightly higher stator current i s= i2
s α + i2
s β ,
than the one without saturation In contrast, tracking control without any included mag-netic saturation required smaller stator current to perform the same task at no-load, but it
responded with a much higher increase in stator current i s , when the motor was loaded with
step changes of the load torque (Fig 5a)
The reason for the described behavior of the controlled IM in Fig 5 can be explained if the controlled system is analyzed together with the observer and, if only for explanation, the
stator currents i s α , is β are transformed to the common dq reference frame, i.e to the stator currents i sd, isq (Fig 6) The observer of electromagnetic state variables, with included magnetic saturation yields a smaller rotor flux linkage moduler for equal stator current value than the linear observer introduced in [15] Accordingly, the input-output linearizing tracking control with included magnetic saturation increases the magnetizing stator current
isd in the direction of the rotor flux linkage vector, to achieve the reference value of the rotor flux linkage module Therefore, the IM with the proposed input-output linearizing control is going to be magnetized in the best possible way to ensure the proper stiffness and optimal dynamic response When the step changes of the load torque are applied on
Trang 4256 Dolinar et al.
Figure 5 Stator currents i s α , is β , and is= i2
sα − i2
(b) saturation is included
the shaft, the input-output linearizing control with included magnetic saturation performs much better than the control with neglected saturation, requiring smaller stator current
isq to produce the necessary torque with the rotor flux linkage vector The transformed
currents are shown in Fig 6 The measured stator current i sdin the case of the input-output linearizing tracking control with and without included magnetic saturation agrees with the
corresponding value of i sddetermined from the nonlinear and linearized magnetizing curve
of the IM, used in the observer with and without included magnetic saturation (Fig C1, Appendix C)
Figure 6 Stator currents i sd and isq
Trang 5II-10 Impact of Magnetic Saturation of Induction Motor 257
Conclusion
Consideration of magnetic saturation in the IM model substantially improves its accuracy, leading to a more efficient and consistent synthesis of the control algorithm, observer, and estimator of load torque The proposed input-output linearizing control of IM with included magnetic saturation improves the dynamic performance of the drive It gives smaller rotor position and speed errors, as well as a higher stiffness and a better load torque rejection, which results in a smaller stator current, when the load torque is introduced An important reason for the improved behavior of the controlled IM is more adequately calculated value
of the rotor flux linkage when magnetic saturation is considered in the observer design
Appendix A
Elements of matrices C, Z, W, D
L i m = L2
l − L l L2rl
1
Lrl + L m
L rl + L +
L¸rl4 (L rl + L m ) (L rl + L)
c11 = − 1
L i m
Rs
L l− L2rl Lqq + R r
Lm
L r
L l − L2
rl
1
Lrl + L m + 1
Lrl + L
−Lsl Lrl
L dd
(L rl + L m ) (L rl + L)
c12 = 1
Li m
Rs L
2
rl Ldq − R r
Lm Lr
Lsl Lrl
L dq
c13 = 1
Li m
Rr
Lr
L l − L2
rl
1
Lrl + L m
L rl + L −
L sl Lrl
L dd
(L rl + L m ) (L rl + L)
c14 = − 1
L i m
Rr
Lr
L sl Lrl
L dq
, c21= c12, c23= c14
c22 = − 1
Li m
R s
L l− L rl2
L dd + R r
L m Lr
L l − L2
rl
1
L rl + L m
Lrl + L
−Lsl Lrl
L qq + L3rl
(L rl + L m ) (L rl + L)
c44 = −Rr
Lr = c33
c24 = 1
Li m
Rr
L r
Ll − L2
rl
1
Lrl + L m + 1
Lrl + L −
Lsl Lrl Lqq
(L rl + L m ) (L rl + L)
c31 = R r
L m
Lr = c42, c33= −Rr
Lr = c44, c42 = R r
L m
Lr = c31
Trang 6258 Dolinar et al.
