68

CHAPTER 4

HARMONIC ELIMINATION IN PWM AC CHOPPER

4.1 INTRODUCTION

Power quality is the ability of the electric utilities to provide

electric power without interruption. Power quality encompasses even minimal

deviation from a perfect sinusoidal waveform. It includes EMI and Radio

Frequency Interference (RFI) noise, transients, surges, sags, brown outs, black

outs and any other distortions to the sinusoidal waveform. One distortion to

the sinusoidal waveform which has dangerous consequences is harmonics.

Harmonic is a sinusoidal component of a periodic wave or quantity

having a frequency that is an integer multiple of the fundamental frequency.

An AC periodic voltage or current can be represented by a Fourier series of

pure sinusoidal waves which contain the basic or fundamental frequency and

its multiple called harmonics. Harmonics can be present in voltage, current or

both. The effects of these harmonics are the failure of electrical and electronic

components, overheating of neutral wires, transformer heating, and the failure

of power factor correction capacitors, losses in power generation and

transmission, interference with protection, control and communication

networks as well as customer loads.

It is forecasted that in future, half of all electrical devices will

operate with nonlinear current. These nonlinear loads are the cause of current

harmonics. The term harmonic currents are a common word in the power

69

quality industry and it needs to be realized that they can be a serious problem

if not treated properly.

The AC voltage regulator is used as one of the power electronic

systems to control output AC voltage for power ranges from a few watts (as in

light dimmers) up to fractions of megawatts (as in starting systems for large

induction motors). Phase angle control line commutated voltage controllers

and integral cycle control of thyristors have been traditionally used in these

types of regulators. Some techniques offer such advantages as simplicity and

the ability of controlling a large amount of power economically. The

controllers suffer from inherent disadvantages, such as retardation of the

ring angle, causing a lagging power factor at the input side, in particular, at

large ring angles and high low order harmonic contents in both load and

supply voltages/currents (Rashid 2005).

The recent developments in the eld of power electronics make it

possible to improve the electrical power system utility interface. Line

commutated AC controllers can be replaced by PWM AC chopper controllers,

which have better overall performance and the above problems can be

improved if these controllers are designed to operate in the chopping mode. In

this case, the input supply voltage is chopped into segments, and the output

voltage level is decided by controlling the duty cycle of the chopper switching

function. The advantages to be gained include nearly sinusoidal waveforms,

better input power factor, better transient response, elimination of the low

order harmonics and consequently, smaller input output lter parameters

(Deib et al 1993).

4.2 EVOLUTION OF AC CHOPPER

The growth in the field of power electronics has made it possible to

improve the performance of electrical system utilities through various means.

The solid state power switching devices are employed in source conditioning

70

by changing either its magnitude or frequency such as converters, inverters,

choppers, regulators and cycloconverter. The advantages of using these

devices are listed out by Addoweesh et al (2003). The advantages are

mentioned below:

Fast response

Compactness

Loss free control

Low power demands of control circuitry

Medium, large motor drives, Uninterruptible Power Supply and

high power inverters in flexible alternate current transmission systems need

switching elements which can bear high voltages and currents. To overcome

the limitations of semiconductor switches, several new techniques and

topologies have been developed, such as multiple switching elements in one

leg of an inverter, series connected inverters, parallel connected inverters,

multilevel reactive power compensators, multiple rectifiers for unity power

factor correction, optimization of motor performance indexes (such as

harmonic current, torque ripple, common mode voltage and bearing currents),

and neutral point clamped inverters. In the present research work the control

technique is based on the SHEPWM. The SHEPWM based methods can

theoretically provide the highest quality output among all the PWM methods.

4.2.1 Advantages of AC Chopper over AC Voltage Controller

The advantages of AC chopper over AC voltage controller are

listed below,

Improved load power factor due to high frequency switching

Control range is wide in terms of firing angles regardless of

load power factor

71

The low order harmonics are eliminated compared to the phase

angle control

The order of the dominant load voltage harmonics can be

controlled through changing chopper frequency.

