Download Bidirectional power flow control of a power converter using passive

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS
Int. J. Circ. Theor. Appl. 2008; 36:769–788
Published online 11 October 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cta.459
Bidirectional power flow control of a power converter using passive
Hamiltonian techniques‡
Carles Batlle1 , Arnau Dòria-Cerezo2, ∗, † and Enric Fossas3
1 Department
of Applied Mathematics IV and Institute of Industrial and Control Engineering,
Universitat Politècnica de Catalunya, Spain
2 Department of Electrical Engineering and Institute of Industrial and Control Engineering,
Universitat Politècnica de Catalunya, Spain
3 Institute of Industrial and Control Engineering, Universitat Politècnica de Catalunya, Spain
SUMMARY
A controller able to achieve bidirectional power flow for a boost-like full-bridge rectifier is presented. It
is shown that no single output yields a stable zero dynamics for power flowing both ways. The controller
is computed using port Hamiltonian passivity techniques for a suitable generalized state space averaging
truncation of the system, which transforms the control objectives, namely specified output mean value of
the voltage dc-bus and unity input power factor in the ac side, into a regulation problem. Simulation and
experimental results for the full system confirm the correctness of the simplifications introduced to obtain
the controller. Copyright q 2007 John Wiley & Sons, Ltd.
Received 13 September 2006; Revised 23 August 2007; Accepted 3 September 2007
KEY WORDS:
ac–dc power conversion; nonlinear systems; Fourier transforms; control systems
1. INTRODUCTION
In this paper we present a controller for a boost-like full-bridge rectifier [1]. The main novel feature
of our approach is the ability to handle a bidirectional power flow. In many applications, such as
the control of doubly fed induction machines [2], power can flow in both directions through the
back-to-back (rectifier+inverter) converter connected to the rotor. Since the aim of the control
∗ Correspondence
to: Arnau Dòria-Cerezo, EPSEVG, Universitat Politècnica de Catalunya, Av. Vı́ctor Balaguer s/n,
08800 Vilanova i la Geltrú, Spain.
†
E-mail: [email protected]
‡
A preliminary version of the results reported in this paper was presented at the 44th IEEE Conference on Decision
and Control, and the European Control Conference, Seville, Spain, 12–15 December 2005.
Contract/grant sponsor: European sponsored project Geoplex; contract/grant number: IST-2001-34166
Contract/grant sponsor: Spanish project; contract/grant number: DPI2004-06871-CO2-02
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2007 John Wiley & Sons, Ltd.
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
scheme is to keep the intermediate dc-bus to a constant voltage, the rectifier’s load current can
have any sign (although it can be supposed to be, approximately, piecewise-constant in time).
Advantages of power converters able to sustain a bidirectional power flow are discussed in
[3], where a topology based on SCR thyristors is presented, instead of the fully controlled IGBT
switches that we consider in this paper.
It is well known that the behavior of switching power converter circuits contains very rich
dynamics, which can include fast-scale bifurcation phenomena [4, 5]. In this case a previous study
of the dynamics can reveal some undesired behavior.
The problem we want to study in this paper involves simultaneous mean value-output voltage
regulation in the dc-side of the inverter and power factor compensation in the ac-side. The most
usual control architecture for this problem is a cascade PI [6], with an inner loop that controls
the current and an outer loop for the bus voltage. However, it can be shown that, no matter which
output is chosen, either capacitor voltage or inductor current, the zero dynamics is unstable for one
of the two modes of operation (see also [7]). To avoid this problem we apply passive techniques
to design a controller able to operate in both cases. In particular, the interconnection and damping
assignment-passivity-based control (IDA-PBC) [8] methodology is used. This technique relies on
the port-controlled Hamiltonian systems (PCHS) [9] properties to show stability.
In this paper we apply PCHS techniques to a generalized state space averaging (GSSA) [10, 11]
model of a boost-like full-bridge rectifier. This problem was already studied in [12] for the case
of a constant sign load current. A variable sign invalidates the solution found in [12], and a new
IDA-PBC controller has to be developed.
