Nonlinear analysis and control of two-phase cell boost dc-dc

Nonlinear analysis and control of
two-phase cell boost dc-dc converters
Mr.Haimeng Wu, Prof.Volker Pickert, Dr.Damian Giaouris
Newcastle University
Email: [email protected]
1. Introduction
2. Objective
In this research, we present a monodromy matrix based nonlinear analysis
method to investigate the influence of parameters for the stability of a twophase cell boost dc-dc converter. The proposed approach applies Filippov
theory to analyse the system behaviour during the switching instant. By using
the knowledge gained from this analysis it is possible to design controllers that
guarantee a satisfactory performance of the converter avoiding fast and slow
scale bifurcations. In this method, all the comprehensive information, such as
system input, load, converter parameters and coefficients of the control loop
are introduced in the derivation of the Monodromy and Saltation matrices.
which can be used for further stability analysis. Numerical and analytical
results validate our work.
L1
iL1
vc(t)
D2
D1
Iref
Δi
iL1
iL2
C
R
iL2
Vi
Δiin
iin
d1
d
L2
iin
Ф1 Ф2 Ф3 Ф 4
d2
S2
S1
Cs1
Cs2
S23 S34
S12
S41
clock
(a)
(b)
Fig.1(b) Key operation waveforms when d>0.5
Fig.1(a) Diagram of interleaved boost converter
3. Nonlinear Analysis
Ф1
Ф2
S12
○ Saltation matrix is used to describe the switching instants
X(t0 )
X(t0  T ) S41
Mcycle
Vc (V)
239.9
S23
239.8
○ Monodromy matrix represents the state transition of perturbation over a
whole clock cycle. The locus of its eigenvalues is used to predict the stability
of system.
239.9
239.7
239.8
0.0599 0.0599 0.0599 0.06
0.06
0.06
0.06
0.06
0.06 V (V)
c
239.7
11
Ф3
S34
1 ~  4
State transition matrix
239.6
9
12
8
10
7
S12 ~ S41
time (s)
10
iL1, iL2 (A)
Ф4
Saltation matrix
0.0599 0.0599 0.0599 0.06 0.06
time (s)
0.06
0.06
0.06
8
7
Vin=110V
(a)
X(t0  T )  Mcycle X(t0 )
iL2 (A)
0.06
8
9
10
11
12
13
4.Simulation
iL1 (A)
(b)
Table1. Simulation parameters
242
M cycle  1  S12 2  S23 3  S34 4  S41 V (V)
240
parameters
Input voltage （V）
Output voltage （V）
Power rating （W）
Inductance （uH）
Output capacitance (uF)
c
238
①A1 x  B1 E

②A 2 x  B 2 E
x
③A3 x  B3 E
④A x  B E

4
4
S1 and
S 2 on
S1 on
and
S 2 off
S1 off
and
S 2 on
S1 and
S 2 off
236
0.0597
M cycle
X
X
X
0.0598
0.0599
0.0599
time (s)
0.0599
0.06
0.06
240
Vc (V)
239
16
parameters
Frequency （kHz）
Slope coefficient
Kp
Ki
value
100
-0.5
5
500
238
X
X 
X 
242
14
iL1, iL2 (A)
12
14
12
0.0598
0.0599
0.0599
time (s)
(c)
Monodromy matrix
0.0599
0.06
0.06
Vin=80V
iL2 (A)
Po=2kW
L=200uH
Vin=80~120V
241
16
0.5
12
10
Eigenvalues2
10
8
8
240
iL1 (A)
Eigenvalues1
Vc (V)
(d)
Imag
10
8
0.0597
1
237
16
14
System differential equation
X
  X
 X
241
value
80~120
240
2000
5~200
20
Vin=96V
0
96V
Eigenvalues3&4:
Conjugate complex eigenvaules
120V
239
Fig.3 (a)~(d) waveforms of inductor current and capacitor voltage
and corresponding phase portrait
-0.5
238
237
80
85
90
95
100
Vin (V)
105
110
115
-1
120
-1
-0.5
0
Real
0.5
1
(b)
(a)
Fig.4 (a)(b) Bifurcation diagram of interleaved boost converter and
corresponding locus of eigenvalues
○The eigenvalues of Monodromy matrix jump out of unit cycle
when Vin equals 96V, which indicates the system will lose stability in
that condition. There is a good agreement with bifurcation diagram.
5. Conclusions / Future Work
1/s
vip(t)
vslope(t)
KI
Vref
Vref
Kp
vco(t)
viL(t)
Clock
Supervising
S2
R Q
S
Q
¯
PWM
S1
kvc
kiL
iL
controller
Vi D
vc
Fig.6 Interleaved boost converter with supervision
control methodology
○ Nonlinear phenomenon is demonstrated and
analysed in the two-cell boost converters. The
influence of some parameters for system stability has
been illustrated and described effectively by the
proposed Monodromy matrix
○ Develop the supervising control methodology to
address these nonlinear behaviors and to improve
the system performance
260
240
Vc (V)
1
200
80
unstable
0.5
L (H)
100
Vin (V)
120
0
(a)
260
1.5
1
0.5
80
-4
90
100
110
x 10
Vin=80V
Po=2kW
L=5~200uH
Vin=110V
Po=2kW
L=5~200uH
250
L=155uH
Vc (V)
230
220
L=55uH
230
220
210
210
Vin=80V
0.5
1
1.5
L (H)
The authors would like to acknowledge the Engineering and Physical Sciences Research
Council (EPSRC) for supporting the Vehicle Electrical Systems Integration (VESI) project
(EP/I038543/1)
0
240
Vc(V)
200
0
130
(b)
260
250
120
Vin(V)
L (H)
240
Acknowledgement
-4
x 10
unstable
220
140
○ Start to prepare the experiments to verify the
effectiveness of the proposed methods
1.5
Stable
Stable
(c)
2
Vin=110V
2.5
-4
x 10
200
0
0.5
1
1.5
2
L (H)
(d)
Fig.5 (a)(b) System stability at different values of input voltage and inductance
(c)(d) Output voltage vs. inductance at Vin=80V and Vin=110V
2.5
-4
x 10

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