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Transcript
4th Power Electronics, Drive Systems & Technologies Conference (PEDSTC2013), Feb l3-14, 2013, Tehran, Iran
Advanced Non-Inverting Step up/down Converter
with LQR Control Technique
l
2
3
Hassan Oehghani , Ali Abedini , Mohammad Tavakoli Bina
123
, , K.N. Toosi University of Technology
123
, , Tehran, Iran
2
3
I [email protected], [email protected], [email protected]
Abstract-This paper proposes a new topology of non inverting
in this paper, is LQR to control the proposed converter. LQR
buck-boost converter. This topology is composed of a boost
control technique has the advantages of more robustness and
converter which is followed by a buck converter through a
faster
magnetic coupling. A series resistor with capacitor is considered
in the boost part to enhance the dynamic of the converter. The
more accuracy. A control algorithm
PI
or
PID
conventional
mathematical model of the converter. This modeling is carried
and AC
out considering losses in the windings, power mosfets and
models of the converter are obtained to analysis the dynamic of
the converter with
to
proposed converter structure. Section III presents DC and AC
higher bandwith and faster response compared to conventional
Non-ideal DC
compared
This paper is organized as follows: Section II introduces
advantages of proposed topology are smaller capacitors size,
non inverting buck-boost topologies.
dynamics
controllers[3].
diodes. In Section IV elements of the converter is designed step
is
by step. Section V introduce control algorithm for the buck­
developed based on LQR method to regulate the output voltage
boost converter. In Section VI a sample buck-boost converter is
of the converter. This controller is more robust compare to
simulated in frequency domain and time domain. Also, The
conventional controller. PSCAD/EMTDC software is used to
evaluate and verify mathematical model results and simulated
converter
circuit model.
PSCADIEMTDC is used for simulation. The last section
losses
and
efficiency
are
calculated.
presents the conclusion of this paper.
Keywords-Cascade buck-boost converter; LQR control; RHP
zeros;Losses; Efficiency.
I.
II.
Fig. I shows the proposed converter circuit diagram. This
INTRODUCTION
converter is consisted of a boost converter cascaded with a
Nowadays, applications of DC-DC converters have been
increased
significantly
because
they
are
widely
used
buck converter. A transformer with turns ratio of 1 is
in
embedded into the boost part, so that the secondary side of
renewable energy systems such as solar systems, fuel cells,
transformer is in series with inductor L of the buck part. Also,
battery chargers and etc[1-8].
a damping resistor Rd is in series with the capacitor C in the
In many applications due to the large variation of input
voltage,
step
up-down
converters
must
be
used.
boost part. Co and Ro show the filter capacitor and load resistor
These
in the buck part, respectively. Vg shows the input voltage. Lm
converters are divided into single inductance i.e. conventional
is the magnetizing inductance of the transformer, Ron displays
buck-boost converter, or two windings Converter i.e. cascade
the on resistance of the power mosfets Ql and Qz, and VD
buck boost groups. Single winding are used to reduce the cost
shows the forward voltage drop of the diodes 01 and O2.
RLillustrates ohmic resistance of the transformer and output
and size of the circuit. However two windings converters are
more for high voltage applications where the size of capacitors
is
more
important.
Other
advantages
of
two
inductor.
windings
Boost Part
1'-'-'-'--;'-'-"
converters are the continuous input and output currents, less
i
EMI noises, and better control of input and output currents
compared to single winding converters[ I].
But
the
disadvantages
converters
are
RHP
of
zeros
response[8,9]. A solution to
conventional
which
restrict
step
up-down
the
controller
L{
bandwidth
a
new
converter that
and
more
topology
composed of
efficiency[1,9].
of
non
a boost
inverting
This
i
Co
paper
buck-boost
converter which
� .-.-.-.-.-.-. �� .-.-.-.-.-.-.-.-.-.
is
Figure 1.
followed by a buck converter through a magnetic coupling in
order to removing RHP zeros. Also, a series resistor with a
capacitor is considered in the boost part to enhance the
dynamic of the converter. The control technique which is used
978-1-4673-4484-5/13/$31.00 ©2013 IEEE
,i
L
resistor with capacitor. Moreover this solution helps to gain
proposes
I
Buck Part
r'-'-'-'-'-'-'-'-'-"
+
remove RHP zeros, is using a
magnetic coupling between two windings along with a series
higher
STRUCTURE
254
Proposed buck-boost converter topology
Ro
Vo
III.
