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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.

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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.