PCHS, with or without dissipation, generalize the Hamiltonian formalism of classical mechanics
to physical systems connected in a power-preserving way [13]. The central mathematical object
of the formulation is what is called a Dirac structure, which contains the information about the
interconnecting network. A main feature of the formalism is that the interconnection of Hamiltonian
subsystems using a Dirac structure yields again a Hamiltonian system [9]. A PCH model encodes
the detailed energy transfer and storage in the system, and is thus suitable for control schemes
based on, and easily interpretable in terms of, the physics of the system [14, 15].
PCHS are passive in a natural way, and several methods to stabilize them at a desired fixed
point have been devised [8]. On the other hand, variable structure systems (VSS), specially in
power electronic applications, can be used to produce a given periodic power signal to feed, for
instance, an electric drive or any other power component. In order to use the regulation techniques
developed for PCHS, a method to reduce a signal generation or tracking problem to a regulation
one is, in general, necessary. One powerful way to do this is averaging [16], and in particular
GSSA. In this method, the state and control variables are expanded in a Fourier-like series with
time-dependent coefficients; for periodic behavior, the coefficients will evolve to constants. In many
practical applications [12], physical consideration of the task to solve indicates which coefficients
to keep, and one obtains a finite-dimensional reduced system to which standard techniques can be
applied.
The paper is organized as follows. Section 2 derives the ordinary differential equations (ODEs)
which give the solution of the full-bridge zero dynamics when either inductor current or bus
voltage is taken as output. These turn out to be nonlinear Abel equations, for which no closed-form
solutions are available. However, it is shown both numerically and performing a linear analysis
around a given solution, that for each output (current or voltage), one of the two modes of operation
yields an unstable zero dynamics. In Section 3 basic formulae of the PCHS description and the
GSSA approximations are presented. Section 4 presents the full-bridge rectifier, its PCHS model
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
BIDIRECTIONAL POWER FLOW CONTROL OF A POWER CONVERTER
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and the GSSA approximation of interest for the problem at hand. Section 5 computes a controller
using IDA-PBC techniques, and Section 6 presents numerical simulations of the controller for the
full model of the converter. Section 7 shows the experimental validation of the controller. Finally,
Section 8 states our conclusions and points to further improvements.
2. ZERO DYNAMICS OF THE FULL-BRIDGE RECTIFIER
Figure 1 shows a full-bridge ac–dc monophasic boost rectifier, where vi = vi (t) = E sin(s t) is a
single phase ac voltage source, L is the inductance (including the effect of any transformer in the
source), C is the capacitor of the dc part, r takes into account all the resistance losses (inductor,
source and switches), i = i(t) is the inductor current and V = V (t), il are the dc voltage and current
of the load/output port, respectively. The states of the switches are given by s1 , s2 , t1 and t2 , with
t1 = s¯1 , t2 = s¯2 and s2 = s¯1 .
The dynamical equations of the full-bridge rectifier are
di
= −SV −ri + E sin(s t)
dt
dV
C
= Si −il
dt
L
(1)
where the discrete variable S takes value +1 when s1 is closed (vs1 = 0), and −1 when s1 is open
(i s1 = 0). Note that for the approximate average models that we are going to consider, S will take
values in a continuum set; the discrete implementation of the switch is then recovered by means
of a suitable procedure, such as a pulse-width modulation (PWM) scheme.
The control objectives are as follows:
• The mean value of the voltage V should be equal to a desired constant Vd .
• The power factor of the converter should be equal to one. This means that the inductor current
should be i = Id sin(s t), where Id is an appropriate value achieving the first objective via
energy balance. The rms balance equation in steady state is
−il Vd + 12 Id E = 12 r Id2
(2)
Figure 1. Full-bridge rectifier with arbitrary load il .
