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CHAPTER 6
INPUT VOLATGE REGULATION AND EXPERIMENTAL
INVESTIGATION OF NON-LINEAR DYNAMICS
IN PV SYSTEM
6.1 INTRODUCTION
The DC-DC Cuk converter is used as an interface between the PV
array and the load, but other types of converters can be used for the same
purpose. The input voltage of the converter is controlled in order to regulate
the operating point of the array. Besides reducing losses and stress because of
the bandwidth-limited regulation of the converter duty cycle, controlling the
converter input voltage reduces the settling time and avoids oscillation and
overshoot, making easier the functioning of maximum power point tracking
methods. The voltage regulation problem is addressed that starts with the
linear modelling of the PV module and the design of controller.
6.2 NEED FOR INPUT VOLTAGE REGULATION
In photovoltaic power systems, both photovoltaic modules and
switching-mode converters present non-linear and time-variant
characteristics, which result in a difficult control problem.
Figure 2.1 illustrates that the changing radiation varies the
photovoltaic current dramatically. The fast dynamics of insolation is usually
caused by a cover of mixed rapid moving clouds. The PV array operating
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point can be adjusted by regulating the voltage or current at the terminals of
the array.
If the photovoltaic current is used as the set point, the MPP tracking
requires fast dynamics to follow a wide operating range from 0 A to the short-
circuit current, depending heavily on weather conditions. Nevertheless, the
changing insolation slightly affects the voltage of MPP (Vm). Figure 2.2
shows the effect of temperature on the I-V characteristics.
Unlike the current of the MPP, the photovoltaic voltage of the MPP
is usually bounded by 70%-82% of the open circuit voltage. This gives a
lower bound and upper limit of the tracking range. When regulation of
photovoltaic voltage is implemented, the MPP tracker can quickly decide the
initial point according to the percentage of the open-circuit voltage. The value
of VMPP is continuously tracked and updated by the MPP tracker. Therefore,
the regulation performance of the photovoltaic voltage is important for MPP
tracking.
The voltage control is preferred because the voltage at the MPP is
approximately constant. The PV current, on the other hand, changes greatly
when the solar irradiation varies.
This research work discussed the voltage control problem as shown
in Figure 6.1 in solar PV-powered Cuk converter MPPT system. The PV array
feeds the DC-DC Cuk converter. The Cuk converter is used as an interface
between solar PV module and load, since the Cuk converter is the good
choice for the maximum power point tracking circuits.
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Figure 6.1 Input voltage regulation for Cuk converter-based PV system
The input voltage of the converter is controlled in order to regulate
the operating point of the solar PV module. However, both photovoltaic
modules and switching-mode converters demonstrate non-linear and
time-variant characteristics, which make a controller design difficult.
This research work proposes to design a voltage controller to
regulate the input voltage of the converter for the change in irradiation. The
voltage controller improves the transient response to the input voltage of the
converter, avoids oscillation, overshoot, making easier the functioning of
MPPT methods and ensures period -1 operation.
6.3 LINEARIZATION OF SOLAR PV MODULE AT MPP
PV module of L1235-37Wp has non-linear I-V charecteristic
which is shown in Figure 6.2. The operating characteristic of a solar cell
consists of two regions: the current source region, and the voltage source
region. In the current source region, the internal impedance of the solar cell is
high and this region is located on the left side of the current-voltage curve.
The voltage source region, where the internal impedance is low, is located on
the right side of the current-voltage curve. As can be observed from the
characteristic curve, in the current source region, the output current remains
almost constant as the terminal voltage changes and in the voltage source
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region, the terminal voltage varies only minimally over a wide range of output
current. The terminal current Ipv remains at a somewhat constant level in the
constant-current region up to MPP voltage. The current decreases if the
voltage is further increased in the constant voltage region eventually
diminishing to zero when the open circuit condition is reached .
Figure 6.2 Non-linear I-V characteristics of L1235-37Wp solar module
The Thevenin’s equivalent circuit at MPP is shown in Figure 6.3.
Table 6.1 shows the values of the Thevenin’s equivalent circuit of the L1235-
37Wp module at MPP.
