An Investigations into the Fastand Slow–Scale Instabilities of an Energy Generation System with a Fuzzy Hysteretic Control
ABSTRACT This paper presents an investigation of the nonlinear phenomena in the current mode controlled boost converter. The boost converter is used into an energy generation system with a PEMFuel Cell as energy source and a battery stack as energy storage device. The bifurcation diagrams with the reference current as the variable parameter have been obtained. The simulation results show that the converter shifts between period one, period two, higher periods and chaos as the some parameter is varied. The bifurcation phenomena have been reported for a clocked hysteretic controller. Using a clocked fuzzy hysteretic controller, a comparison of the effects on the DC/DC Boost converter response in state space, under reference current variations, load variations, and different component variations is performed.
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Conference Paper: Intermittent chaos in switching power supplies due to unintended coupling of spurious signals
[Show abstract] [Hide abstract]
ABSTRACT: This paper studies the "intermittent" chaos and subharmonics observed in switching DC/DC converters using a simple circuit model that describes possible unintended coupling of some weak spurious signal to the converter. The study shows that the signal strength and frequency of the intruding signal are vital parameters that affect the type of intermittent behavior and the period of intermittency. Simulations and experimental results are presented.Power Electronics Specialist Conference, 2003. PESC '03. 2003 IEEE 34th Annual; 07/2003  SourceAvailable from: Chi Kong Tse[Show abstract] [Hide abstract]
ABSTRACT: This paper describes the bifurcation phenomena of a system of parallelconnected dc/dc buck converters. The results provide useful information for the design of stable current sharing in a masterslave configuration. Computer simulations are performed to capture the effects of variation of some chosen parameters on the qualitative behavior of the system. These are summarized in a series of bifurcation diagrams. In particular, it is found that while variation of the voltage feedback gains leads to standard perioddoubling bifurcation, variation of the current sharing ratio leads to border collision bifurcation. Analysis is presented to establish the possibility of the bifurcation phenomena and to locate the current sharing ratio at which border collision occursIEEE Transactions on Circuits and Systems I Fundamental Theory and Applications 03/2001;  SourceAvailable from: Chi Kong Tse[Show abstract] [Hide abstract]
ABSTRACT: Power electronics circuits are rich in nonlinear dynamics. Their operation is characterized by cyclic switching of circuit topologies, which gives rise to a variety of nonlinear behavior. This paper provides an overview of the chaotic dynamics and bifurcation scenarios observed in power converter circuits, emphasizing the salient features of the circuit operation and the modeling strategies. In particular, this paper surveys the key publications in this field, reviews the main modeling approaches, and discusses the salient bifurcation behaviors of power converters with particular emphasis on the disruption of standard bifurcation patterns by border collisions. Author name used in this publication: Chi K. TseProceedings of the IEEE 06/2002; · 6.91 Impact Factor
Page 1
Advances in Intelligent Systems and Technologies
Proceedings ECIT2006 – 4th European Conference on Intelligent Systems and Technologies
Iasi, Romania, Septembrie 2123, 2006
An Investigations into the Fast and Slow –
Scale Instabilities of an Energy Generation
System with a Fuzzy Hysteretic Control
Nicu Bizon, Emil Sofron, Mihai Oproescu
University of Pitesti
nbizon@upit.ro
Abstract. This paper presents an investigation of the nonlinear
phenomena in the current mode controlled boost converter. The boost
converter is used into an energy generation system with a PEMFuel Cell as
energy source and a battery stack as energy storage device. The bifurcation
diagrams with the reference current as the variable parameter have been
obtained. The simulation results show that the converter shifts between
period one, period two, higher periods and chaos as the some parameter is
varied. The bifurcation phenomena have been reported for a clocked
hysteretic controller. Using a clocked fuzzy hysteretic controller, a
comparison of the effects on the DC/DC Boost converter response in state
space, under reference current variations, load variations, and different
component variations is performed.