4.2.2 Switching Pattern for PWM AC Chopper

Normal switching is resumed at the instant when the load current

reverses its direction, by such switching patterns; a continuous current path

always exists, regardless of the load current direction. Since only a single

switch is modulated and due to the fact that a single freewheeling switch is

turned on during the majority of the half period of the voltage source, the

switching losses are significantly reduced and consequently high efficiency

can be approached. The switching sequence for PWM AC chopper is

presented in Figure 4.1.

Figure 4.1 Switching sequence of PWM AC chopper

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4.2.3 Advantages of PWM

The advantages of practicing the PWM switching for PWM AC

choppers are listed below,

The output voltage control can be obtained without any

additional components.

With this type of control, lower order harmonics can be

eliminated .or minimized along with its output voltage control.

4.3 PWM AC CHOPPER

The PWM AC chopper has been widely used to drive the induction

motors. It is a very well known fact that the industrial processes for day to day

activities are carried out by induction motors. In a world where power

shortage is so acute it necessary to improve the performance of electrical

systems which drives the induction motors. This work considers the harmonic

elimination in the output parameters of AC chopper. The comparison is also

done between open loop and closed loop models of AC chopper and the

harmonics are eliminated with the application of evolutionary algorithms

namely DE.

Line frequency AC choppers are widely used in applications such

as industrial heating, lighting control, soft starting of induction motors, and

speed controllers for fans and pumps. Many of these are conventional phase

controlled AC controllers using thyristors, which have the advantages of

simplicity of the control circuit and large power capability. However, these

have the inherent drawbacks that power factor decreases when the firing angle

increases and control by switching is often accompanied by extra losses due

to time harmonics presented in output voltage waveforms, added to lower

values of system power factor.

73

The performance of the regulator can be improved if it is designed

to operate as a chopper (Choe et al 1989). In this case, input supply voltage is

chopped into segments and output voltage level is decided by the ratio

between ON/OFF. Chopper mode of operation can be realized by using two

ac switches, one connected in series and the other in parallel with the load as

shown in Figure 4.2. The operation of the AC voltage regulator in chopping

mode has the following advantages:

Phase controlled AC choppers are well known and are being widely

used to obtain variable AC voltage from a fixed AC voltage source. They are

used for the speed control of ac series motors, fan motors, industrial heating,

light dimming and for obtaining the regulated ac supplies. The output voltage

of the phase controlled ac chopper is load dependent and contains a good

amount of odd order harmonic components. Hence, adequate filtering may be

required at the output. If the output voltage is not filtered, an increased

heating will take place in the load. The presence of harmonics also introduces

distortion components in the supply current and reduces the input power

factor and efficiency of the system.

Sundareswaran et al (2004) proposed a method of voltage harmonic

elimination in a PWM AC to AC voltage converter using GA. The output

voltage of the AC chopper with k pulses per half cycle is written in terms of

switching angles using Fourier series, and the best switching angles are

identified with the dual objectives of harmonic elimination and output voltage

regulation.

The performance of the PWM AC chopper is observed under open

loop and closed loop conditions in the way of simulation through

MATLAB/Simulink and the development of hardware. The PWM AC

chopper is developed using two IGBT switches whose switching pulses are

generated by the DSP processor.

74

4.3.1 Configuration of PWM AC Chopper

In 1822, J.B.J. Fourier postulated that any continuous function

repetitive in an interval T can be represented by the summation of a DC

component, a fundamental sinusoidal component and a series of higher order

sinusoidal components (called harmonics) at frequencies which are integer

multiples of the fundamental frequency. Harmonic analysis is then the process

of calculating the magnitudes and phases of the fundamental and higher order

harmonics of the periodic waveform. The resulting series, known as the

Fourier series, establishes a relationship between a time domain function and

that function in the frequency domain.