Figure 6.3 Linear equivalent circuit valid at the linearization point
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Table 6.1 Thevenin’s equivalent circuit values for solar PV at MPP
Maximum power at MPP (Pmax) 37WShort circuit current (Isc) 2.5AVoltage at MPP (Vpv) 16.4Current at MPP (Ipv) 2.25Thevenin’s equivalent voltage (Veq) 34.6V Equivalent resistance (Req) 7.289Open circuit voltage (Voc) 21V
The linear equivalent circuit of Figure 6.3 is valid at the
linearization point (V, I) and is a good approximation of the solar PV module
for the computer simulation. The dynamic behavior of the solar PV powered
MPPT system depends strongly on the point of operation of the module.
6.4 SMALL SIGNAL MODELLING FOR INPUT VOLTAGE
CONTROL
In Figure 6.4, the small signal model of the Cuk converter-based
solar PV system is analyzed to obtain small-signal converter transfer function.
The small signal model describes the behavior of Vpv with respect to the duty
cycle of the Cuk converter
Figure 6.4 Cuk converter with Solar PV module linear model
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The bar over a variable name (e.g. v, ) means the discrete-time
average value of the variable within one switching period of the converter. By
writing the circuit state equations with average variables, the high-frequency
components are eliminated and natural system behavior is analyzed.
The average capacitor state equation is
c v - 1= 0 (6.1)
The average output inductor state equation is
V - Vo - L - RL = 0 (6.2)
The circuit constituting L1, C1, MOSFET, and diode may be
replaced by the average equivalent quatripole with terminals 1-2-3-4, which is
equated with following equations, where d is the duty cycle of the transistor
V = v (-d / 1-d) = v K (6.3)
1 = (-d / 1-d) = k (6.4)
where k= -d/ 1-d
The relation between the input voltage of the converter and the
capacitor voltage is given by
v = R v +v (6.5)
To obtain the input voltage control for Cuk converter-based solar
PV system, the small signal variables are introduced into the state equations.
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The steady state DC values are capitalized and the small signals are marked
with a hat.
v = Vc + v
v = Vpv + v
= IL + (6.6)
k = K - k
By substituting 6.4, 6.5 in 6.1 and eliminating non linear product
terms, the small signal equation is derived using Laplace transformation
IL k(s) - v (s) – s C Rc v (s)- (s)K –s C v (s) =0 (6.7)
Similarly, solving 6.2, 6.3, 6.5, 6.8,
-Vc k(s) - RL (s)-sL (s)- s C Rc K v (s)+ v (s)K- s C Rc Vc k(s)=0
(6.8)
Solving 6.5, 6.7, and 6.8, the small signal input voltage to duty
cycle is obtained which is given below:
Gvd(s) =( )
=( )( ( )
( )
( )( ) (6.9)
For analyzing the input voltage regulation of Cuk converter-based
solar PV system, the converter parameters are chosen as follows:
L1=L2=500e-6H, C1=C2=220e-6F, Vpv = 16.4V, d=0.442, K= -0.792,
IL= 2.84A, Vo = 13V load resistance R= 2 , switching frequency fs =25kHz,
diode (BY129). The switch S in the power stage is realized using a MOSFET
(IRF 840). The component values used in linear model of solar PV module
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circuit are C = 2200 F, Rc= 0.05 , RL =0.1 , Req =7.29 , d=0.442,
K= -0.792, Veq = 34.6V
6.5 DESIGN OF SINGLE FEEDBACK LOOP VOLTAGE
CONTROLLER
The voltage controller (PI controller) actuates on the converter duty
cycle and directly regulates the input voltage (Vpv) of the Cuk converter.
Figure 6.5 shows how the controller is constituted. Figure 6.6 shows the
single feedback loop voltage controller to regulate the input voltage of the
Cuk converter.