AC AND DC MODELING
(3)
Assume Ts as switching period, QJ and Q2 conduct during
Where in the above terms R 1 and R 2 are as follows:
dJ(t)Ts and d2(t)Ts intervals, respectively, while DJ and D2 are
off in that times. Fig.
modes
of
the
2 and Fig. 3
converter,
represent the boost and buck
respectively.
Assume
that
the
)Ron +(�)
Rd
R1 = (�
Dz-D'l
Dz-D'l
(D2D'2) Rd +Rl (4)
R2 = (D, 1 +D, 2 +D1D2
�) Ron + D1 +
converter operates in continuous conduction mode(CCM), and
switching frequency is much higher than the converter natural
frequencies[8], the nonlinear state differential equations are as
�
follows:
Considering x as the ac state vector of the converter, then:
dILm(t)
dt
vg(t)-d1 (t) Ron h(t)+tLm(t) -d� (t)[VC(t)+Rdhm(t)+vDl-d� (t)RdIL(t)
Lm
vg(t)-Vo(t)-Rlh(t)-d
(t)
Ron lL(t)+hm(t) +RdlL(t)-vc(t)1
1
I
d L(t)
_
[ (
[ (
=
�
(5)
)l
the small signal model of the converter is equation
(6).
)
L
x
)
[ (
+----------------���---=��----------�
dCit = + B1dl + B2d2 + B3Vg + B4rO
-d� (t)(Ron[L(t)+VD)-d� (t) Rd h(t)+lLm(t) +vC(t)+VD-RonlL(t)1
L
A
Ax
A
(6)
Where ,B1, B2, B3 and B4 are as follows:
c
dvo(t)
dt
h(t)
-
-
A=
Vo(t)
---
(1)
-(D1Ron+D'lRd)
Lm
-(D1Ron+D' ZRd)
-(D1Ron+D' ZRd)
Lm
-(DzRon+(D1+D'z)Rd+Rl)
D'l
Lm
Dz-D'l
D'l
C
D'z-D1
C
0
L
+
Ro
1
0
Vo
B1 =
Figure 2.
L
The boost mode ofthe converter
1
L
0
Co
o
L
o
-1
Ro Co
[(Vc+RdlLm+VD-Ronlg) (VC+VD-RonlLm-Rdh) -lg O ] T
L
C
Lm
T
[Rd1L (VC+VD-Ron1L+Rd1g) -IL O ]
B2 =
Lm
B3
+
C
L
=
[2..�
Lm L
0
O
]
T
Vg
Ro
Vo
(7)
Figure 3.
In the above terms,ILm + IL = Ig, is the converter input
The buck mode ofthe converter
current. Fig. 4 and Fig. 5 illustrate DC and ac small signal
transformer model of the proposed converter, respectively. In
Where Zj is mean of the Zj in the switching period T"
the ac model,
a
and
f3 are as follows:
a
= Vc
+ VD -RonIg +D'lRdIg -D2Rdh
where Zj is any variable i.e. vg,iL,d1,... :
(2)
Where the Zj is the DC part of Zj' and Zj is the ac part of it,
around Zj. By substituting
(2)
in
(1)
f3 = Vc
and separating the ac part
+
VD - RonfL + D'lRdfg - D2RdfL
(8)
The voltage conversion ratio is gained as follow:
from the DC part, the operating point of the converter is gained
as following:
(9)
Fig.
IL
m
=
6
shows M as function of D1+D2 , as shown M is a
continuous curve between the boost and buck modes of
(DD,z_l ) h
operation.
1
255
In the last equations, £::"iLm,Boost,p-p and £::"iLm,Buck,p-p are
peak to peak magnetizing inductor current ripples in the boost
Ro
and buck modes respectively. Lm is determined with regard to
the maximum allowed of the input current ripple.
Figure
4.
Output Capacitor(Co) Selection
DC transformer model of the converter
Co
Capacitor
is calculated as follow:
Co
CO
Where
C
Figure
5.
l
and
CO
=
l:J.iL
0,
Max{Col' CO2}
=
(14)
are:
2
BoostP-P Ts
8l:J.vo
BoostP-P
£::"VO,Boost,
£::"VO,BUCk,
In above equations,
P-P and
P-P are
peak to peak output capacitor voltage ripples in the boost and
ac transformer model of the converter
buck modes respectively. COl and CO2 are determined
with
regard to the maximum allowed output voltage ripples.
Input Capacitor(C) and Damping Resistor(RcJ
Selecting values of L, Lm and C o is accomplished with
regard to desired peak to peak their current and voltage ripples.
Rd
But C and
determine the location of the converter zeros and
converter dynamic. Therefore they must be selected so that the
zeros are in the left half plane(LHP) with good dynamic
converter response.