1. Introduction
Nonlinear behavior in switching power converters, such as bifurcation and chaos, has
attracted much attention from both the engineering and the applied science communities in
the past two decades [1–3]. Control applications of switched mode power supplies have
been widely investigated. The main objective of research and development (R&D) in this
field is always to find the most suitable control method to be implemented in various
DC/DC converter topologies. In other words, the goal is to select a control method capable
of improving the efficiency of the converter, reducing the effect of disturbances (line and
load variation), lessening the effect of EMI (electro magnetic interference), and being less
effected by component variation.
The main objective of this paper is to study clocked hysteretic control methods
implemented in switched mode power supplies (namely the peak current control or Clocked
Page 2
N. Bizon, E. Sofron, M. Oproescu
2
2
Basic Hysteretic Control – CBHC, and Clocked Hysteretic Fuzzy Control – CHFC [4,5]).
Only this two control methods (CBHC and CFHC) are selected to be used for controlling an
Energy Generation System (EGS), and their effects on DC/DC Boost converters are
examined with Matlab. The other ones (without clock) have many drawbacks that where
reported in [47].
2. The energy generation system
Switched mode power supplies (SMPSs) are needed to convert electrical energy from
one form to another. SMPSs are widely used in DC/DC conversions, where the input is a
DC voltage that can be, for example, a rectified line voltage or fuel cell voltage, an output
voltage of a power factor correction (PFC) circuit or a battery. In this paper we investigate
the EGS topology presented in figure 1.
Fig. 1. A typical energy generation system
The advantages and drawbacks of each control method are seen into the fast and
slowscale instabilities of an energy generation system with a PEMFuel Cell as energy
source and a battery stack as energy storage (slow processes). The boost converter gives the
fastscale instabilities of the system in order to optimize the power conversion. Power
converters exhibit a wealth of nonlinear phenomena [810]. The prime source of
nonlinearity is the switching element present in all power electronics circuits. Nonlinear
components (e.g. power diodes) and control methods (e.g. pulse width modulation) are
further sources of nonlinearity. In this paper we experimentally investigate such phenomena
in the current mode controlled boost converter [1112].
Since any circuit involving diodes is nonlinear, chaotic behavior is expected to be
widespread throughout power electronics. This has been borne out by numerous studies,
which have revealed the possibility of chaotic operation in many power converters [1316].
Chaos can be loosely defined as apparently random behavior which is found in a wide
variety of deterministic nonlinear systems. Bounded oscillations of chaos may seem like
random noise (witch usually appear in hysteretic control), but they have a different origin.
They are not caused by chance factors such as thermal noise, but are inherent in the
equations underlying the ideal, noiseless system. In other words, chaos is deterministic: it is
inherent in the solution of the system’s differential equations, just as sine waves result from
solving d2x/dt2 +ω2x =0 equation. Though deterministic, the oscillations of chaos are
unpredictable: their exact course cannot be known ahead of time. Chaotic systems are
Page 3
An Investigation into the Fast and SlowScale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control
highly sensitive to perturbations, so the slightest error in the initial conditions will quickly
blow up into a large error in the predicted waveform. This is what makes it impossible to
predict the weather more than a few days in advance.
The power interface studied in this paper is the hysteretic currentmode controlled
boost converter shown in figure 1. Bifurcations and chaos in this system have been
reported. The boost converter interface model shown in figure 1 is described by four
differential equations that expands the system to a four order system (with four variables:
output voltage  vout, PEMFC voltage – Vin, voltage over the Cstorage  : vC_storage and inductor
current  iL ).
3
3
3. The EGS control
In current control mode we must allow for the situation where a light load produces a
low average inductor current that causes the converter to operate in discontinues conduction
mode (DCM). So, there are three possible states or circuit configurations that depend by the
state (q1) of the electronic switch (controlled with command voltage vcommand) and the diode
conduction state (q2). When operating chaotically, this system may operate in
discontinuous conduction mode (DCM) for some switching cycles, in addition to the usual
continuous conduction mode (CCM) of operation. Thus, to study the chaotic behavior of
such a system, we must take into account the possibility of occasional DCM operation in
which the inductor current may fall to zero in some switching periods.