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
from which, selecting the solution such that Id = 0 when il = 0, one obtains
2
E
2il Vd
E
Id = −
−
2r
2r
r
(3)
An analysis of this twofold problem for the more stringent condition of constant-output voltage
can be found in [17], together with a general formulation of the associated circuit theory and a
discussion of the limitations of the possible topologies.
Note that il is considered constant,§ but in order to get a bidirectional power flow no assumption
is made on its sign, i.e. il ∈ R. The zero-dynamics studies the behavior of the internal variables
assuming that the control objective is achieved. We will first study the zero dynamics taking the
inductor current i as output and after that the zero dynamics when the control goal is the capacitor
voltage V .
2.1. Current-output analysis
As mentioned before, the inductor current control specification is
i(t) = Id sin(s t)
(4)
with Id given by (3). Substituting (4) in (1) we obtain the control law S = u(t):
u(t) =
(E −r Id ) sin s t −s L Id cos s t
V (t)
and the remaining dynamics for the dc bus voltage
dV
1
=
((E −r Id )Id sin2 s t −s L Id2 sin s t cos s t − V (t)il )
dt
C V (t)
(5)
The zero dynamics is the solution of this differential equation (5), which, upon using (2), can be
expressed as
1
dV
g(t)
= − il +
dt
C
V
(6)
with
g(t) =
1
(2il Vd sin2 s t −s L Id2 sin s t cos s t)
C
(7)
Equation (6) is a class A, 2nd type, Abel ODE, for which numerical simulations show the
existence of unstable solutions when il <0. These simulations are performed using the parameters
of the rectifier of Section 6. Solutions of (5) are displayed in Figure 2, with Id = ±4.425 (depending
on the sign of the power flow) and initial condition V (0) = 200 V. Abel equations are known to
appear in tracking problems for power converters [18]. Note that this differs from the case with a
resistive load [19], il = il (t) = V (t)/R, for which the solution is a stable Bernoulli ODE.
§ This
assumption is made to facilitate the analysis and, moreover, it corresponds to a practical scenario where the
dc-link is connected to an inverter to drive an electrical machine [2].
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
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BIDIRECTIONAL POWER FLOW CONTROL OF A POWER CONVERTER
DC Bus Voltage
600
500
i >0
l
V [V]
400
300
200
il <0
100
0
0
0.5
1
1.5
2
2.5
3
time [s]
Figure 2. Zero dynamics of the current-output analysis, stable for il <0 (solid line) and
unstable for il >0 (dotted line).
In order to confirm these numerical results, a linear analysis around a solution V0 (t) can be
performed. Writing V (t) = V0 (t)+ z(t), one obtains
ż = −
1
V02 (t)
g(t)z
(8)
From this it follows that the stability of V0 (t) is determined by the sign of g(t). Although, in
principle, the sign of g(t) depends on the values of the several parameters appearing in (7), it
turns out that for sensible values it is determined by 2il Vd sin2 s t and hence by il . Hence, V0 (t)
is unstable for il <0 and stable for il >0.
2.2. Voltage-output analysis
Now we set V = Vd and, using (1), the control law is obtained as
u(t) =
il
i(t)
The remaining equation for the inductor current is then
di −ri 2 +i E sin s t −il Vd
=
dt
Li
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2007 John Wiley & Sons, Ltd.
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Int. J. Circ. Theor. Appl. 2008; 36:769–788
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
This equation is, again, a class A, 2nd type, Abel ODE. Given a solution i 0 (t) to (9), write
i(t) = i 0 (t)+ z(t) and perform a linear analysis. One obtains
r
il Vd
ż = − + 2
z ≡ A(t)z
(10)
L Li 0 (t)
If il <0 one has A(t)<0 for all t and hence i 0 (t) is stable, while for il >0 the stability depends on
the size of i 0 (t), i.e. i 0 (t) will be stable as long as
ri 02 (t)>il Vd
∀t
(11)
However, as explained before, from an energy-balance point of view one is interested in solutions
i 0 (t) going to zero when il goes to zero, which means that i 0 (t) = il a(t)+higher powers of il .