The output transfer function of the converter based on Fig.
Figure
IV.
6.
Voltage conversion ratio M, as function of
3
0,+02
is this:
(16)
DESIGNING ELEMENTS OF THE CONVERTER
Output Inductor(L) Selection
Where:
inductor L is calculated as follow:
(10)
Where L1 and L2
are:
2
bo D'1 Ro + D1D'lRd
2
2
b1 D1 Lm + D'1 L + D1D'lRdRoCo + D'lRdD'lRd + D1D'lRd2C
2
b2 D12LmRoCo + D\ LRoCo + D1D'lRd2C + D1LmRdC +
D'lRdLC + LmCRo
b3 LmLC + D1LmRdCRoCo + D'lRdCLRoCo
b4 LmLCRoCo
(20)
=
=
=
In the above equations, £::"iL,Boost,P-P and £::"iL,BUCk,P-P are
peak to peak output inductor current ripples in the boost and
=
=
buck modes respectively.In the each mode, with regard to
To eliminate RHP zeros the following conditions must be
maximum value of these current ripples, minimum value for L1
met:
and L2must be selected separately, in order to obtaining CCM
conditions for current of the output inductor L.
ao
Magnetizinglnductor(LmJ Selection
Having
Magnetizing inductor Lmis gained of:
Lm
Where Lm 1 and Lm 2
Lm
=
=
=
(12)
Having
using
(Vg-RonIg)D1Ts
(l1igBoostP P -l1iLBoostP
-p)
(Vg-VD-RdI Lm -Vc DzTs
)
(l1i9BuCkP P -l1iLB CkP P
- )
U
> 0 and
a2
al
> 0,
a2
> 0
> 0, with respect to
(21)
(3)
must be
following condition satisfied:
are:
l
L z
m
Max{Lml,LmJ
ao
> 0,
(3),
a
Rd=< Ro
1 > 0, since 10
IL, by replacingit in
(22)
(18),
and
then:
(23)
(13)
256
Considering
where a1
=
(23),
as shown in Fig.
0, then:
I
a Critical
=
7,
IO
Critical
R dCVg
Rd 2 C +E.!..L
D'1 m
v.
is defmed as
LQR
control
CONTROLALGORITHM
techniqueis
used
to
design
the
control
algorithm of the proposed converter. The theory of optimal
(24)
control is concerned with operating a dynamic system at
minimum cost. The case where the system dynamics are
described by a set of linear differential equations and the cost is
To have all zeros in LHP, the following conditions must be
described by a quadratic functional is called the LQ problem.
met:
One of the main results in the theory is that the solution is
provided by the linear-quadratic regulator (LQR)[lO]. If the
state equations of the system are these:
x =Ax + Bu
Ro
y=Cx +Du
1
2
(27)
2
According
By using LQR:
(27) must:
u = -Kx
Then:
J =
(29)
By taking derivative of
opt
(24)
(32)
Where matrix K must be selected so that, J is minimized:
(28)
equaling to zero, Rd
(31 )
f (xTQx + uT Ru)dt
(33)
Where J is the cost function which must be minimized and
Q(state weighting matrix)? 0 and R(inputweighting matrix)
with respect to Rd and
>
O.
is achieved as follow:
K is gained as follows:
(30)
(34)
Fig. 8 illustrates IO
as function of the Rd for different
Critical
values of the duty cycle D1.
And P is obtained via
the
following
algebraic ricatti
equation:
Alp + PA + -PSR -ISlp + Q = 0
12XIO"
(35)
Fig. 9 illustrates the control block diagram of the converter.
As shown, because the system type is zero, an integrator must
be used to have zero steady-state error.
OJ.
·�O�-.7--.'!;--;!;----7:--+'5
-t"
.-----.--.:,-�
--,
,,,
1 -----16c'-------;'
lo(Ampere)
Figure 7.
Coefficient a1 as function ofIo
Figure 9.
control block diagram for converter
[
Q and R are selected as:
Q=
0
�m
Q
0
Q1L
0
0
0
0
Qfjc
0
0
-
-
dz
(36)
And K is this:
DO��O.�
5 -�-�1.5'-�--+".5'--+-�--+-�'�5 -�
Rd(Ohm)
Figure 8.
d,
0
�l
Q.
R [R� R� ]
'OCritical as function of the Rd for different values of the duty cycle
(37)
Dl and showing Rdopt in the each them
Thus the closed-loop system representation is given by:
257
Ac
=
(38)
A - B1Kl - B2K2
VI.
