9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
SLIDING MODE CURRENT CONTROLLER FOR THE THREE PHASE
SINGLE-STAGE AC/DC BUCK-BOOST CONVERTERS
2
V. Fernão Pires1,2, J. Fernando Silva
Escola Superior Tecnologia de Setúbal, Instituto Politécnico de Setúbal, Portugal
2
CAUTL, IST, SMEEP (24), Universidade Técnica Lisboa; Av. Rovisco Pais 1049, Portugal
1
Košice
Slovak Republic
2000
Abstract. This paper presents a sliding mode space vector DE current modulator for three-phase
single stage AC/DC Buck-boost rectifiers. Using this fast and robust control method, this type of
converters draw near sinusoidal input currents with high power factor. A Proportional Integral (PI)
controller can, then, be adopted to regulate the rectifier output voltage. The references of the sliding
mode current controller are sinusoidal waveforms whose amplitude is modulated by this external
voltage controller. Simulation and experimental results show the high power factor and low harmonic
distortion obtained.
Keywords: AC/DC converters, Power factor correction, Power quality, Sliding mode control.
1.
voltages or load parameter [9], [10], [11], [12].
Moreover, since the sliding mode controller actively
shapes the input line currents, it is possible to obtain a
near unity power factor and input currents with nearly
sinusoidal shape, even with high ripple in the DC
inductor current. Therefore, these AC to DC current
converters do not require a DC inductor with high
inductance, reducing the size, weight and cost. As will be
shown in this paper, this control scheme results in a very
simple overall controller that can be implemented directly
in hardware.
Some simulation and experimental results of two
prototype converters are presented for demonstration.
INTRODUCTION
In the last years, with the remarkable progress in the
development of power semiconductor devices, more and
more high frequency switching power source have been
used in industrial and home electronic equipment.
Usually, such systems use a combination of diode
rectifier or phase controlled rectifier with a bulk capacitor
as the primary dc source voltage. The advantages of these
circuits are their inherent ruggednes and simplicity.
However, due to combined effects of the smoothing filter
and phase control action, the input current waveform
becomes quite nonsinusoidal and phase shifted with
respect to the sinusoidal input voltage. These drawbacks
cause a number of problems in the power distribution
network and other electrical systems, such as heating of
core of transformers and electrical machines, large losses
on the power lines, the supply voltage distortion, high
reactive componets size and malfunctions in the electrical
equipment.
To overcome the above shortcomings, in recent years AC
to DC converters with input power factor have been
proposed. The main function of these converters is to
shape the input line current ir order to have a sinusoidal
waveform and a near unity power factor operation. The
Boost type [1], [2], [3] and the Buck type [4], [5], [6] AC
to DC converters with input power factor have been the
dominant design. Only few three-phase Buck-Boost type
AC to DC converters with input power factor have been
proposed [7], [8], despite advantages such as inherent
short circuit protection, ruggedness and step-up/down
output voltage characteristic. Therefore, this paper
focuses on two three-phase single stage AC to DC BuckBoost converters and their input current controller.
To control the input line currents, a new control system
based on a sliding mode approach with a space vector DE
current modulator is proposed. This current controller
takes into account the real time perturbations of line
2. THREE-PHASE SINGLE-STAGE BUCKBOOST AC/DC CONVERTER TOPOLOGIES
Fig. 1. shows the three-phase single-stage Buck-Boost
type ac-dc converter with six switches, used to obtain
high power factor and sinusoidal source currents. Since
the switches must have bipolar voltage blocking
capability, with devices such as punch-through IGBTs or
MOSFETs, additional series diodes will be required,
implying greater conduction loss. Fig. 2. shows the threephase single-stage Buck-Boost type ac-dc converter with
reduced number of switches (four switches). Therefore,
this topology has the advantage of decreasing the number
of switches, thus increasing the reliability and the
efficiency since the conduction losses in the switches are
reduced.