z12 = 1
L i m
Ll−L rl2
L r Ll− L2rl
L qq
z13 = − 1
L i m
L m
L r
L2
rl
L dq + L sl L rl
L dq
z21 = − 1
L i m
L l− L2rl
L r Ll− L rl2
L dd , z22 = z11
z14 = − 1
L i m
−L m
L r
Ll− L2rl
L qq +
Ll − L2
rl
1
L rl + L m
L rl + L
−L sl L rl
L dd + L3rl
(L rl + L m ) (L rl + L)
z23 = − 1
L i m
L m
L r
L l− L2rl
L dd +
−L l + L2
rl
1
L rl + L m
L rl + L
+L sl L rl
(L rl + L m ) (L rl + L)
z24 = z13, z34 = 1, z43= −1, w13= 1
L i m
Lsl Lrl Ldq
, w13 = 1
L i m
Lsl Lrl
L dq
w14 = 1
Li m
L l − L2
rl
1
Lrl + L m
Lrl + L −
Lsl Lrl
(L rl + L m ) (L rl + L)
w23 = − 1
L i m
Ll − L2
rl
1
L rl + L m
L rl + L −
L sl Lrl
L qq + L3rl
(L rl + L m ) (L rl + L)
w24 = −w13, w34 = −1, w43 = 1
d11 = 1
L i m
Ll− L2rl
L qq
, d12= 1
L i m
L2
rl
L dq
d21 = d12, d22 = 1
L i m
Ll− L2rl
L dd
Appendix B
Lie derivatives
L3f φ1= ∂
∂x
L2f φ1
dx
dt = ∇L2f φ1
[f + Gu]
= p L m
L r
1
J
c22+ c33− f
J is βψr α−
c11+ c44− f
J is αψr β + c12
i s α ψr α − i s β ψr β− ω r
i s α ψr α + i s β ψr β + (c24+ c13− 2w13ωr)ψr αψr β
+ (c23+ w23ωr)ψ2
r α − (c14+ w14ωr)ψ2
r β
+ f
J2tl+ f2
J2ωr
L2f φ2 = ∂
∂x
L f φ2
dx
dt = ∇L f φ2
[f + Gu]
= 2R r
L m L
c11+ c33− 2c31
L is αψr α+
c22+ c44− 2c31
L is βψr β
Trang 7II-10 Impact of Magnetic Saturation of Induction Motor 259
+ c12
is βψr α + i s αψr β
+ ω r
is β ψr α − i s αψr β
+ (c14+ c23+ w14ωr + w23ωr)ψr α ψr β+
c13− 2c33
Lm ψ2
r α
+
c24− 2c33
Lm ψ2
r β + w31ωrψ2
r α − ψ2
r β
+ c31
i s2α + i2
s β
L g α L2f φ1= ∂x ∂ L f φ1
g α = p Lm Lr
1
J
d21ψr α − d11ψr β
Lg β L2f φ1= ∂
∂x
L f φ1
g β = p L m
L r
1
J
d22ψr α − d12ψr β
L g α L f φ2= ∂
∂x
L f φ2
g α = 2R r
Lm Lr
d11ψr α + d21ψr β
Lg β L f φ2= ∂x ∂ L f φ2
g β = 2R r
Lm
L r
d12ψr α + d22ψr β
Appendix C
Table 1 Parameters of the 3 kW induction motor with wound rotor Sever ZPD112MK4:
Figure C 1 Rotor flux linkage and corresponding stator current in the case of linear and nonlinear
magnetizing curve
Trang 8260 Dolinar et al.