In the design of a nonlinear control, a nonlinear plant to be controlled and certain
specifications of the closedloop system behavior are given. The task is to construct a
controller so that the closedloop system meets the desired characteristics (figure 2). The
analysis of nonlinear systems studies the effect of limitcycle, soft and hard selfexcitation,
hysteretic, jump resonance and sub harmonic generation. In addition, the response to a
specific input function must be determined. Several tools are available for the analysis of
nonlinear systems. It may be mentioned:
• The Linearization approximation,
• The Describing function concept,
• The Piecewiselinear approximation,
• The phase plane,
• The Lyapunov’s stability criterion,
• Popov’s method, and
• The Sliding mode control (SMC).
In this paper we chose the
clocked hysteretic control (namely in
the literature like the peak hysteretic
control) and clocked fuzzy hysteretic
control, that are presented on detail in
[5].
Fig. 2. Control methods for a energy
generation system
Page 4
N. Bizon, E. Sofron, M. Oproescu
The classical clocked hysteretic model (using a specific boundary control law) is
shown in figure 3.
4
4
Fig. 3. Clocked Hysteretic Controller Fig. 4. Clocked Hysteretic Fuzzy Controller
The structure of the Clocked Hysteretic Fuzzy Control (CHFC) is well defined in [5]
so only a short presentation is following made (figure 4):
•
the inputs:
LLL
Iii
−=∆
(input 1) on [
[0,
MM
V
], respectively, where
LMM
I
∆
•
the output: out on [0, m],
ℜ∈
m
;
•
trapezoidal membership functions for all variables (table 1):
•
the rule list: (N, L, B), (ZE, L, M), (P, L, S), (N, N, M), (ZE, N, M), (P, N, S),
(N, H, S), (ZE, H, S), (P, H, S).
•
Zadeh fuzzy connectives (maxmin) and Mamdani implication [17].
LMM
I
∆
I
,
LMM
I
∆
] and
out
v
(input 2) on
max
L
>
max
VVMM>
;
*
TABLE 1.
Membership functions for fuzzy variables
Fuzzy variables
I
v
LLL
ii
−=∆
out
out
Membership functions
Negative (N)=
, 
I
∆
(
LMM
Zero Equal (ZE)=
, 0, 0,
I
∆
LMM
, 
Lnom
I
∆
, 0),
(
Lnom
Positive (P)=
I
,
∆
I
∆
Lnom
I
∆
),
(0,
Lnom
∆
LMM
I
,
LMM
I
∆
)
Low (L)=
V
(0, 0,
offset
,
kneet
V
),
Normal (N)=
,
offset
V
(
offset
V
,
kneet
V
,
kneet
V
),
High (H)=
,
max
V
(
kneet
V
,
MM
V
,
MM
V
)
Small (S)=
(0, 0, m/4, m/2),
Medium (M)=
(m/4, m/2, m/2, 3m/4),
Big (B)=
(m/2, 3m/4, m, m)
Page 5
An Investigation into the Fast and SlowScale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control
4. EGS modeling
5
5
Battery model
Usually, a leadacid battery for the Energy Storage Device (ESD) is a low price
solution. Generally, a battery model is complex because the storage device has many model
parameters such as capacity, deadcell voltage, discharge impedance, selfdischarge
impedance, and shunt capacitance. In order to simplify the simulations it is used a simple
model for a sealed lead acid battery (SLA) [18,19]. The battery is modeled as a capacitor
for energy storage Cstorage, a DC offset voltage Voffset and a series resistance Rs to limit the
short circuit current (figure 1). In this paper, the 60V/7Ah battery pack structure (5
batteries, 6 cells/battery and 2,45V max/cell) it is used [20]. The value of series resistance
is taken as 80mΩ/cell (as suggested in [21]). The calculated equivalent series resistance of
the pack is RS=5⋅80 mΩ=0,4 Ω. The typical “dead cell” voltage for SLA battery technology
is about 1,75V. Therefore the total offset voltage is Voffset=5⋅6⋅1,75 V=52,5 V. Finally, the
energy stored in the capacitor can be calculated. First we calculate the maximum battery
pack voltage: Vmax=5⋅6⋅2,45 V=73,5 V. So, the maximum storage capacitor voltage must be
the difference between the maximum expected battery voltage and the deadcell voltage:
VC_storage= Vmax  Voffset =21 V. For the 7Ah batteries we obtain Q=7
Ah⋅3600sec/hour=25200 C and the value for the modeled storage capacitance is Cstorage=Q/
VC_storage =1200 F. Obviously, the addition of the battery to the boost converter output
change the control characteristic that must consider the required battery charging
parameters in the control law generation [20].
