A synthesis of my Ph.D. defense. It's about modeling and steady state analysis of switched electronic power converters, both in open and closed loop, by using the complementarity framework. Application examples: Buck, Boost, Resonant and Modular Multilevel Converters
2. Research activities outline
Complementarity models for switched electronic systems:
Modeling of switches
Power converters as linear complementary systems
Steady-state analysis for switched electronic systems through complementarity:
Analysis of accuracy for LCP approach
Uniqueness of solution: preserving positive realness under discretization
Detecting of multiple solutions
Operating conditions of high power converters with minimum total energy
Focus on modular multi-level converters
3. The main idea
2
e
2
x1
1
1
x2
+
-
• Power electronics converters represent an interesting class of
switched systems. They can be assumed to consist of piecewise
linear elements, external sources, and electronic devices such as
diodes and electronic switches.
• The behavior of the converter is obtained by the commutations of
the electronic devices which determine the switchings among the
different converter modes.
4. Main ingredients
Power converters
+
-
x Ax Bz Ee
w Cx Dz Fe
w
Complementarity models
0 w z 0
z
Linear Complementarity Problem: Given a
real vector q and a real matrix M, find a real
vector z such that:
5. Why complementarity models?
j
e
j
i
…..
+
-
i
e
x Ad x Bd Ed e
Cd x Dd Fd e
As u Bs u z g s u
w Cs u Ds u z hs u
0w z0
= vector of switches configuration
x Ad x Bd Ed e
Cd x Dd Fd e
As Bs z Gs u g s
w Cs Ds z H s u hs
0w z0
u
u
7. Boost DC-DC converter
e
1
Lx1 e 1
2
x1
1
2
1
Cx2 x2 2
R
1 x1 2
x2
2 x2 1
2
2
2 z
2 w
0w z0
… and what about
the switch model?
8. Switch complementarity model
z1 z2
F 1
F
w1
1 F R
1 F R
1R
1
w2
1 F R
1 F R
0w z0
Ebers-Moll model
R 0
z1 z2
w1 F 1 F m axu
w2 m axu
0w z0
Unidirectional switch model
G. Angelone, F. Vasca, L. Iannelli, K. Camlibel, “Complementary Modeling”, chapter on the book
“Dinamics and Control of Switched Electronic Systems”, Springer Verlag, to be published.
9. Closed loop linear complementarity model
j
x Ax Bz Ee
j
i
…..
+
-
w Cx Dz Fe
0w z0
i
circuit equations
x Ad x Bd Ed e
Linear
Complementarity
System
e
Cd x Dd Fd e
As Bs z Gs u g s
w Cs Ds z H s u hs
0w z0
electronic devices
x
e
xc Ac xc Bc x Ec e
u
u Cc xc Dc x Fc e
controller
10. Computation of periodic oscillation (I)
xk A xk 1 B z k E ek
x Ax Bz Ee
w Cx Dz Fe
0w z0
Discretization
wk C xk 1 D z k F ek
0 wk z k 0
Periodicity
x(T ) x(0)
T Ts N
~
w q M~
z
~ z
0w~0
Sampling
Switching
Oscillation
Linear
Complementarity
Problem
11. Strictly Positive Realness (S. P. R.)
Continuous time
Gc ( s ) has no poles in Re s 0
Gc ( s ) is real for all positive real s
0
Re Gc ( s ) 0 for all Re s 0
Discrete time
G (z )
has no poles in
G (z )
is real for all positive real z
Re G (z ) 0
for all
z 1
z 1
0 1
12. Uniqueness of solution: preserving S. P. R.
If the discrete transfer function
xk A xk 1 B z k E ek
G ( z ) D C ( zI A) 1 B
wk C xk 1 D z k F ek
0 wk z k 0
Is strictly positive real then then the
LCP admits a unique solution.
If the continuous transfer function is strictly positive real does the
discrete counterpart is positive real also?
If we use the “backward zero-order-hold” discretization technique, is
proven that for “sufficiently” small sampling time the strictly positive
realness is preserved:
xk e A xk 1
k
e A( k ) ( Bzk Eek ) d
( k 1)
L. Iannelli, F. Vasca, G. Angelone, “Computation of Steady-state Oscillations in Power
Converters through Complementarity”, IEEE Trans. on Circuits and Systems I: 58(6), 2011.