Figure 4.2 PWM controlled AC chopper

Figure 4.3 Output voltage

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The circuit of a PWM AC chopper is shown in Figure 4.2 and the

output voltage in Figure 4.3. In this circuit, IGBT are used as switches and are

marked as S1 and S2. The switch S1 is connected to the load. The switch S2 is

included in the circuit to allow freewheeling of the load current when switch

S1 is turned OFF. The switch S1 is turned ON at various firing angles such as

1, 3, k-1, and are turned OFF at the switching angles 2, 4, 6,…. k. The

switching pulse with m number of pulses per half cycle is shown in the above

figure. They are symmetrical with respect to /2.

4.3.2 Analysis of PWM AC Chopper

The output voltage can be expressed using Fourier series as,

n

0 0 n n

1

V a (A cos n t B sin n t) (4.1)

Where,

n = 1, 2, 3,……….

The Fourier series basically consists of odd and even harmonics.

The even harmonics are absent and the coefficients An and a0 are reduced to

zero as the output voltage waveform is purely symmetrical. The above

equation is thus reduced to,

n

0 n

1

V B sin n t (4.2)

Where,

n=1, 3, 5,…………

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The value of Bn is computed as,

, ,.......2 42

mn

n 1, ,.......1 3 k

2V sin(n 1) t sin(n 1) tB

(n 1) (n 1) (4.3)

Where,

Vm is the maximum value of the input sine wave.

The fundamental component can be computed as,

, ,................2 4

m 21

, ,................1 3 k

2V sinn tB t-

2 (4.4)

The objective is to trace out the switching angles to make B1= V0*

and to perform selective harmonic elimination, where V0* is the reference

output voltage. For m number of switching pulses, let F ( ) be the objective

function then the optimization problem can be mathematically stated as,

F ( ) = F ( , ,................1 3 k

) = er + hc (4.5)

subject to the condition

0 1 2…….. k-1 k /2

Where,

er = |V0*-B1/ 2 | (4.6)

hc = |B3|+|B5|+|B7|………. + |Bk-1| (4.7)

77

Thus the DE algorithm has to be framed as MATLAB program for

the objective function mentioned in Equation 4.5. subject to the condition

mentioned in the Equation 4.6 and Equation 4.7.

4.4 HARDWARE MODULE

The hardware model of the AC chopper is developed using the

below mentioned modules. This section deals with components and

techniques which are used in hardware and the explanations of each

components is presented in this section.

Pulse Generator

IGBT

Diode

DSP Processor

Fast Fourier Transform

The MATLAB/Simulink software is used to verify the simulation

results based on the hardware model developed for PWM AC chopper.

4.4.1 Pulse Generator

Pulse Width Modulation, as it applies to motor control, is a way of

delivering energy through a succession of pulses rather than a continuously

varying (analog) signal. By increasing or decreasing pulse width, the energy

flow to the motor shaft can be regulated. The motor’s own inductance acts

like a filter, storing energy during the ON cycle while releasing it at a rate

corresponding to the input or reference signal. In other words, energy flows

into the load not so much the switching frequency, but at the reference

frequency.

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The energy of each push is stored in the inertia of the heavy

platform, which accelerates gradually with harder, more frequent, or longer

lasting pushes. The riders receive the kinetic energy in a very different

manner than how it’s applied.

4.4.2 IGBT

The Insulated Gate Bipolar Transistor is a minority carrier device

with high input impedance and large bipolar current carrying capability.

Many designers view IGBT as a device with MOSFET input characteristics

and bipolar output characteristic that is a voltage controlled bipolar device. To

make use of the advantages of both Power MOSFET and Bipolar Junction

Transistor (BJT), the IGBT has been introduced. It’s a functional integration

of Power MOSFET and BJT devices in monolithic form. It combines the best

attributes of both to achieve optimal device characteristics.

4.4.3 DSP Processor

Digital signal processing is carried out by mathematical operations.