PEMFC model
The fuel cell electrical equivalent model has a rather large capacitance shunting the
device. The equivalent circuit for this model is shown in figure 5, where Re is the
electrolyte and contact resistance (=Rohm), Rct is the charge transfer resistance (it is this the
same as activation loss), Cd is the double charge layer capacitance, Zd is the diffusion
impedance also called the concentration loss or Warburg impedance [22,23].
Fig. 5. The PEMFC equivalent circuit Fig. 6. The PEMFC ui characteristic shape
Page 6
N. Bizon, E. Sofron, M. Oproescu
The proposed model it is based on well know and simple analytical description [22
25]. The losses due to irreversibility’s will be determined in terms of three main groups:
Activation losses, Ohmic losses and Concentration losses.
By adding, the cell voltage is determined in terms of the drawn current (iL) and PEM
area (APEM):
(
0
6
6
)
max
1lnln
0
+
−+
+
i
−+−=
i
nili
conc
B
nili
act
A
ohmic
R
nili
EE
,
A/
L
i
li
PEM
=
TABLE 2.
Value of model coefficients
Constant
Eo / [Volts]
in / [mA/cm2]
i0 / [mA/cm2]
imax / [A/cm2]
ROhmic / [Ω]
Aact / [Volts]
Bconc / [Volts]
TPEMFC / [ms]
τPEMFC / [ms]
Value from [3]
1.2
2
0.067
0.9
0.03
0.06
0.05
100
10
A first order system (with death time τPEMFC and time constant TPEMFC) it is used for
the PEMFC dynamic behavior (where
CELL
n
is stack cells number):
t
PEMFC
⋅
=⋅=
1
Using the above data, the fuel cell voltage is expected to be in the range of 42V to 52V
based on nominal load current and normal operating conditions (see figure 6, where ncell=60
and APEM=60 cm2). However, the open cell voltage can be as high as 72V if the preload
fails and must be designed for or protected against in the inverter design.
To speed up the simulations (time equivalent systems are presented in [26]), the
battery and PEMFC time constants are reduced by 10, so the values used in simulations are
C = 120 F, and TPEMFC =10 and τPEMFC = 1 ms, respectively (see figures 7).
sT
e
sEn
PEMFC
CELL
⋅+
−
V
)(V,V
/
PEMFC
inPEMFC
τ
Fig. 7. The PEMFC dynamic
Page 7
An Investigation into the Fast and SlowScale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control
Inherent in the fuel cell is a high intolerance for current ripple at low frequency or
slower load transients [27].