[n
this
paper
a
study
has
been
1
0
VD
-�
--
r---.-
1! (a)
SIMULAT10NRESUL TS
case
50.50
_
Where in that, Ac is new state matrix of the converter.
49.50
accomplished.
00
50
40
30
20
10
0
Simulation of the buck-boost converter has been considered for
the design parameters which are mentioned in table I. By using
equations given previously in the paper the elements values are
obtained in table [I. There are 3 columns in table II: boost
I
VQ2
--
1
12. a -,---"
iQ
::.:-- A----- A-----'-- -='---:;
mode, buck mode and fmal selection. The boost and buck
"
modes columns show the obtained value of the elements in the
� (e)
�
boost and buck modes respectively, and the fmal selection is
made based on the worst case conditions. Table III shows input
O.
parameters of the converter shown in Fig. 1. Assuming as
follows:
Q1Lm
=
0,
=
0,
a
=
QIL
R
1
Qvc
=
0,
Raz
=
1
Qvo
=
0.03
time (5) G.G22515
(39)
K is gained as follows:
Kll
=
0,
K12
K21
=
0,
K22
=
=
K13
=
0,
K14
=
0.1062
0.0340,
K23
=
0,
K24
=
0.1242
Dl=O,Dz= 1&wc=31400
K[
=
0.022535
0. .0022540
0.022545
().G22555
0..0.22550
voltage OfQ2 with gray color; (c):current OfQl with black color and current
(40)
=
50.50
31400) .
=
1
=
-�
--
r--
� (a)
° db
49.50
,,,, Jb Db l�
i (CI :V1r1 �
(41)
8102
r:
OfQ2 with gray color; (d): ig with black color and iL with gray color.
order that cut off frequency of the open loop transfer function
be a twentieth of the switching frequency(wc
0
().G2253G
mode(Vg=35V)(a):output voltage; (b):voltage OfQl with black color and
K( as an integrator coefficient according Fig. 9, is achieved in
CJW)
0..0022525
Figure 10. Voltgaes and currents ofthe converter in the boost
0.0291,
I�Gv aUw)1
0.022520.
Fig. 10 and Fig. 11 illustrate Voltgaes and currents of the
converter in the boost(V g=35V) and buck(V g=65V) modes,
respectively,
A, Converter analysis results in MATLAB software
In this section, the time and frequency domain performance
of the proposed converter is investigated. Fig. 12 illustrates
70
step response of the proposed converter in the boost and buck
modes. As shown, the system responds quickly with little
CI)
�'"
overshoots, and zero steady-state errors and the short settling
times in the both modes, but the overshoot and settling time in
(d)
6.0
5.0
40
3.0
2.0
1.0
;--"'-----�.----- --�<___"=-----�.----- --�
time (5) O.GnS15
the boost mode is more than the buck mode, Fig, 13 illustrates
O.G2252G
(lOnS25
G.(}2253D
o. a 22S35
0- . 022:540
O.(}22545
O.(}2255G
()'021555
bode diagram of the proposed converter in the boost and buck
modes. As shown, Desirable phase margin(about 60°), and cut­
Figure II. Voltgaes and currents ofthe converter in the buck
off frequency(about 5kHz), is obtained with used control
mode(Vg=65V)(a):output voltage; (b):voltage OfQl with black color and
algorithm.
H
voltage OfQ2 with gray color; (c):current OfQl with black color and current
OfQ2 with gray color; (d): ig with black color and iL with gray color.
Losses and efficiency calculation of converter
The conduction losses of the converter regardless of the
equivalent series resistance (ESR) of the capacitor is obtained
PLOSSSWQl
as following equation:
2
2
2
R1h,rms + Ron(!Ql,rms + IQz,rms ) +
2
RdIc,rms + VD(!Dvrms + IDz,rms)
PLoss,cond
converter
switching
losses
which
is
due
to
PonswQl +POffswQl
Q I,minIQ bmintr + V:Q I,maxIQ I,maxtf)Boost
&(V:
2
=
(42)
The
=
=
Mode
(43)
PLoss,SwQz - Pon,SwQz +Poff,SwQz -
�
the
switching of the power MOSFETS, are obtained as follows:
(VQz,minIQz,mintr + VQz,maxIQz,maxtf) Buck
ModJ44)
Total losses of the converter is as follows:
PLosses
258
=
PLoss,Cond +PLoss,Sw
(45)
And finally, efficiency of the buck-boost converter is:
__
_
.PS ,-ns _._�_
o�
P R_ --,
1.4,---_�-___,_--,_____ St�.'---,-----,
----,
1.2
Pin - PLosses
7]Converter ==
Where Pin
VgIg,rms' Fig.