In comparison, word processing and similar programs merely rearrange stored

data. This means that computers designed for business and other general

applications are not optimized for algorithms such as digital filtering and

Fourier analysis. DSP are microprocessors specifically designed to handle

digital signal processing tasks. These devices have seen tremendous growth in

the last decade, finding use in everything from cellular telephones to

advanced devices. The majority of the DSP can interface with external

converters through serial or parallel ports.

79

4.5 OPEN LOOP AND CLOSED LOOP PWM AC CHOPPER

The circuit of a PWM AC chopper is shown in Figure 4.2 and the

output voltage in Figure 4.2. The operation of the PWM AC chopper is

explained in section 4.5.1.

4.5.1 Open loop PWM AC Chopper

The performance of the PWM AC chopper is observed under open

loop condition in this section. The PWM AC chopper circuit is simulated

using MATLAB/ Simulink. The simulation results are presented.

4.5.1.1 Results of Simulation

The simulation circuit for open loop controlled PWM AC chopper

is shown in Figure 4.4. The input voltage waveform for the AC chopper is

shown in Figure 4.5. The power spectral density waveform is shown in the

Figure4.6.

Figure 4.4 Open loop simulation circuit of PWM AC chopper

80

Figure 4.5 Voltage waveform of open loop PWM AC chopper

Figure 4.6 Power spectral density waveform of open loop PWM AC

chopper

81

4.5.1.2 Experimental Results

The hardware model is developed for the induction motor drive fed

by PWM AC chopper. The top view of the hardware and the rear view of the

hardware are shown in Figures 4.7 and 4.8. The associated waveforms are

also presented in this section.

Figure 4.7 Top view of the hardware

Figure 4.8 Rear view of the hardware

82

The above figure shows the hardware developed for induction

motor drive. The hardware developed consists of the power circuit, switching

circuit along with the sufficient arrangements for the measurement of input

and output waveforms.

Figure 4.9 Output from cathode ray oscilloscope

The output waveform of induction motor drive is presented in the

Figure 4.9. It is in the form of pulsed sine wave. The harmonic content is

present in the waveform. The output waveforms measured using Mixed

Signal Oscilloscope (MSO) for open loop AC chopper is presented in Figure

4.10 and Figure 4.11.

83

Figure 4.10 Output current waveform measured using MSO for open

loop AC chopper

Figure 4.11 Output voltage waveform measured using MSO for open

loop AC chopper

84

4.5.2 Closed loop PWM AC Chopper

The performance of the PWM AC chopper is observed under

closed loop condition in this section. The PWM AC chopper circuit is

simulated using MATLAB/Simulink. The simulation results are presented.

4.5.2.1 Results of Simulation

The simulation circuit for closed loop controlled PWM AC chopper

is shown in Figure 4.12. The output voltage waveform for the AC chopper is

shown in Figure 4.13.

powergui

Continuous

Voltage

Measurement

v+

-

Speed

Single Phase

Asynchronous Machine

Tm

mM+

M

capacitor

-start-run

Scope 6

Scope

Repeating

Sequence

Relational

Operator

<=

Power Spectrum

-1

PI Controller

eu

Logical

OperatorNOT

IGBT 1

g

C

E

IGBT

g CE

Gain

-K-

Display

6537

Diode 7 Diode 6

Diode 5 Diode 4

Diode 3 Diode 2

Diode 1

Diode

Current Measurement 1

i+-

Constant 1

1

Constant 800

230 V

AC

Voltage

Source

<Rotor speed (rad/s or pu )>

Figure 4.12 Closed loop PWM AC chopper simulation circuit

85

Figure 4.13 Voltage waveform of closed loop PWM AC chopper

The induction motor is operated under closed loop and its

performance is observed. The output voltage waveform is measured under

closed loop and is presented in figure 4.13. It is inferred from the Figure 4.14

that the speed of the induction motor has settled down to a value as it is

operated under closed loop.

Figure 4.14 Speed waveform of closed loop PWM AC chopper

86

Figure 4.15 Power spectral density waveform of closed loop PWM AC

chopper

The power spectral density waveform is shown in the Figure 4.15.