Boost converter model
Figures 14 show the complete circuit diagram of the boost converter with different
current mode control. The current mode controlled boost converter is designed to operate in
continuous conduction mode. The converter equation when operate in continuous
V
, where
7
7
conduction mode (CCM) is
τ−=1
in
out
V
T
ton
=
τ
is the duty ratio. For example, if
input voltage, output voltage and output power are
VVin
P
48
=
,
VVout
60
=
and
WPout
900
=
,
respectively, the duty ratio is
2 , 0
=τ
and
Ω=
⇒
==
4 15
load
R
out
out
load
A
V
I
. The average
inductor current <iL>= IL is
A
I
I
out
−
L
75, 18
1
==
τ
. Series resistance of the inductor (RL) is
0.05 Ω in simulations, and the switching frequency
T
f
1
=
is constant (10 kHz).
In the boost converter when the switch (MOSFET – ideal modeled) is closed (q1=1),
the input provides energy to the inductor, and the diode is reverse biased (q2=0). When the
switch is open (q1=0), energy stored in the inductor is transferred to the output (q2=1). No
energy is supplied by the input during q1=q2=0 interval [28]. Clearly, depending upon the
value of iL at the beginning of the period and some circuit parameters, the switch may or
may not turn off during a switching period, as is following presented:
q1(k)=1;
if iL(k+1)>Iref(k+1)
q1(k+1)=0;q2(k+1)=1;
elseif iL(k+1)<0
q1(k+1)=0;q2(k+1)=0;
iL(k+1)=max(0,iL(k+1));
else
q1(k+1)=q1(k);q2(k+1)=q2(k);
end
Thus, there are three possible types of operation in any switching period, as illustrated
in figure 8 where in =iL(nT) and Iref =ILMAX.
Fig. 8. Inductor current waveforms corresponding to three possible types of operation
Page 8
N. Bizon, E. Sofron, M. Oproescu
Linear and nonlinear dynamics
The corresponding fourdimensional map (usually named as Poincare´ map) can be
derived using following equations:
kicel=max(kdcel,1); TS=RS*CS; tcel=exp(Ts/Tcel);
Vcel(k+1)=ncel*(E0(iL(kicel)/Scel+in)*RohmicAact*log((iL(kicel)/Scel+in)/i0)+
+Bconc*log((p^0.5)*(1(iL(kicel)/Scel+in)/((f^0.5)*imax))));
Vin(k+1)=tcel*Vin(k)+(1tcel)*Vcel(k);
iL(k+1)=iL(k)+(T/L1)*(Vin(k)iL(k)*RLq2(k)*vC(k));
vC(k+1)=vC(k)(T/(C*RS))*(vC(k)vCS(k)Vof)(T/(C*Rout1))*vC(k)+ +(T/C)*q2(k)*iL(k);
vCS(k+1)=vCS(k)+(T/TS)*(vC(k)vCS(k)Vof);
We define the Poincare´ map as an iterative function that expresses the state variables at the
end of a switching period xn+1 = f(xn, p), where xn is a vector consisting of the state
variables at t = nT, T being the switching period, and p is a variable control or circuit
parameter:
(
n L,n C_storage,n PEMFC,n out,n
ivvvx
==
To illustrate the behavior of this system, we
consider the following set of PEMFC parameters:
ncel=60 cells; APEM=600 cm2, and Iref as a variable
parameter. With above mention set of parameters
and iLM=21.75 A, the converter is supposed to
operate in CCM under CBHC currentmode
control, with a duty cycle of 0.2 in the steady
state. The system becomes unstable when it is
under CBHC currentmode control. A chaotic
attractor from the Poincare´ map of this system is
shown in Fig. 9. The same relations are used to
analyze the time system dynamic behavior using
a sample time Ts=1µs. For example, the inductor
current dynamic is presented in figure 10.
The changes in the shape of the inductor current are well correlated with the
bifurcation phenomena from the Poincare´ map (converter shifts between period one,
period two, higher periods and chaos).