Substitution in (11) shows that the inequality cannot be satisfied for il small enough and this hints
at the instability of the desired solution. Numerical simulation confirms this behavior, since it is
seen that the stable solution for il >0 does not go to zero with il .
To summarize, from numerical evidence and a linear stability analysis it is concluded that,
taking V as output, the zero dynamics is unstable for il <0 and stable for il >0. Taking i as the
output yields a zero dynamics which is stable for il <0, while for il >0 the stability depends on the
solution; however, the solution i 0 (t) which goes to zero with il , which is the interesting one, is
unstable. Hence, a control scheme based on a cascade structure does not guarantee, in principle,
the stability of the closed-loop system. This situation also appears in the zero-dynamics study of
a dc–dc boost converter, which can be considered as a zero frequency limit of our rectifier, and
where all the computations can be done analytically [20].
3. GENERALIZED AVERAGING FOR PCHS
As explained in the Introduction, this paper uses results that combine the PCHS and GSSA
formalisms. Detailed presentations can be found in [8, 9, 14, 21] for PCHS, and in [10, 11, 22, 23]
for GSSA.
A VSS system in explicit port Hamiltonian form is given by
ẋ = [J(S, x)−R(S, x)](∇ H (x))T + g(S, x)u
(12)
where S is a (multi)-index, with values on a finite, discrete set, enumerating the different structure
topologies. The state is described by x ∈ Rn , H is the Hamiltonian function, giving the total energy
of the system, J is an antisymmetric matrix, describing how energy flows inside the system,
R = RT 0 is a dissipation matrix, and g is an interconnection matrix that yields the flow of energy
to/from the system, given by the dual power variables u ∈ Rm and y = g T (∇ H )T .
Averaging techniques for VSS are based on the idea that the change in a state or control variable
is small over a given time length, and hence one is not interested on the fine details of the variation.
Hence, one constructs evolution equations for averaged quantities of the form
1 t
x(t) =
x() d
(13)
T t−T
where T >0 is chosen according to the goals of the problem.
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2007 John Wiley & Sons, Ltd.
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The GSSA expansion tries to improve on this and capture the fine detail of the state evolution
by considering a full Fourier series. Thus, one defines
xk (t) =
1
T
t
x()e−jk d
(14)
t−T
with = 2/T and k ∈ Z. The time functions xk are known as index-k averages or k-phasors.
Under standard assumptions about x(t), one obtains, for ∈ [t − T, t] with t fixed,
x() =
+∞
xk (t)ejk (15)
k=−∞
If xk (t) are computed with (14) for a given t, then (15) just reproduces x() periodically outside
[t − T, t], so it does not yield x outside of [t − T, t] if x is not T -periodic. However, the idea of
GSSA is to let t vary in (14) so that we really have a kind of ‘moving’ Fourier series:
+∞
x() =
xk (t)ejk ∀
(16)
k=−∞
If the expected steady state of the system has a finite frequency content, one may select some of
the coefficients in this expansion and get a truncated GSSA expansion. The desired steady state can
then be obtained from a regulation problem for which appropriate constant values of the selected
coefficients are prescribed. A more mathematically advanced discussion is presented in [23].
In order to obtain a dynamical GSSA model we need the following two essential properties:
d
dx
xk (t) =
(t)−jkxk (t)
(17)
dt
dt k
x yk =
+∞
xk−l yl
(18)
l=−∞
Note that xk is in general complex and that, if x is real,
x−k = xk
(19)
We will use the notation xk = xkR +jxkI , where the averaging notation has been suppressed. In
terms of these real and imaginary parts, the convolution property (18) becomes (note that x 0I = 0
for x real, and that the following expressions are, in fact, symmetric in x and y)
x ykR = xkR y0R +
x ykI
Copyright q
=
∞
R
R
I
I
{(xk−l
+ xk+l
)ylR −(xk−l
− xk+l
)ylI }
l=1
∞
I
I
R
R
xkI y0R + {(xk−l
+ xk+l
)ylR +(xk−l
− xk+l
)ylI }
l=1
2007 John Wiley & Sons, Ltd.
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
Moreover, the evolution equation (17) splits into
R
dx
R
ẋk =
+kxkI
dt k
I
dx
I
ẋk =
−kxkR
dt k
(21)
If all the terms in (12) have a series expansion in their variables, one can use (21) and (20) to
obtain evolution equations for xkR,I , and then truncate them according to the selected variables.