=
(46)
Pin
14 represents the conduction
and switching losses, and Fig. 15 represents the efficiency of
the converter as functions of the input voltage, in the boost and
buck modes.
2.5
1.5
Time (sec)
TABLE1.
3.5
THE CONVERTER DESIGN PARAMETERS
Vg = 3SV-6SV
Input voltage range
Va = SOV
Reference value of the output voltage
Figure 12. The Step responses of the proposed converter in the boost and
buck modes
llvop_p =
�
!1.
0.16Va
peak to peak output voltage ripple
or-------------
__�
� ·50
�
:el-100
po•max =
Mgp_p =
9A
peak to peak input current ripple
lliLp_p =
4A
peak to peak inductor L current ripple
SOOW, Po.rated = 2S0W
Nominal and Maximum output power
fs =
:.: �"""'....-......I
.270 L,-����������-��...::::'t
�
�
�
�
�
100kHZ
Switching frequency
Frequency (Hz)
THE CONVERTER CIRCUIT ELEMENTS VALUES
TABLEII.
Figure 13. Bode diagram of the converterin the boost and buck modes
Element
20 ,----.---,----,---�
Boost mode
Magnetizing
inductor
Output
inductor
Output
capacitor
Input
capacitor
'8
16
'4
4
Lmi
Input Voffage(Volt)
Lm,
=
L2 = 27.8J.lH
( =
( =
62.5J.lF
0,
(, = 17.IJ.lF
62.5J.lF
( =
0
(=
66J.lF
16J.lF
VALUES AND TYPE OF THE CONVERTER CIRCUIT ELEMENTS
Symhol(type)
Quantity
Rd
0.8 il
Damping
resistor
Transformer and output inductor
ohmic resistance
input voltage
Lm= 25J.lH
L= 30J.lH
(2 = 8.3J.lF
Element
Figure 14. The converter conduction and switching losses as function of the
Final selection
18.47J.lH
L, = 26.32J.lH
0,
TABLElll.
��5--��--�--�-�·
23. 86J.lH
=
Buck mode
70 mil
RL
Q,&Q2
Power mosfet
IRFB4115PbF
D,&D2
Schottky diode
40CPQ080G
tr =
tf =
73ns
39ns
Ron = 10mil
VD=O.6V
C. Converter simulating results in PSCAD/EMTDC software
Simulating is accomplished with considering 1 microsecond
delay time to make gate signals
Ul
and
U2.
Fig. 16 shows the
case which the input voltage changes from 35V to 65V, with
frequency of 50Hz, at nominal load. This test is done to
��5--��
�---�=---�5=O---=55�--�
60�--�65
evaluate the dynamic stability of the proposed converter, and
Input Voltage(Volt)
respond
of
the
designed
control
to
such
small
signal
disturbances. As shown, in the both boost and buck modes,
Figure 15. The converter efficiency as function ofthe input voltage for the
nominal and maximum loads
gates signal
259
Ul
and
U2
are made correctly, so that the output
voltage is regulated at SOY in the allowed ripple range. Fig. 17
and Fig. 18show the casesin which the converter is on the
minimum input voltage(3SY) in the boost mode and the
maximum input voltage(6SY) in the buck mode, respectively.
Suddenly, load resistor changes from 1012 to 512 and after a
short time returns to the original state. These tests done to
evaluate transient stability of the proposed converter to these
large signal perturbations. As shown in the both modes, the
designed controller regulates the output voltage in the reference
value,among allowed ripple range with appropriate overshoot
and settling time similar to step response of Fig. 12.
I.
CONCLUSION
A new non-inverting step up/down converter topology is
discussed here. LQR technique is applied to regulate the output
voltage of the converter. The DC model and dynamic model of
the converter are gained to compare the performance of the
Figure 16. Changing the input voltage at nominal load, and keeping output
converter with conventional converter. The Dynamic model
showed that there is no any RHP zero in the transfer function
of the proposed converter.
voltage in the allowed range for converter dynamic stability evaluation;
(a):input voltage; (b):output voltage; (c):gate signal
Simulation results also match
closely with the responses of mathematical model gained in
MATLAB.
"'u
5
� (a)
efficiency has not been scarified in this converter compared to
�
(Q
(b:' 0 �
-Yo
REFERENCES
Carlos RestrepoO, Javier Calvente, Angel Cid-Pastor, Abdelali El
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Figure 18. Converter transient stability evaluation in the buck mode(Vg=65V)
(a):load current ; (b):output voltage; (c):gate signal
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