8
8
)()t
L,C_storage
v
PEMFC
v
out
t
)nT(i) nT() nT() nT(v
Fig. 9. The bifurcation diagram
Fig. 10. The inductor current dynamic (a zoom are presented in the right side)
Page 9
An Investigation into the Fast and SlowScale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control
5. Simulation results
9
9
We analyze the system dynamic behavior using a sample time Ts=1µs, a CBHC and a
CHFC control method, and different control or circuit variable parameter that are mention
in every set of figures. Apparently, for a nominal regime with or without PEMFC energy
source (see figures 1112) the dynamic of the variables is almost the same for different
control implementations used in this paper (clocked variants: CBCT and CHFC). For a step
nominal to light the PEMFC dynamic appear in evidence when a long simulation time is
used (figure 13).
Fig. 11. The system dynamic with
PEMFC as energy source
Fig. 12. The system dynamic with a
constant input voltage as
energy source
Fig. 13. The system dynamic with PEMFC as energy source and a load step
Figures 1416 show the effect of the
reference current parameter Iref, when Iref
linearly decreases from the nominal value to
zero. Inductor current ripple is also PEMFC
output current ripple. If PEMFC output
current is over the corner value of the
PEMFC ui characteristic, the current ripple
is up to IN/4, and after that value decreases.
The current ripple is smaller if an ESD it is
used to the EGS output port.
Fig. 14. First reference current shape
Page 10
N. Bizon, E. Sofron, M. Oproescu
10
10
Fig. 15. The system dynamic with CBHC control and with (right)/without (left)
ESD, and Iref as parameter
Fig. 16. The system dynamic with CHFC control and with (right)/without (left)
ESD, and Iref as parameter
Figures 1719 show the effect of the reference
current parameter Iref , when Iref increases from
nominal value (IN) to 4x IN. If Iref rise over the 40A
the CBHC can’t catch this reference and the
inductor current rise uncontrollable. The boost
converter operate in DCM and the output voltage
rise and the inverter can’t safety operate. The CHFC
can operate using a bigger current reference with
penalty in ripple current. The fuzzy controller must
be optimized for this kind of operation, especially
for the case without ESD, when the inductor current
can be rise uncontrollable.
Fig. 17. Second reference current
shape
Page 11
An Investigation into the Fast and SlowScale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control
11
11
Fig. 18. The system dynamic with CBHC control and with (right)/without (left)
ESD, and Iref. as parameter
Fig. 19. The system dynamic with CHFC control and with (right)/without (left)
ESD, and Iref as parameter
Figures 2022 show the effect of the reference current
parameter Iref when a triangle shape is used. The system will
be either stable or oscillatory depending on the initial
condition. For the second case of reference current variation
initial condition is nominal condition. Now the initial
condition is zero for inductor current and the evolution in
state space is unstable even with CHFC control. It is true
that we have the same parameter for the fuzzy controller,
and into simulation time sometime appear message “outside
of range”. So, if the variable ranges are large according to
dynamic of the variables the CHFC will be able to limit the
inductor current. Different initial conditions may give rise
to different stability boundaries.
Fig. 20. Triangular reference
current shape
Page 12
N. Bizon, E. Sofron, M. Oproescu
12
12
Fig. 21. The system dynamic with CBHC control and with (right) /without (left)
ESD, and Iref as parameter
Fig. 22. The system dynamic with CHFC control, without ESD and with (right)
/without (left) variable ranges adapted to the variation of the input fuzzy
variables, and Iref as parameter
Figures 2325 show the effect of the boost
converter inductance parameter L, when L
linearly decreases from nominal value to zero.
If the inductance value is bigger then the
critical value, the boost converter operates in
CCM mode and the current ripple is
controllable. After that value the current ripple
is bigger than accepted PEMFC current ripple
value, so this regime (DCM) must be avoided.
Fig. 23. L variation shape
Page 13
An Investigation into the Fast and SlowScale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control
13
13
Fig. 24. The system dynamic with CBHC control and with (right)/without (left)
ESD, and L as parameter
Fig. 25. The system dynamic with CHFC control and with (right)/without (left) ESD,
and L as parameter
Finally, figures 2628 show the effect
of the load parameter. Load value linearly
increases from the nominal value (RN) to
12x RN (light load). For a light load the
boost converter operate in DCM and the
same mention problems appear. The
PEMFC current ripple is bigger without
ESD connected to EGS output port. For
different initial condition of the output load,
the boost converter present the nonlinear
phenomena, but with different levels and
oscillatory frequencies.