Figure 6.5 Input Voltage-controlled Cuk converter-based solar PV system
Figure 6.6 Voltage controller with single feedback loop
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The Cuk converter-based solar PV system is compensated with
voltage controller of (24s+420)/s and the feedback gain (H) is 1/26. The
crossover frequency of the compensated system is 2.2×103 rad/sec with phase
margin of 104 . The voltage controller makes the Cuk converter-based solar
PV system operate at points other than the point at which the I-V curve was
linearised. The bode plots of open loop system (Gvd) and closed loop system
are illustrated in Figures 6.7 and 6.8. To verify the validity of small signal
modelling in time domain the closed loop system is tested with unit step
input.
Figure 6.7 Bode plots of the open loop system Gvd(s) and the closed loop
compensated system
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Figure 6.8 Bode plot of compensated system
At low frequency, the compensated system gain is 71dB which is
high enough to minimize the steady state error. The unit step response of the
transfer function is shown in Figure 6.9.
Figure 6.9 Unit step response of the compensated system
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6.6 SIMULATION RESULTS
The diagram of the input regulated closed loop system designed in
MATLAB/Simulink is presented in Figure 6.10 that includes linearised model
of solar PV array which consists of voltage source in a series with the
equivalent panel resistance, Cuk converter, voltage controller and a load.
Figure 6.10 Input voltage regulation of solar PV powered Cuk converter
in MATLAB /Simulink
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To test the system operation under change in irradiation condition,
the system is modeled in which indent to change the solar panel resistance.
Initially, the solar panel resistance is kept as 7.289 which corresponds to
irradiation of 1000 W/m2. At t=0.5 sec, the panel resistance is reduced by
50% which corresponds to 500 W/m2. The voltage controller is designed in
such a way that converter input voltage regulation of 16.4V is achieved for
both the conditions. The simulated input voltage regulation for the change in
irradiation is shown in Figure 6.11.
Figure 6.11 Regulated Input voltage waveform due to change in
irradiation levels
6.7 IMPLEMENTATION OF INPUT VOLTAGE REGULATION
FOR CUK CONVERTER-BASED SOLAR PV SYSTEM
The experimental setup is shown in Figure 6.12. It consists of a
power circuit and a control circuit. The power circuit consists of inductors L1
and L2 made of ferrite core, and capacitors C1 and C2 are of plain polyester.
Power MOSFET (IRF840) is used as active switch S. The converter is
assumed to operate in continuous conduction mode. The control circuit
consists of the following blocks: voltage divider, Vref generation, difference
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amplifier, inverting amplifier, and a Schmitt trigger. A reference voltage is
generated and fed to non-inverting input of the difference amplifier.
Voltage from the divider circuit is given to the inverting input of
the difference amplifier LM358. This input voltage is regulated irrespective of
the temperature and irradiation change. The experimental setup and the Piece
spice PCB layout for PID controller are shown in Figures 6.12 and 6.13.
Figure 6.12 Photography of an experimental setup
Figure 6.13 PCB layout for PID controller
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Voltage measurement is required at the point where the solar PV
module output is connected to the input of Cuk converter. The voltage at this
point is the operating voltage of the PV module. The unregulated input voltage of
the converter, i.e., solar PV panel output voltage is shown in Figure 6.14.
Figure 6.14 Unregulated input voltage without voltage controller
(Horizontal scale: 5*10-6sec/div, Vertical scale= 5V/div)
The input voltage is regulated using PI voltage controller. The
regulated input voltage is shown in Figures 6.15 and 6.16.
Figure 6.15 Regulated input voltage with PID controller (Horizontal
scale: 5*10-6sec/div, Vertical scale= 5V/div)
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Figure 6.16 Regulated input voltage with voltage controller (Horizontal
scale: 50*10-6sec/div, Vertical scale= 5V/div)
The unregulated input voltage takes a time of 30ms to reach steady
state voltage of 17V without voltage controller. Using PI voltage controller,
the input voltage of the converter takes a time of 5ms to reach the steady state
voltage of 16.4V. The input voltage is regulated as constant for the change in
irradiations using single feedback loop voltage controller.
6.8. EXPERIMENTAL INVESTIGATION OF NON-LINEAR
DYNAMICS IN CUK CONVERTER-BASED SOLAR PV
SYSTEM
Power electronics is a field spawned by many real-life applications
in industrial, commercial and aerospace environments. At the same time, it is
also a field rich in nonlinear dynamics. As one of the most popular members
in power electronics circuits, the DC-DC converter has found in widespread
application for many decades. In non-linear circuits and systems a great
variety of strange phenomena have been observed, including sub-harmonics,
quasi-periodic oscillations, and chaotic behaviors.