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
iDo
D1
Vs1
Vs2
Vs3
Lf , R f
is1
irec1
iC
Lf , R f
is2
f1
S1
D3
S3
D3
Cf
iC f3
D2
S2
VC
VC
f1
dt
VLo
VCo
Co
Ro
f2
D4
S4
Vo
+
2 J 12 J 22 J 32
1
iLo VC
(8)
dt
Co Ro o
2 Lo
where the time-dependent variable J k represent the states
dVCo
f3
Fig. 1. Three-phase six switches single stage BuckBoost AC/DC converter
iDo
V s1
V s2
V s3
Lf , Rf
is1
irec1
iC
Lf , Rf
is2
f1
iC
f2
VLo1 2
Cf
V
C f1
Cf2
of the switches of the k th , k{1,2,3} bridge leg, defined
as:
io
iCo1 1
Co1
VC o1
iLo2
irec3
D2
f3
S2
V
Lo1
\o1
S3
Do
Ro
iC
Cf
Cf
iLo2
irec2
Lf , Rf
is3
S1
D3
V
D4
Lo2
\
o2
Vo
+
VC o2
V Lo2 2
J1
­ 1 , if S1 is ON and
°
( S 4 is ON or S 6 is ON )
°
°
J2
® 1 , if S 2 is ON and
°
( S 3 is ON or S 5 is ON )
°
°¯ 0 , other combinatio ns
J3
­ 1 , if S5 is ON and
°
°° ( S 2 is ON or S 4 is ON )
® 1 , if S 6 is ON and
° ( S is ON or S is ON )
1
3
°
¯° 0 , other combinations
Co2
S4
C f3
Do
Fig. 2. Three-phase four switches single stage BuckBoost AC/DC converter
3.
MODELLING THE PROPOSED
TOPOLOGIES
disD
dt
dt
Rf
Lf
i s1 2
1
1
VC f 1 VC f 2 VC f 3 3L f
3L f
3L f
dis E
(1)
2
1
1
V s1 Vs 2 V s3
3L f
3L f
3L f
dt
dVC f D
dt
di s2
dt
Rf
1
2
1
VC f 1 VC f 2 VC f 3 3L f
3L f
3L f
1
2
1
V s1 Vs 2 Vs 3
3L f
3L f
3L f
Lf
i s2 (2)
dVC f E
dt
diLo
di s3
dt
Rf
1
1
2
VC f 1 VC f 2 VC f 3 3L f
3L f
3L f
1
1
2
Vs1 Vs 2 Vs 3
3L f
3L f
3L f
Lf
i s3 dVC f 1
dt
dVC f 2
dt
dVC f 3
dt
­ 1 , if S3 is ON and
°
° ( S 2 is ON or S 6 is ON )
°
® 1 , if S 4 is ON and
° ( S is ON or S is ON )
1
5
°
°¯ 0 , other combinations
(9)
Using the concordia transformation, the new simplified
state-space model of the circuit of Fig. 1. is:
Switched state-space model of the six-switch rectifier
Applying Kirchhoff laws to the circuit of Fig. 1. (using
the displayed state variables), a simplified switched statespace model can be obtained:
di s1
(7)
D4
S6
VC
D1
J
J1
J
VC f 1 2 VC f 2 3 VC f 3 Lo
Lo
Lo
2 J 12 J 22 J 32
VCo
2 Lo
irec3
f2
Cf
diLo
i Co
S5
Lo
iC
Cf
io
irec2
Lf , R f
is3
Do
iLo
dt
J
1
i s 1 i Lo
Cf 1
Cf
(4)
J
1
i s 2 i Lo
Cf 2
Cf
(5)
J
1
i s 3 i Lo
Cf 3
Cf
(6)
Rf
Lf
isD 1
1
VC f D VsD
Lf
Lf
(10)
is E 1
1
VC f E VsE
Lf
Lf
(11)
J
1
isD D iLo
Cf
Cf
(12)
JE
1
is E iL
Cf
Cf o
(13)
dVCo
2 J D2 J E2
dt
2 Lo
iLo 1
VC
Co Ro o
(15)
Switched state-space model of the four-switch rectifier
Using the displayed state variables of the circuit of Fig.