Fig. 26. Load variation shape
Page 14
N. Bizon, E. Sofron, M. Oproescu
14
14
Fig. 27. The system dynamic with CBHC control and with (right)/without (left) ESD,
and load as parameter
Fig. 28. The system dynamic with CHFC control and with (right)/without (left) ESD,
and load as parameter
Changing in the load dynamic is the most frequently case and the controller must be
designed to give controllability for the EGS in any situation.
6. Conclusions
The simulation results shows that, in comparison with the CBHC control, the CHFC
provides better dynamic response, robustness against system uncertainty disturbances, and
an implicit stability proof. The increased stability is obtained by a proper designing of the
CHFC controller.The bifurcation phenomena into an EGS using reference current as the
variable parameter is reported. For different values of the output load, inductor and
reference current we present the nonlinear phenomena into an EGS.
The EGS will be either stable or oscillatory depending on the initial condition. The
implication of this finding is relevant to practical operation of the system since stability
Page 15
An Investigation into the Fast and SlowScale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control
information obtained from linear models or any method that involves perturbation around
the operating point can be unreliable. Specifically, stability information obtained from
linear methods has been shown overoptimistic. In fact, the basin of attraction of an
operating point is an important piece of control design information, and stability boundaries
in parameter space have to be interpreted in conjunction with the initial conditions.
Different initial conditions may give rise to different stability boundaries.
The behaviour of the EGS with CBHC controller, without an ESD can be summarized
by means of the presented phase diagrams for state variable:
? when Iref is up to a specific value (initial condition depending) the EGS operate in stable
conditions; in conventional engineering terms, for a given Iref in this range the EGS can
catch the imposed steady state.
? if the Iref is increased further the behaviour bifurcate to a periodn subharmonic
oscillation; after 1 up to 5 of these stages, depending by L value and controller
parameters, chaotic operation is entered;
? chotic operation is maintained if Iref is up to maximum reference current: Iref(max)=200A;
The behaviour of the EGS with CHFC controller, without an ESD can be summarized
by mean of the presented simulation results, too:
? when Iref is up to a specific value (initial condition depending) the EGS operate in
stable conditions;
? if L value is over to 1000µH the EGS behaviour goes directly from stable operation to
chotic operation;
? chotic operation is maintained if Iref is up to Iref(max)<200A; over this value the inductor
current jump to a bigger value.
If the EGS have an ESD at the output port the EGS behaviour is more stable as an
effect of the ESD pole presence into the EGS transfer function, indifferently what type of
the controller is used. The obtained results are very promising, validating the model of the
proposed hysteretic fuzzy control for a boost converter supplied by a PEMFC source.
15
15
Acknowledgments. The Grant #570/20062008 of the National University Research
Council (CNCSIS) has supported part of the research for this paper.
References
[1]. Banerjee S, Verghese G, editors. Nonlinear phenomena in power electronics. IEEE
Press (2000).
[2]. C.K. Tse, Y. Zhou, F.C.M. Lau and S.S. Qiu, Intermittent chaos in switching power
supplies due to unintended coupling of spurious signals, IEEE Power Electron. Spec.
Conf. Rec, (2003) 642647.
[3]. CK.Tse, Complex behavior of switching power converters. CRC Press (2003).
[4]. N. Bizon, M. Oproescu  Hysteretic Fuzzy Control of The Boost Converter, ECAI’05,
ISSN 1453–1119, nr. 5, Piteşti, S2 (2005) 110.
[5]. N. Bizon, M. Oproescu  Clocked hysteretic fuzzy control of the boost converter,
ECAI’05, ISSN 1453–1119, nr. 5, Piteşti, S2 (2005) 1120.