136
The non-linear behaviors have been intensively studied in the cross-
disciplinary science of chaos. In particular, it has recently been observed that
a large number of power electronic circuits can exhibit deterministic chaos.
Power converters can work under linear control or non-linear control. Most
research works focus on linear feedback controlled converters, which may
exhibit interesting bifurcation and chaos when some parameters are varied.
Period-doubling bifurcation, Hopf bifurcation, border-collision bifurcation,
and chaos have been reported in these converters. On the other hand, non-
linear controlled converters can also exhibit bifurcation and chaos, although
little is known about these nonlinear phenomena.
From the experimental point of view, the chaos may be defined as
bounded steady-state behavior which is not an equilibrium point, not periodic,
and not quasi-periodic. In time domain, a chaotic trajectory is neither periodic
nor quasi-periodic but looks “random”.
Also DC-DC converters exhibit different non-linear phenomena
including bifurcations, quasi-periodicity and chaos under both voltage mode
and current mode control schemes. Due to the non-linear dynamics in the
power electronic circuits, their operation is characterized by the cyclic
switching of circuit topologies, which gives rise to a variety of non-linear
behaviors like bifurcation and chaos. The behavior of a chaotic system is a
collection of many orderly behaviors, none of which dominates under
ordinary circumstances. Since current and voltage mode controlled converters
have wide industrial application, control of chaos has an important
significance.
Non-linear phenomena jeopardize the performance of converters,
and suppression of bifurcation and chaos has been an important subject in
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designing converters. Non-linear phenomena in DC-DC converter used for
solar PV system have drawn attention only recently. Studying the non-linear
behavior in DC-DC converter used for solar PV system is not only interesting,
but also very useful. To track the maximum power from the solar PV module,
the output voltage of the solar PV module (input voltage of the DC-DC
converter) has to be chaos- free and the solar PV voltage to ensure period-1
operation so that the oscillation near to maximum power point is nil. An
attempt to control chaos in the Cuk converter-based solar PV system is made
in this research by adopting a conventional PID controller. Hence, Cuk
converter used for solar PV system is designed to operate period -1 operation.
Different methods are proposed for controlling chaos in non-linear
systems which can be classified into two general categories namely, feedback
control methods and non-feedback control methods. Feedback methods
include the Ott-Grebogi-Yorke (OGY) method, Variable Ramp Compensation
(VRC), Time-Delayed Feedback Control (TDFC) method, etc. Examples of
non-feedback methods include adaptive control and Resonant Parametric
Perturbation (RPP). In adaptive control, conventional controllers such as PID
controller and sliding mode controllers are used to control chaos in non-linear
system.
Even though most of the approaches proposed until now are very
interesting, they mainly present theoretical or simulated results. As a
consequence, there is a lack of experimental analysis on the parameter
domains for which chaotic behavior may occur.
Therefore, this research work aims to bridge this gap by presenting
an experimental study of some dynamic phenomena that can occur in solar
PV powered voltage mode-controlled Cuk converter system.
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In particular, this research illustrates a novel hardware
implementation able to show some pathways through which the solar PV
powered Cuk converter may enter into chaos. The analysis of chaos in a
voltage mode-controlled Cuk converter-based solar PV system has been
performed and the use of conventional control method to suppress chaos has
been discussed.
The chaotic behavior of voltage mode controller Cuk converter-
based solar PV system in continuous conduction mode is analyzed using the
block diagram shown in Figure. 6.17. The non-linear dynamics are analyzed
by varying the Vref.
6.8.1 System Description
Figure 6.17 describes the block diagram of experimental setup of
the proposed voltage-controlled Cuk converter-based solar PV system, which
is constituted by a power stage and a control circuit. The power stage includes
an inductor L1, L2, capacitor C1, C2, a switch S, a load resistance, a solar PV
module ( L1235-37Wp).