2., and using the concordia transformation, the following
switched state-space model is obtained:
disD
dt
1 - 156
Lf
2 J D2 J E2
JE
JD
VC f D VC f E VC o (14)
Lo
Lo
2 Lo
(3)
Rf
Rf
Lf
isD 1
1
VC f D VsD
Lf
Lf
(16)
9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
dis E
dt
dVC f D
dt
dVC f E
dt
d\ o
dt
Rf
Lf
is E 1
1
VC f E VsE
Lf
Lf
(17)
JD
1
\o
is Cf D
C f Lo
(18)
JE
1
\o
i sE Cf
C f Lo
(19)
J D VC f D J E VC f E dVCo
2 J D2 J E2
dt
4 Lo C o
\o 2 J D2 J E2
VCo (20)
2
1
V Co
C o Ro
(21)
Vs E R f i s E VC f
(24)
E
Lf
written as (25), using as new state variables the errors
between the references and the actual variable values,
eis D i sref D i s D ,
T ref D T D and eT E
i sref E i s E , eT D
f1 E ,
ª
«
«
«
«
«
¬
f 2 E ,
T ref E T E .
eTD
f1 E p1 t f 2 t J D
eT E
f 3 E p 2 t f 4 t J E
f 3 E ,
º
»
»
»
»
»
¼
(25)
f 4 E are in general
function of the error vector E
>e
is D
, eTD , eis E , eT E
@,
T
while p1 t and p 2 t are perturbations and J D , J E are
the control actions.
(22)
Considering i s D and i s E currents as the controlled
From the state space model in the controllability
canonical form (25), and knowing the strong relative
degree of each output variable ( i s D and i s E ), it can be
concluded that the sliding surfaces ensuring the
robustness of the closed loop system are:
outputs, the input-output linearization of the state-space
model (10-15) gives the state-space equations in the
controllability canonical form of (23).
ªis D º
« »
d «TD »
dt «is E »
« »
«¬T E »¼
ª TD
º
«
»
1
« R f T »
is D D
« Lf
»
Lf C f
«
»
«
»
JD
3Z
\o »
Vs1 max coswt «
L f C f Lo
2 Lf
«
»
«
»
« TE
»
« R
»
1
« f T E »
is E Lf C f
« Lf
»
«
»
JE
3Z
«
»
\o»
Vs1 max senwt S « L f C f Lo »
2 Lf
¬«
¼
Lf
two [10] (as only its second-time derivative contains the
control variables J D and J E ). The model (23) can be
where
V s1 max sinZ t V s 3 max sinZ t 2S 3
D
From this new state-space equations, it can be concluded
that i s D and i s E currents have a strong relative degree of
SLIDIND-MODE CURRENT
CONTROLLER
V s 2 max sinZ t 2S 3
V s D R f i s D VC f
ªeis D º
«
»
d «eTD »
«
»
dt «eis E »
«
»
e
¬« T E ¼»
The main task of the current controller is to set the
switching sequence for every switch of the rectifier, so
that the input source currents are approximately
sinusoidal and in phase with the respective line voltage.
Defining the input source voltages as:
­V s1
°
®v s 2
°v
¯ s3
­
° TD
°
®
°
° TE
¯
e is E
The developed models for the circuits of Fig. 1. and Fig.
2. will be used to define the sliding mode controllers.
4.
where
­
° S eis D , eTD , t
°
®
°
°¯ S eis E , eT E , t
e is D k D
e is E k E
deTD
dt
deT E
dt
0
(26)
0
where kD and k E are the parameter related to the time
constant of the desired first order response of input
source currents i s D and i s E ( k D ! 0 and k E ! 0 ). In
(23)
1 - 157
practice the derivative of the currents can be obtained
using the voltage across the inductors of the LC filter,
therefore:
9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
di sref D
­
i sref D i s D k iD
°S eis D , eTD , t
dt
°
°
k
iD V s D R f i s D V C f D
0
°
Lf
°
(27)
®
di sref E
°S e , e , t
i sref E i s E k iE
° is E T E
dt
°
k iE
°
V s E R f i s E VC f E
0
°
Lf
¯
Since isref D and isref E are two sinusoidal signals shifted
90q, a new simplified control law can be obtained:
i sref D i s D k iD Z i sref D ­S eis D , eTD , t
°
k
°
0
iD V s D R f i s D V C f D
°
Lf
°
(28)
®
i sref E i s E k iE Z i sref E °S eis E , eT E , t
°
k iE
°
V s E R f i s E VC f E
0
°
Lf
¯
The reaching mode and sliding mode stability conditions,
or simple energy flow considerations (Tab. 1.), will give
the switching strategy relating the switching laws (2) with
the switching states J k .
1
1
0
1
1
1
1
1
1
* - For topology presented in Fig. 1.