The result is a PCHS description for the truncated GSSA system, to which IDA-PBC regulation
techniques can be applied. General formulae for the PCHS description of the full GSSA system,
as well as a discussion of the validity of the controller designed for the truncated system, can be
found in [24].
4. PCHS MODEL FOR THE GSSA EXPANSION OF THE FULL-BRIDGE RECTIFIER
As the control design uses the Hamiltonian structure, in this section the dynamical equations
are obtained by the Hamiltonian variables, fluxes and charges q. The relationship between
Hamiltonian and Lagrangian variables are
= Li
q = CV
From (1), the system can be rewritten in Hamiltonian variables as
S
r
˙ = − q − +vi
C
L
S
q̇ = −il
L
(22)
It is sensible for the control objectives of the problem to use a truncated GSSA expansion with
= s , keeping only the zeroth-order average of the dc-bus voltage, q0 , and the two components
of the first harmonic of the inductor current, 1R and 1I . As explained in [12], this selection of
coefficients can be further justified if one writes it for z = 12 q 2 instead of q, and uses the new control
variable v = −Sq. In fact, these redefinitions are instrumental in order to fulfill the conditions
(see [24]) under which the controller designed for the truncated system can be used for the full
system.
With all this, one obtains the PCHS:
0
−r v
* H
1
˙
=
+il
+vi
(23)
√
−v 0
0
*z H
ż
− 2z
with Hamiltonian
H (, z) =
Copyright q
2007 John Wiley & Sons, Ltd.
z
2
+
2L C
(24)
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BIDIRECTIONAL POWER FLOW CONTROL OF A POWER CONVERTER
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Now we apply an GSSA expansion to this system and set to zero all the coefficients except for
x1 ≡ z 0 , x2 ≡ 1R , x3 ≡ 1I , u 1 ≡ v1R and u 2 ≡ v1I . Using that il is assumed to be locally constant, and
I = −E/2, one obtains
that the only nonzero coefficient of vi is vi1
2
2
ẋ1 = −il 2x1 − u 1 x2 − u 2 x3
L
L
r
1
ẋ2 = − x2 +s x3 + u 1
L
C
ẋ3 = −s x2 −
(25)
r
E 1
x3 − + u 2
L
2 C
This system can be given a PCHS form
ẋ = (J (u)− R)(∇ H )T + g1 (x1 )il + g2 E
with
⎛
0
⎜
⎜
⎜
J = ⎜u1
⎜
⎝
u2
and
−u 1
−u 2
0
−
s L
2
⎞
⎟
s L ⎟
⎟
,
2 ⎟
⎟
⎠
0
⎛ ⎞
− 2x1
⎟
⎜
⎜
g1 = ⎝ 0 ⎟
⎠,
0
⎛
0
0
r
2
⎜
⎜0
R =⎜
⎜
⎝
0
⎛
0
0
0
⎞
⎟
0⎟
⎟
⎟
r⎠
2
⎞
⎜ ⎟
⎟
g2 = ⎜
⎝ 0 ⎠
− 12
and the Hamiltonian function
1
1
1
x1 + x22 + x32
C
L
L
√
This model differs from [12] in the −il 2x1 term that now is included in the g1 matrix.
√ This
change is instrumental in achieving a bidirectional power flow capability, since in [12] il 2x1 was
included in the dissipation matrix, for which il 0 was necessary.