For analyzing the chaotic behavior in Cuk converter-based solar PV
system, the converter parameters are chosen as follows: L1=L2=500e-6H,
C1=C2=220e-6F, Vin = 16.4V, load resistance R= 2 , switching frequency fs
=25kHz, diode (BY129). The switch S in the power stage is realized using a
MOSFET (IRF840). The control circuit consists of a voltage divider, a
comparator LM311, PID controller, Schmitt trigger-gate drive circuit (555
timer).
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Figure 6.17 Block diagram of the experimental setup
The LM311 compares the reference voltage Vref with the voltage
across solar PV module (input voltage of Cuk converter) using a voltage
divider which is proportional to the input voltage of the Cuk converter.
Therefore, the output of the comparator is high when the input voltage
reaches the value Vref, whereas it is low when the input voltage is less than
Vref.
In order to generate a square wave with amplitude of 5 V,
frequency fs = 1/T = 25 kHz and duty cycle d =0.4, the integrated device 555
Timer (NE555N) IC (along with proper resistors and capacitors) is used. The
measurements have been recorded by using a RIGOL digital storage
oscilloscope.
The Cuk converter has two modes of operation. The converter is
assumed to operate in continuous conduction mode, that is, the input inductor
current of Cuk converter never falls to zero. Hence, there are two possible
constituent linear circuit configurations.
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When deriving the state equations for the Cuk converter, all
capacitor voltages and inductor currents are chosen as state variables. The
converter can be modeled by the following equation :
= A1 x + B1 Vin the switch S is on
= A2 x + B2 Vin the switch S is off (6.10)
where x=
vvii
and Vin is the input voltage of the Cuk converter.
In mode 1, the switch is on and the diode is off. During this mode,
the system matrices A1 and B1 are given by
A1 =
0000
0011
0100
0101
22
1
22
LL
C
CRC
, B1 =
1
1000
L
In mode 2, the switch S is off and the diode is on. In this case, A2
and B2 are
A2 =
0010
0001
1000
0101
1
2
1
22
L
L
C
CRC
, B2 =
1
1000
L
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Also the Cuk converter can work in discontinuous mode which can
be either discontinuous-inductor-current or discontinuous-capacitor-voltage
mode. For discontinuous-inductor-current operation, it is characterized by the
presence of a duration in which both the switch S and diode are open, i.e.,
i + i = 0. This happens when the inductances are relatively small. During
this mode of operation, the system matrices are given by
A3 =
0011
0011
1000
0101
2121
2121
1
22
LLLL
LLLL
C
CRC
, B3 =
21
211
100
LL
LL
The discontinuous –capacitor –voltage mode is characterized by the
presence of a duration in which the capacitor voltage, V , is zero. This
happens when the capacitance of the capacitor C1 is relatively small, such that
the value of V drops to zero within the switch on-time, introducing a
duration in which both the switch and diodes are conducting.
During this duration, the system matrices are given by
A4 =
0000
0001
1000
0101
2
1
22
L
C
CRC
, B4 =
2
1000
L
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6.8.2 Route to Chaos
For analyzing the dynamic behavior in solar PV-powered Cuk
converter system, the reference voltage Vref is varied. The input voltage of
the Cuk converter is kept as 16.4 V.
6.9 INVESTIGATION OF DYNAMIC BEHAVIOR FOR
PARAMETER VARIATION
The dynamic behavior of input voltage of the solar PV-powered
Cuk converter system is experimentally analyzed by varying Vref. The
fundamental period 1- waveform has been found with Vref =5.68V. The solar
PV powered Cuk converter system has stable periodic behavior. The input
voltage of the period-1 operation is shown in Figure 6.18.
Figure 6.18 Experimental period-1 waveform in the input voltage when
Vref =5.68V (Horizontal scale: 50*10 – 6 sec/div, Vertical
scale= 50*10mV/div)
When the reference voltage (Vref) is 5.2V, the system has unstable
periodic behavior. The 1-periodic orbit loses its stability via flip bifurcation
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and gives a 2-periodic waveform. The input voltage of the period -2 operation
is shown in Figure 6.19.