** - For topology presented in Fig. 2.
1
0
1
3,2
6,4
1,1
Tab. 2. Selected vector and switch according VC f D ,
VC f E , S (eis D , eTD , t ) and S (eis E , eT E , t )
Depending on the states of the Jk variables (representing
the states of the kth switch of the rectifier), the bridge
rectifier input currents can assume only 8 (for the
converter with 6 switches) or 6 (for the converter with 4
switches) possible distinct states, represented as current
vectors in the DE reference frame (Fig. 3. and 4).
To accurately select all the 8 or 6 available current
vectors, from (2) it is essential to read two input line
currents and two AC voltage capacitors in the DE
reference frame, and use two hysteresis comparators at
the output of the current sliding mode controller and other
two for evaluate the semiconductors bias. Therefore, the
switches that must be on are listed in Tab. 2.
E
2
3
1
0,7
If SD ! H then i sref D ! i s D , hence i s D must increase,
Therefore choose J k to increase the i s D current;
D
4
6
5
If SD H then i sref D i s D , hence i s D must decrease,
Fig. 3. Space-vector diagram of the converter with
six switches
Therefore choose J k to decrease the i s D current;
If S E ! H then i sref E ! i s E , hence i s E must increase,
Therefore choose J k to increase the i s E current;
E
2
1
If S E H then i sref E i s E , hence i s E must decrease,
Therefore choose J k to decrease the i s E current;
0,5
D
3
Tab. 1. Reaching and sliding mode stability conditions
VC f D
0
0
0
0
0
0
0
0
1
1
1
1
1
VC f
0
0
0
0
1
1
1
1
0
0
0
0
1
E
S (eis D , eTD , t )
S (eis E , eT E , t )
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
4
Fig. 4. Space-vector diagram of the converter with
four switches
Vector
* , **
4,3
3,2
6,4
0,0
4,3
3,2
0,0
1,2
5,4
0,0
6,4
1,1
0,0
5.
CONTROLLING THE OUTPUT VOLTAGE
To control the output voltage of the proposed topologies,
a proportional plus integral compensator (PI) is chosen,
in order to have a stable system with good steady-state
and dynamic response. As the frequency of the system is
low (50Hz or 60Hz), and the current is controlled at high
switching frequency, the rectifiers can be considered as
current sources, with a small delay td. With this
simplification, equations (29) give the parameters of the
PI controller.
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
1 ;
2 Ro td
KP
6.
Co
2 td
KI
(29)
SIMULATED AND EXPERIMENTAL
RESULTS
To verify the control algorithms, the design of the control
parameters, and study the static and dynamic
performances of the proposed topology, simulation
results were performed. Therefore, the proposed
topologies (with Co=Co1=Co2=10000uF, Ro=200 :, Lo=10
mH, Lf=10 mH, Rf=1.5 :, Cf=10 uF) were simulated
using the software package "MATLAB / SIMULINK"
[14]. The laboratory implementation of the proposed
topologies allowed the validation of the theoretical results
presented and the computer simulations. Figs. 5. and 6
show simulation and experimental results of the input line
currents and input source voltage (Vs1max = 60 V) with the
current controller parameters kD = kE = 1.3e-4 and a
hysteresis width of 0.001. These results show the almost
sinusoidal input currents and high power factor provided
by both rectifiers. The input source currents are
dominated by the mains frequency sinusoid and the
switching frequency components are greatly attenuated.
Figs. 7. and 8 show simulation and experimental results
of the DC reactor current iLo (in the topology presented in
Fig. 1.). These results show that this converter with the
proposed current sliding mode control works in continuos
conduction mode. Fig. 9. shows the input line current in
phase 1 of the topology presented in Fig. 2., and the
amplitude of the current reference. As can be seen in this
fig., for a step change in the current reference, there is a
fast response of the current controller.
Fig. 10. shows the experimental transient responses with a
PI output voltage controller for the step change in
reference value of output voltage, from 60V to 80V and
back to 60V. This figure shows that it is possible to
achieve a sufficiently fast voltage regulation with a good
quality AC current with a high power factor. The
behaviour of the output voltage is close to the
theoretically expectable. Therefore, the PI compensator,
with the parameters presented in (29), is adequate to be
used in this topologies, as it presents a good steady-state
and dynamic response.
7.
CONCLUSIONS
Sliding mode controllers for input line currents of the
three-phase single stage AC/DC Buck-boost converters
were presented and tested. Phase canonical state-space
models of the circuits were used to design the sliding
mode controllers. The proposed input current control
structure allow a near unity power factor operation, input
currents with low harmonic content, fast dynamic
response and robustness to parameter and load variations.