The control objectives for this rectifier are a dc value of the output voltage V = (1/C)q equal to
a desired point, Vd , and the power factor of the converter equal to one, which in GSSA variables
translates to x2∗ = 0. From the dynamical equations we can obtain the equilibrium points:
H=
x ∗ = [x1∗ , 0, x3∗ ]
where
Copyright q
x1∗ = 12 C 2 Vd2
(26)
x3∗
(27)
−E L/2r − (E L/2r )2 −(2L 2 /r )il Vd
=
2
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
where we have chosen the smallest of the two possible values of x 3∗ . This is in accordance with
(3), obtained from power balance argument.
5. CONTROLLER DESIGN
The central idea of IDA-PBC [8] is to assign to the closed-loop a desired energy function via
the modification of the interconnection and dissipation matrices. The desired target dynamics is a
Hamiltonian system of the form
ẋ = (Jd − Rd )(∇ Hd )T
(28)
where Hd (x) is the new total energy and Jd = −JdT , Rd = RdT 0 are the new interconnection and
damping matrices, respectively. To achieve stabilization of the desired equilibrium point we impose
x ∗ = arg min Hd (x). The matching objective is achieved if and only if the following PDE
(J − R)(∇ H )T + g = (Jd − Rd )(∇ Hd )T
(29)
is satisfied, where, for convenience, we have defined Hd (x) = H (x)+ Ha (x), Jd = J + Ja , Rd =
R + Ra and g = g1 (x1 )il + g2 E.
Fixing the interconnection and damping matrices as Jd = J and Rd = R, Equation (29) simplifies to
−(J − R)(∇ Ha )T + g = 0
and, defining k(x) = (k1 , k2 , k3 )T = (∇ Ha )T , one obtains
0 = u 1 k2 +u 2 k3 −il 2x1
(30)
r
s L
k3
0 = −u 1 k1 + k2 −
2
2
0 = −u 2 k1 +
(31)
s L
r
E
k2 + k3 +
2
2
2
Equations (31) and (32) can be solved for the controls:
r k2 −s Lk3
u1 =
2k1
u2 =
(32)
(33)
s Lk2 +r k3 + E
2k1
(34)
and replacing (33) and (34) in (30), the following PDE is obtained:
r (k22 +k32 )+ Ek3 −2il 2x1 k1 = 0
(35)
If one is interested in control inputs u 1 and u 2 which only depend on x1 , one can take k1 = k1 (x1 ),
k2 = k2 (x1 ) and k3 = k3 (x1 ), and consequently, using the integrability condition
*k j
*ki
(x) =
(x)
*x j
*xi
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2007 John Wiley & Sons, Ltd.
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BIDIRECTIONAL POWER FLOW CONTROL OF A POWER CONVERTER
779
one obtains that k2 = a2 and k3 = a3 are constants. Then, from (35),
k1 =
1
√
2il 2x1
(r (a22 +a32 )+ Ea3 )
(36)
The equilibrium condition
∇ Hd |x=x ∗ = (∇ H +∇ Ha )|x=x ∗ = 0
is
1
+k1 (x1∗ ) = 0
C
2 ∗
x +a2 = 0
L 2
(37)
2 ∗
x +a3 = 0
L 3
and, since x2∗ = 0, one obtains a2 = 0 and a3 = −(2/L)x3∗ . Substituting these values of a2 and a3
in (36) yields
1 x1∗
k1 = −
(38)
C x1
which satisfies the equilibrium condition (37). One can now solve the PDE (35) and find Ha
2 x1∗ √
2
Ha = −
x1 − x3∗ x3
(39)
C
L
from which
1
1 2 1 2 2 x1∗ √
2
Hd = x1 + x2 + x3 −
x1 − x3∗ x3
C
L
L
C
L
(40)
In order to guarantee that Hd has a minimum at x = x ∗ , the Hessian of Hd has to obey
2
* Hd >0
*x 2 ∗
x=x
From (40)