4.5.2.2 Experimental Results

The PWM AC chopper produces pulsed sine wave signal when AC

supply is fed to it. This output signal generated is fed to the induction motor.

The output from load is analyzed using Fast Fourier Transform. The outline

of the work involved in hardware development for the PWM AC chopper is

presented in Figure 4.16.

87

Figure 4.16 Block diagram representation of the implementation of

hardware

Figure 4.17 Output waveform measured using MSO for closed loop

PWM AC chopper

88

The Figure 4.17 represents the output waveforms measured using

MSO for closed loop AC chopper. This section presents the overall view of

closed loop operation of the PWM AC chopper and also its performance

analysis.

4.5.3 Fast Fourier Transform

FFT is a mathematical algorithm that is used to indicate any

algorithm attempting to determine the power versus frequency graph for a

signal. FFT is also used for digital signal processing and to remove or

enhance patterns of periodic noise in an image or other related signals.

4.5.3.1 Implementation of FFT

Under some operating conditions, power system devices exhibit

nonlinear characteristics such as magnetic saturation, resulting in distorted

voltage and current waveforms that can interfere with other devices on the

power system. The first step in controlling the distortions is measuring them

accurately.

This seems like a relatively easy application for Fourier analysis.

For example, collect 256 samples that span one power cycle at the

fundamental frequency. A 256 point FFT would then deliver an analysis of

the first 128 harmonics, more than enough to cover the 40 or so needed for a

harmonic distortion analysis. The FFT analysis is carried out for PWM AC

chopper. The analysis reveals that the THD for closed loop operation of the

PWM AC chopper is around 3.15%. The FFT analysis of the closed loop

PWM AC chopper is presented in Figure 4.18 and Figure 4.19.

89

Figure 4.18 FFT analysis of closed loop output current in PWM AC

chopper

Figure 4.19 FFT calculation window of output current in closed loop

PWM AC chopper

The THD for closed loop PWM AC chopper is 3.72% which is

below the prescribed limits of harmonics.

90

4.6 COMPARSION OF HARMONIC CONTENT IN OPEN

LOOP AND CLOSED LOOP TECHNIQUE

The harmonic contents of open loop and closed loop AC chopper is

presented in the Table 4.1. The results present in the table prove that the

harmonic contents are comparably lower in closed loop operation of PWM

AC chopper.

Table 4.1 Comparison of harmonic content in open loop and closed loop

PWM AC chopper

Frequency

(Hz)

Amplitude (%)

Closed loop Open loop

50 Hz(fundamental) 100 100

100 Hz (h2) 2.95 0

150 Hz (h3) 10.66 50

200 Hz (h4) 2.84 29.33

250 Hz (h5) 5.76 31.89

300 Hz (h6) 1.73 24.78

350 Hz (h7) 4.3 25

400 Hz (h8) 0.28 2

450 Hz (h9) 2.9 3

500 Hz (h10) 0.76 1.8

550 Hz (h11) 2.06 2.7

600 Hz (h12) 1.03 1.5

This performance analysis for AC chopper can be improved further

by following techniques,

Genetic Algorithm

Differential Evolution Algorithm.

91

4.7 OPTIMIZING THE PERFORMANCE OF PWM AC

CHOPPER

As optimization methods evolve, new methods appear which are

based on other methods but may outperform them for specific problems. DE

is one of these relatively new techniques, which is an adapted version of a GA

incorporating vector differences for perturbing the vector population.

The DE algorithm is tested against the widely used GA for a

number of different interplanetary orbit optimization problems. For both

algorithms, different settings are tested and the settings leading to the best

results are compared. An attempt is made to find generic optimal parameter

settings of the DE algorithm. The general evolutionary algorithm procedure is

shown in Figure 4.20.

Figure 4.20 Block diagram for implementing evolutionary algorithm

The DE convergence, like the GA, may depend on the settings of

the DE parameters. Some settings are similar to the GA, like the as with the

GA, the DE Population Size, the crossover chooses a target vector to

reproduce with probability and the maximum number of another vector. As

with the GA, the DE chooses a target vector to reproduce with another vector.

