An Investigations into the Fast-and 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 PEM-Fuel 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.

Full text

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Advances in Intelligent Systems and Technologies
Proceedings ECIT2006 – 4th European Conference on Intelligent Systems and Technologies
Iasi, Romania, Septembrie 21-23, 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 PEM-Fuel 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

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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 [4-7].

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
slow-scale instabilities of an energy generation system with a PEM-Fuel Cell as energy
source and a battery stack as energy storage (slow processes). The boost converter gives the
fast-scale instabilities of the system in order to optimize the power conversion. Power
converters exhibit a wealth of nonlinear phenomena [8-10]. 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 [11-12].
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 [13-16].
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

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An Investigation into the Fast- and Slow-Scale 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 current-mode 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 closed-loop system behavior are given. The task is to construct a
controller so that the closed-loop system meets the desired characteristics (figure 2). The
analysis of nonlinear systems studies the effect of limit-cycle, soft and hard self-excitation,
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 Piecewise-linear 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

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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 (max-min) 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)

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An Investigation into the Fast- and Slow-Scale Instabilities of an
Energy Generation System with a Fuzzy Hysteretic Control


4. EGS modeling


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Battery model
Usually, a lead-acid 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, dead-cell voltage, discharge impedance, self-discharge
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 dead-cell 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 u-i characteristic shape