⎛
2
* Hd *x 2 Copyright q
2007 John Wiley & Sons, Ltd.
1
⎜ 2C x ∗
⎜
1
⎜
⎜
=⎜ 0
⎜
x=x ∗ ⎜
⎝
0
⎞
0
2
L
0
0⎟
⎟
⎟
⎟
0⎟
⎟
⎟
2⎠
L
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
which is always positive definite, so the minimum condition is satisfied. Substituting everything
in (33), (34), the control laws can be expressed in terms of the output voltage V :
u1 = −
2s C x3∗ V
Vd
(41)
u2 = −
C Lil V
2x3∗
(42)
Writing (41) and (42) in real coordinates, using the inverse GSSA transformation
u = 2(u 1 cos(s t)−u 2 sin(s t))
√
and taking into account that u = −S 2x1 , the control action simplifies finally to
S=
2s x3∗
Lil
cos(s t)− ∗ sin(s t)
Vd
x3
(43)
6. SIMULATIONS
In this section we implement a numerical simulation of the IDA-PBC controller for a full-bridge
rectifier. We use the following parameters: r = 0.1 , L = 1 mH, C = 4500 F, s = 314 rad s−1 and
E = 68.16 V. The desired voltage is fixed at Vd = 150 V, and the load current varies from il = −1 A
to 3 A at t = 1 s. Figure 3 shows the bus voltage V . It starts at V = 140 V and then goes to the desired
value, for different load current values. The small static error corresponds to the non-considered
DC Bus Voltage
160
155
V [V]
150
145
140
135
130
0
0.5
1
1.5
2
time [s]
Figure 3. Simulation results: bus voltage V waveform.
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
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Source Voltage and Current
100
80
60
vi [V], i [A]*4
40
20
0
–20
–40
–60
–80
–100
0.9
0.95
1
1.05
1.1
time [s]
Figure 4. Simulation results: source voltage vi and current i waveforms, showing the change in power flow.
Control Output
1
0.8
0.6
0.4
s
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
0
0.5
1
time [s]
1.5
2
Figure 5. Simulation results: control action S remains in [−1, 1].
harmonics in the control design using GSSA. The ac voltage and current are depicted in Figure 4.
Note that when il >0 (for t<1), current i is in phase with voltage vi and power flows to the load;
when il <0 (for t>1), i is in opposite phase with vi and power flows from the load to the ac main.
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
Finally, Figure 5 shows that the control action S remains in [−1, 1], which allows its discrete
experimental implementation using a PWM scheme.
7. EXPERIMENTS
7.1. Experimental setup
The experimental setup has the following parts:
• A full-bridge boost converter with IGBT switches (Siemens BSM 25GD 100D) and parameters: r = 0.1 , L = 1 mH, C = 4500 F. The switching frequency of the converter is 20 kHz
and a synchronous centered-pulse single-update PWM strategy is used to map the controller’s
output to the IGBT gate signals.
• The analog circuitry for the sensors: The ac main source, PMW and dc bus voltages and
currents are sensed with isolation amplifiers. All the signals from the sensors pass through
the corresponding gain conditioning stages to adapt their values to A/D converters.
• Control hardware and DSP implementation: The control algorithm can be implemented using
the analog devices DSP-21116 and DSP-21992 processors. The processing core of this device
runs at 100 MHz and has a 32bit floating-point unit. The sampling rate of the A/D channels has been selected at 20 kHz, the same as the switching frequency of the full-bridge
system.
• The nominal RMS ac mains voltage is Vs = 48 V RMS, and its nominal frequency is 50 Hz,
with total distortion THD = 1.3%.
In order to achieve a bidirectional power flow, a current source has been connected to the dc
side of the converter.
The parameters’ values of the control law (43) were obtained off-line and the input voltage
amplitude computed via its RMS value. Note that no synchronization is needed, since the control
law can be directly computed from the normalized value of vi and its derivative.