However, in contrast with the GA, the DE chooses not one but two other

vectors in the population. From these two vectors, a new vector is created

which is a weighted difference between the two vectors. This new vector is

92

then added to a third randomly chosen vector. This is then the new vector to

‘mate’ with the first randomly chosen vector. After applying only a crossover

between these two vectors, the newly obtained vector is then compared to the

original (first that was randomly chosen) vector. The vector leading to the

highest objective is the one that passes on to the next generation. No mutation

is applied, which is another difference with the GA.

Evolutionary algorithms are optimization techniques that solve

problems using a simplified model of the evolution process. These algorithms

are based on the concept of a population of individuals that evolve and

improve their fitness through probabilistic operators like recombination and

mutation. These individuals are evaluated and those that perform better are

selected to compose the population in the next generation. After several

generations these individuals should improve their fitness as they explore the

solution space for the optimal value.

One extremely powerful algorithm from evolutionary computation

due to convergence characteristics and few control parameters is differential

evolution. Differential evolution is an optimization algorithm that solves real

valued problems using natural evolution using a population P of Np floating

point individuals that evolve over G generation s to reach an optimal solution.

Each individual or candidate solution is a vector that contains as many

parameters as the decision variables D. In differential evolution, the

population P remains constant throughout the optimization process. The steps

that are involved in differential evolution are being summarized here.

Step 1

Select the control variables, population size Np, scaling factor F,

crossover constant CR

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Here the control variables are the switching angles of the switch S1

in the circuit. There are m number of switching angles that are chosen which

will act as the control variables, population size Np is five, scaling factor can

range from 0 to 1, crossover constant also ranges from 0 to1. Based on all

these data the initial population matrix is generated based on the given

formula below. The matrix size depends on the control variables D and

population size Np.

(0) min max min

j,i j j j j PX X X X ,i 1,....N ; j 1,...,D (4.6)

Step 2

The initial population matrix (different values of optimum angles)

consists of number of vectors. Select the target vector from the initial

population matrix. The target vector that is chosen should be satisfying a

condition i.e., the target vector that is chose should be different from the other

vectors that constitute the initial population matrix.

Step3

Three other random indices should be chosen which are different

from the target vector. The random indices are also vectors form the initial

population matrix (which forms another set of switching angles).

Step 4

A mutant vector represented in Figure 4.21 is created based upon

the random indices that are chosen. The mutant vector is created based upon

the formula stated below.

'(G) (G) (G) (G)

i a b c PX X F X X , i 1,....N (4.7)

94

Figure 4.21 Mutation operator

Step 5

The target vector is being compared with the mutant vector that is

generated in the earlier step. The result is, the generation of trial vector

represented in Figure 4.22.

Figure 4.22 Crossover operator

95

The random number vector is generated and its value ranges in the

interval 0 and 1. If the target vector switching angle is greater than the

random number, the trial vector that is being generated will take into

consideration the value from the mutant vector or else it will be replaced by

the value from the target vector. Now new vectors of switching angles are

generated that are being named as trial vector.

'(G) '

j,i j R"(G)

j,i P(G)

j,i

X if C or j qX , i 1,..., N , j 1,...,D

X otherwise (4.8)

Step 6

Return to step 3 until the next generation population is filled using

a different target vector each time.

Step 7

Return to step 3 and repeat till convergence criteria is satisfied.

4.8 SIMULATION RESULTS FOR DIFFERENTIAL

EVOLUTION

4.8.1 Parameter Setting

The parameter setting for GA (Sundareswaran et al (2004)) is stated

below,

Population Size : 30

Coding : Binary

Number of generations : 500

Selection Scheme : Roulette wheel selection with elitism

Crossover operator : Multipoint crossover

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Crossover probability : 0.8

Mutation probability : 0.06

Termination criterion : 500

The parameter setting for DE is stated below,

Decision variables : 5

Population size : 75

Scaling constant : 0.9

Crossover constant : 0.8

Number of generations : 10000

The optimization task of reduction of harmonics in the output

voltage is realized with the help of coding written in C language in

MATLAB. The parameters of DE such as crossover constant, mutation

probability, population size and the number of generations are selected based

on the values by trial and error process, which is discussed in the following

section.