Figure 6.19 Experimental period-2 waveform when Vref=5.2V
(Horizontal scale: 50*10-6sec/div, Vertical scale=
50mV*10/div)
Varying Vref further, it is observed that the converter changes from
a stable operation to an unstable operation. The input voltage of Cuk
converter-based solar PV system has unstable aperiodic behavior. Chaotic
waveform is observed for the Vref = 4.8V as shown in Figure 6.20.
Figure 6.20 Experimental chaos waveform when Vref =4.8V (Horizontal
scale: 200*10-6sec/div, Vertical scale= 500mV/div)
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6.10 ANALYSIS OF INPUT VOLATGE OF PV-POWERED CUK
CONVERTER WITH PI CONTROLLER
The control of non-linear dynamics in the input voltage of the solar
PV module fed Cuk converter is implemented using PI controller. The
feedback resistance Rf =10k potentiometer, input resistance Ri=1k
potentiometer are selected for P (Proportional) controller and feedback
capacitor = 0.1 F, Ri =1k are selected for I (Integral) controller. The input
voltage of the Cuk converter is regulated and its ripple is experimentally
analyzed. The PI (Proportional plus integral) controller improves the transient
response of the input voltage. The time taken to reach the regulated input
voltage is 10ms. The fundamental period-1 waveform shown in Figure 6.21
has been found with Vref=5.68V.
Figure 6.21 Experimental stable Period-1 waveform when Vref =
5.68V using PI controller (Horizontal scale: 20*10-6 sec/div,
Vertical scale= 10mV*10/div)
But the converter reference voltage is decreased as Vref = 4.8V, and
chaotic unstable behavior is observed as shown in Figure 6.22.
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Figure 6.22 Experimental unstable chaotic waveform when Vref=4.8V
(Horizontal scale: 500*10-6sec/div, Vertical scale=
500mV/div)
The input gate pulse to switch (S) corresponds to an unstable
chaotic mode as shown in Figure 6.23.
Figure 6.23 Gate pulse to switch – during the chaos mode of operation
with PI controller (Horizontal scale: 100*10-6sec/div,
Vertical scale:5V/div)
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6.11 ANALYSIS OF INPUT VOLATGE OF PV POWERED CUK
CONVERTER WITH PID CONTROLLER
The control of non-linear dynamics is investigated experimentally
in the input voltage of Cuk converter-based solar PV system with PID
(Proportional plus Integral plus Derivative) controller which is shown in
Figure 6.24. The feedback resistance RP1=10k potentiometer, input
resistance RP2 =1k potentiometer are selected for P controller and feedback
capacitor Ci =0.1 F, input resistance Ri =1k potentiometer are selected for I
controller. Input series resistance Rc =22 , input capacitor Cd = 0.1 F,
feedback resistance RD=10k potentiometer are selected for D controller. The
input voltage is regulated and the non-linear dynamics are experimentally
analyzed for the supply disturbances using PID controller.
Figure 6.24 Diagram of the PID controller
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The Vref is varied from 4.00V to 5.76 V and it is observed that the
converter is always operated in period-1 stable region for all the parameter
variations which is given in Figure 6.25.
Figure 6.25 Experimental period-1 waveform for 3V< Vref < 5.76V
(Horizontal scale:20*10-6sec/div, Vertical scale:10mV*10/div)
The gate pulse corresponds to input voltage regulation of Cuk
converter with PID controller as shown in Figure 6.26.
Figure 6.26 Gate pulse to switch(S)-during the period -1 mode of
operation with PID controller (Horizontal scale: 20*10-
6sec/div, Vertical scale= 5V/div)
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6.12 CONCLUSION
The input voltage of the Cuk converter is regulated using a voltage
controller (PI compensator) in order to control the solar PV module output
voltage for the change in irradiation. The small signal modelling is analyzed
for input regulated Cuk converter-based solar PV system. The non-linear
dynamics such as chaos is investigated experimentally in Cuk converter-
based solar PV system. The PID controller is designed to regulate the input
voltage of Cuk converter and to operate the solar PV- powered Cuk converter
system is chaos-free with “period-1 operation” in which all the waveforms
repeat at the same rate as the driving clock for the parameter variations.
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