With this current controller a simple PI controller can
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7HQVm R>9@&RUUHQWH>$@
7HPSR>V@
7HPSR>V@
Fig. 5. Simulated results of
1 - Input source voltage(Vs1),
2,3,4 - Input line currents (is1,is2,is3).
Fig. 6. Experimental results of
1 - Input source voltage (Vs1),
2,3,4 - Input line currents (is1,is2,is3).
Fig. 7. Simulated results of the DC
reactor current ( iLo)
Fig. 8. Experimental results of the
DC reactor current ( iLo)
Fig. 9. Experimental results of
2 – Amplitude of the input current
reference (is1 ref),
4 - Input line current ( is1)
Fig. 10.Experimental results of
1 - Output
voltage
reference
(20V/Div),
2 - Rectifier output voltage (20V/Div)
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
[10] W. Gao, J. Hung, “Variable structure control of
nonlinear systems: a new approach,” IEEE
Transactions on Industrial Electronics, vol. 40, no.
1, pp. 45-44, 1993.
[11] A. Groef, P. Bosch, H. Visser, “Multi-input
variable structure controllers for electronic
converters,” EPE Conference, September 1994, pp
30-35.
[12] W. Gao, J. Hung, “Variable structure control: A
Survey,”
IEEE Transactions on Industrial
Electronics, vol. 40, no. 1, pp. 2-22, February
1993.
effectively control the output voltage of the rectifiers.
Laboratory prototypes were implemented and computer
simulations were performed. Experimental and simulation
results show a close agreement and confirm the high
power factor operation, fast response and low distortion.
8.
REFERENCES
[1] B. T. Ooi, J. C. Salmon, J. W. Dixon, A. B.
Kulkarni, “A three-phase controlled-current PWM
converter with leading power factor,” IEEE
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pp 78-84, Jan./Fev. 1987.
[2] A. W. Green, J. T. Boys, G. F. Gates, “3-phase
voltage source reversible rectifier,” Proc. Inst.
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[3] John C.Salmon, “Techniques for minimizing the
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[4] Y. Sato, T. Kataoka, “State feedback control of
current-type PWM AC-to-DC converters,” IEEE
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6, pp. 1090-1097, Nov./Dec. 1993.
[5] D. Vicenti, H. Jin, “A three-phase regulated PWM
rectifier with on-line feedforward input unbalance
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1994.
[6] R. Oruganti, M. Palaniapan, “Inductor voltage
controlled variable power factor Buck type ac-dc
converter,” IEEE Power Electronics Specialists
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[7] R. Itoh, K. Ishizaka, “Three-phase flyback AC-DC
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THE AUTHORS
V. Fernão Pires, received the B.S. and
M.S. Degrees in Electrical and
Computer Engineering from Technical
University of Lisbon, Portugal, in 1988
and 1995, respectively. Since 1991 he
is a member of the teaching staff at
Electrical Engineering Department of Escola Superior
Tecnologia Setúbal – I.P.S. Presently he is Joined
Professor, teaching Power Electronics and Automation in
the Electrical Engineering Course. His Main interests are
in the areas of Low-Distortion Rectifier Topologies,
Converter Control, Modelling and Simulation.
V. Fernão Pires, ESTS, IPS, R. Vale de Chaves,
Estefanilha, 2914-508 Setúbal, Portugal
email: vpires@est.ips.pt
J. Fernando A. Silva, born in Monção
Portugal in 1956, received the Dipl. Ing. in
Electrical Engineering (1980) and the
Doctor Degree in Electrical and Computer
Engineering (Power Electronics and
Control) in 1990, from Instituto Superior
Técnico (IST), Universidade Técnica de
Lisboa (UTL), Lisbon, Portugal. He is currently
Associate Professor of Power Electronics at IST, teaching
Power Electronics and Control of Power Converters, and
researcher at Centro de Automatica of UTL. His main
research interests include power semiconductor devices,
modelling and simulation, new converter topologies and
sliding mode control of power converters. He has
published more than a hundred papers.
J. Fernando Silva, IST, SMEEP, UTL, Av. Rovisco Pais,
1, 1049-001 Lisboa, Portugal
email: fernandos@alfa.ist.utl.pt
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