The parameters of DE such as population size, scaling constant and

cross over constant are selected by trial and error, such that they may yield the

best optimal solution. The decision variables are selected depending on the

problem under consideration.

In the Table 4.2 the crossover constant is varied from 0.7 to 0.9

with an interval of 0.05. It is found that the best value for the objective

function was obtained for a crossover constant of 0.8. These results are

obtained for a population of 100, number of generations 100, and a scaling

constant of 0.9.

97

Table 4.2 Comparison of number of generations and F ( )

Cross over

constant

Population

size

Number of

generations

Scaling

constant

Best F( )

value

0.7 100 100 0.9 0.0272

0.75 100 100 0.9 0.0510

0.8 100 100 0.9 0.0206

0.85 100 100 0.9 0.0319

0.9 100 100 0.9 0.0253

From the Table 4.2, it is found that the best optimal solution could

be reached for a cross over constant value of 0.8. In the Table 4.3. the

population size is varied between 50 and 150 with an interval of 25. The best

optimal solution is achieved for a population size of 75.

Table 4.3 Comparison of population size and F ( )

Cross over

constant

Population

size

Number of

generations

Scaling

constant

Best F( )

value

0.8 50 100 0.9 0.0471

0.8 75 100 0.9 0.0399

0.8 100 100 0.9 0.1223

0.8 125 100 0.9 0.0949

0.8 150 100 0.9 0.0651

Figure 4.23 Number of generations versus F ( )

98

In the Figure 4.23 the graph is plotted between number of

generations taken along the x axis and F ( ) along the y axis. It is found that

the F ( ) value significantly drops as the number of generations increase. It

has a faster convergence rate when compared to the earlier methods of

evolutionary computation such as GA for the task of extenuating the voltage

harmonics.

Figure 4.24 Population Size versus F ( )

In the Figure 4.24, the graph is plotted between population size and

F ( ). It is inferred from the graph that as the population size increases, the

value of F ( ) gradually decreases after a population size of 100. Also it has a

marginal increase for the population size in between 75 and 100. Therefore

the population size is fixed to be 75, as it has achieved the lowest F ( ) value.

Figure 4.25 Crossover constant versus F ( )

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In the Figure 4.25, the graph is plotted between crossover constant

and F ( ). It is inferred from the graph that the F ( ) value drops down for

crossover constant value between 0.75 and 0.8 and after 0.85.

The results obtained using DE reiterates extenuation of voltage

harmonics for AC chopper through quicker convergence rate and lesser

computational burden because of lesser computational parameters. The

optimization task is for AC chopper but it can also be applied to the rest of the

power converters.

4.9 CONCLUSION

The PWM AC chopper is considered in this work. The primary aim

of the PWM AC chopper is to eliminate the lower order harmonics by

generating appropriate switching pulses for the operation of switches. The

switching pulses are generated with the help of evolutionary algorithm

namely DE. The same task was earlier performed using GA, Rosenbrocks

rotating coordinates method, Newton Raphson method. The DE algorithm has

given the best results compared to the other methods of generating the

switching pulses. The results obtained from the DE algorithm have attained

the objective in less than 100 generation which is clearly evident though the

results obtained. The DE has performed better over the GA algorithm.

The hardware model was developed for open loop and closed loop

PWM AC chopper. The switching signals were generated for the switches

using DSP. The hardware waveforms were measured using MSO and the

waveforms are presented. The performance of the PWM AC chopper has

improved in its closed mode of operation which is evident with the THD

reducing to 3.72%. The objective of reducing the total harmonic distortion

under permissible limit is achieved in the closed loop operation of PWM AC

chopper.