Acknowledgment
This work was supported in part by the Slovene Ministry of Education, Science and Sport, Project No P2-0115
References
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Induction Motor Drives”, PESC’90 Record, San Antonio, TX, USA, pp 591–598, 1990
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matching, Lect Note Contr Inform Sci., Vol 160, pp 1435–1454, 1991
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motor with magnetic saturation, IEEE Trans Contr Syst Technol., Vol 7, No 3, pp 315–327, 1999
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Trang 9II-11 DIRECT POWER AND TORQUE CONTROL SCHEME FOR SPACE
VECTOR MODULATED AC/DC/AC
CONVERTER-FED INDUCTION MOTOR
M Jasinski, M P Kazmierkowski and M Zelechowski
Warsaw University of Technology, Institute of Control & Industrial Electronics,
ul Koszykowa 75, 00-662 Warszawa,
mja@isep.pw.edu.pl, mpk@isep.pw.edu.pl,
Abstract A novel control scheme for PWM rectifier-inverter system is proposed Fast control
strate-gies such as line voltage Sensorless Virtual Flux (VF) based Direct Power Control with Space Vector Modulator (DPC-SVM) for rectifier and Direct Torque Control with Space Vector Modulator (DTC-SVM) for inverter side are used These strategies lead to good dynamic and static behaviour of the proposed control system—Direct Power and Torque Control- Space Vector Modulated (DPTSVM) Simulations and experiment results obtained show good performance of the proposed system Addi-tional power feedforward loop from motor to rectifier control side improved dynamic behaviours of the power flow control As a result, better input-output energy matching allows decreasing the size of the dc-link capacitor
Introduction
The adjustable speed drives (ASD) with diode rectifier nowadays is the most popular on the marked Large electrolytic capacitor is used as an energy-storing device to decouple rectifier and the inverter circuits The capacitors have some drawbacks: low reliability, high size, weight and cost Hence, reliability of the dc-link capacitor is the major factor limiting the lifetime of the ASD systems [1]
Development of control methods for Pulse Width Modulated (PWM) boost rectifier (active rectifier) was possible thanks to advances in power semiconductors devices and Digital Signal Processors (DSP) Therefore, the Insulated Gate Bipolar Transistors (IGBT) AC/DC/AC converter controlled by PWM is used in motor drive systems (Fig.1) Thanks to active rectifier the dc-link capacitor can be reduced [2] Farther reduction of the capacitor can be achieved by power feedforward loop from motor side to the control of the PWM rectifier A lot of works are given attention to reduce the dc-link capacitor However, a small capacitance leads to a high dc-voltage fluctuation To avoid this drawback various dc-voltage control schemes have been proposed Some of them take into account the inverter dynamics
2006 Springer.
Trang 10262 Jasinski et al.
Figure 1 Representation of three-phase PWM rectifier—inverter system; vector diagram and
coor-dinate system for: a) PWM rectifier side b) inverter side
to improve the PWM rectifier current control by feedback linearization [3] and master-slave [1] manner Another control methodology proposed a fast dc-link voltage controller which works with dc-voltage and motor variables as inputs [4] Moreover, various methods of the output power estimation have been discussed in [5]
In the mentioned methods active and reactive powers of the PWM rectifier are indirectly controlled via current control loops Besides, stator current controllers control the torque and flux of the motor too
In this paper a line voltage sensorless Virtual Flux (VF) based Direct Power Control with Space Vector Modulator (DPC-SVM) is applied to control of the PWM rectifier
The inverter with induction motor is controlled via Direct Torque Control with Space Vector Modulator (DTCSVM) Contrary to the scheme proposed in [6], our solution includes not stator flux controller but space vector modulator
Hence, an AC/DC/AC converter of Fig 1, is controlled by Sensorless Direct Power and Torque Control-Space Vector Modulated (DPT-SVM) scheme In comparison to methods that control an active and reactive power, torque and flux in indirect manner the coordinates transformation and decoupling are not required Moreover, the current control loops are avoided
In respect of dynamic, of dc-voltage control the power balance between line and motor
is very important Therefore, to improve instantaneous input/output power matching, the additional feedforward power control loop is introduced
Thanks to better control of the power flow the fluctuation of the dc-link voltages will be decrease So the size of the dc-link capacitor can be reduced
... loop is introducedThanks to better control of the power flow the fluctuation of the dc-link voltages will be decrease So the size of the dc-link capacitor can be reduced
... data-page="8">260 Dolinar et al.
Acknowledgment
This work was supported in part by the Slovene Ministry of Education, Science and Sport, Project No P 2-0 115... performance of the proposed system Addi-tional power feedforward loop from motor to rectifier control side improved dynamic behaviours of the power flow control As a result, better input-output