7.2. Experimental results
Experimental results are shown in Figures 6–13. First a test with a resistive load Rl connected was
performed. Figures 6–8 show the waveform of the grid current i and voltage vi , the dc bus voltage
V and the load current il . Figure 6 shows that the dc-bus voltage remains close to the desired value
with an acceptable small oscillation. The grid voltage and current are nearly in phase (as shown
by the power factor in Figure 8) but the waveform of the current displays a noticeable distortion
with respect to the desired sinusoidal form, and some higher-order components of i do appear
(Figure 7). This can be attributed to the sampling time and the dead-time of the IGBTs, which
introduce third and fifth harmonic components. The main problem using the GSSA approach is
that the controller is transparent to these disregarded harmonics.
Figures 10–12 display the results for il <0. The experimental results are similar to the il >0 case,
but the inductor current i has a triangular shape. This problem comes from the fact that the third
current harmonic is not controlled and it has the same sign as in the il >0 case. For il >0 the third
harmonic is added to the first harmonic component, while in the il <0 case, due to the current sign
inversion, the third harmonic is subtracted.
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
BIDIRECTIONAL POWER FLOW CONTROL OF A POWER CONVERTER
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V
Vi
i
Figure 6. Experimental results for il >0. V (20 V/div), vi (100 V/div), i (5 A/div). Time scale: 5 ms/div.
Figure 7. THD of the ac current i for il >0.
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
Figure 8. Power factor for il >0.
V
Vi
i
Figure 9. Experimental results for il = 0. V (20 V/div), vi (100 V/div), i (5 A/div). Time scale: 5 ms/div.
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
BIDIRECTIONAL POWER FLOW CONTROL OF A POWER CONVERTER
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V
Vi
i
Figure 10. Experimental results for il <0. V (20 V/div), vi (100 V/div), i (5 A/div). Time scale: 5 ms/div.
Figure 11. THD of the ac current i for il <0.
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2007 John Wiley & Sons, Ltd.
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DOI: 10.1002/cta
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C. BATLLE, A. DÒRIA-CEREZO AND E. FOSSAS
Figure 12. Power factor for il <0.
V
Vi
i
Figure 13. Experimental results for the transient response from il >0 to il <0. V (20 V/div), vi (100 V/div),
i (5 A/div). Time scale: 20 ms/div.
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2007 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. 2008; 36:769–788
DOI: 10.1002/cta
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Finally, the transient behavior of the system when the power flow is inverted is shown in
Figure 13. Note that the system comes quickly to the steady-state values, i.e. i is in opposite phase
with vi and V reaches its constant value (with an error introduced by the disregarded harmonics)
in less than 20 ms.
8. CONCLUSIONS
A controller able to achieve bidirectional power flow in a full-bridge boost-like rectifier has
been presented, and it has been tested in a real plant. The controller has been designed using
interconnection and damping assignment-passivity-based control (IDA-PBC) techniques for a portcontrolled Hamiltonian system (PCHS) model of a suitable generalized state space average (GSSA)
truncation of the variables of the rectifier. The use of truncated GSSA variables allows to formulate
a nontrivial control task, namely a combination of mean value voltage regulation and power factor
compensation, as a pure regulation problem, for which a closed-loop Hamiltonian can be designed.
The method is motivated in part by the difficulty of using standard control-output techniques,
due to the instability of the zero dynamics for at least one of the modes of operation, no matter
which output is chosen. Owing to the properties of the IDA-PBC method, stability is built-in in
the solution presented in this paper.
The control scheme achieves good regulation of the dc bus and high power factor from the ac
side. Minor discrepancies can be related to the discrete-time implementation of the real system
and neglecting of higher harmonics in the voltage bus.
These results validate both the IDA-PBC method and the GSSA decomposition and truncation
of variables. Further improvements of the controller could be obtained by considering higher
harmonics of the dc voltage and inductor current and robustifying the control action in front of
parameter variations (basically E, r and L).
ACKNOWLEDGEMENTS
This work has been done in the context of the European sponsored project Geoplex (IST-2001-34166).
Further information is available at http://www.geoplex.cc. The authors have been partially
supported by the Spanish project DPI2004-06871-CO2-02.
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DOI: 10.1002/cta