13. Oscillations for resonant converters
R2 6.0 (), 4.5 (), 1.5 (*)
Vdc 42V ; R1 200m;
C1 138nF ; L1 7.6H ;
n 1.64; C2 100F ;
Average output voltage vs. the
switching frequency, for different
values of the load resistance
Time-stepping simulations with
PLECS (continuous line) provide
results with negligible differences but
spending more time than LCP
F. Vasca, L. Iannelli, G. Angelone, “Steady State Analysis of Power Converters via a
Complementarity Approach” 10th European Control Conference, August 2009, Budapest, Hungary.
14. PWM boost converter
20
[A]
[V]
L 0.1mH
rL 0.1
15
e 10 V
10
R 7
C 0.2 mF
5
vref 15 V
Ts 0.2 ms
0
k p 0 .1
1
2
3
4
5
6
[s]
-3
x 10
14
13.8
R 20
13.5
[V]
13.7
[V]
0
13.6
13
13.5
12.5
13.4
12
0
0.5
1
1.5
[A]
2
2.5
3
0
1
2
3
4
5
[A]
F. Vasca, G. Angelone, L. Iannelli, “Linear Complementarity Models for Steady-state Analysis of
Pulse-width Modulated Switched Electronic Systems”, 19th Mediterranean Conference on Control
and Automation, Corfu Island, Greece, June 2011.
15. Multiple solutions
L 20 mH C 47 F
R 2 22
R1 0
Vdc 30 V vc Vref k p x2
Vref 11 .3 V k p 8.4
The closed loop system is not passive, and there are three solutions: two
stable with period 2Ts (blue), and one unstable with period (2)Ts (red)
L. Iannelli, F. Vasca, G. Angelone, “Computation of Steady-state Oscillations in Power
Converters through Complementarity”, IEEE Trans. on Circuits and Systems I: 58(6), 2011.
16. Industrial case study: modular multilevel
converters
Cell
Idealization
of DC-Link
Idealized Load
17. Closed loop control results
Capacitor voltages balancing: a typical approach
First, determine the arm voltage (how many cells? ) by using the nominal voltage:
varm j m
Then, select and sort:
the m capacitors whit the smallest voltage when the arm current is positive
the m capacitors whit greatest voltage when the arm current is negative.
5
200
4.5
180
4
160
3.5
140
3
120
V arm 2 [ V ]
Vk
,m n
6n
2.5
2
100
80
1.5
60
1
40
0.5
20
0
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0
0.01
0.02
0.03
0.04
0.05
t [ s ]
Carriers (APO)
Modulating
signal
Phase 1: Lower Arm
0.06
18. Conclusions
• The use of a suitable switch model allows to construct a non-switched
complementarity model, without explicitly detailing all modes of the
converter and without a priori knowledge of the sequence of modes.
• The backward zero-order-hold discretization technique preserves the
positive realness of the transfer function and then the uniqueness of
the solution, for sufficiently small sampling period.
• The complementarity model can be exploited in order to capture the
steady-state behavior also in the presence of: closed
loop, resonances, multiple solutions, unstable orbits, complex
topologies.
19. List of publications
International Conferences
• F. Vasca, L. Iannelli, G. Angelone, “Steady state analysis of power converters
via a complementarity approach”, 10th European Control
Conference, Budapest, Hungary, August 2009.
• F. Vasca, G. Angelone, L. Iannelli, “Linear Complementarity Models for
Steady-state Analysis of Pulse-width Modulated Switched Electronic
Systems”, 19th Mediterranean Conference on Control and Automation, Corfu
Island, Greece, June 2011.
International Journals
• L. Iannelli, F. Vasca, G. Angelone, “Computation of periodic steady state
oscillations in power converters through complementarity”, IEEE Trans.
on Circuits and Systems I: 58(6), June 2011.
Chapters in books
• G. Angelone, F. Vasca, L. Iannelli, K. Camlibel, “Complementary
Modeling”, chapter on the book “Dinamics and Control of Switched
Electronic Systems”, Springer Verlag, to be published.
Da AggiustareThey consist of a modular structure where each leg is composed by a group of capacitors connected by means of switches to form a series of submodules (“cells”) and two inductors, possibly coupled in general. The main characteristics of these kinds of converter are the possibility to extend their voltage range only by adding a suitable number of cells. No central element - e.g. a capacitor on the DC.link - is present Fault tolerant operation is possible with a sufficient number of cells, because the “arm voltages” can be synthesized also when some of them fail as short circuit .