active power filter

DESIGN AND SIMULATION OF THREE PHASE SHUNT ACTIVE POWER FILTER

USING THE p-q THEORY

BE (EE) PROJECT REPORT

Prepared By:

Muhammad Jawwad Iqbal EE-09-129

Muhammad Waleed Khan EE-09-094

Muhammad Danish Shaikh EE-09-101

Hafiz Muhammad Furqan EE-09-122 Farhan Rajput EE-09-074

Project Advisor:

Muhammad Ali Baig (Internal Advisor)

Dr. Abdul Qadir (External Advisor)

N.E.D. University of Engineering & Technology, Karachi-75270

i

ABSTRACT

Harmonics created by nonlinear loads such as arc furnaces, cycloconverters and

motor drives destroys the power quality in the system. They not only affect the

working of adjacent loads but also shorten the life of power equipment by creating

excessive losses. ‘Shunt Active Power Filter’ is a modern addition to family of

compensating devices. It has superior qualities over its contemporaries namely

SVCs and STATCOMs. It not only mitigates harmonics within the allowable limits

defined by IEEE Std 519-1992, but also compensates unbalancing and reactive

power in the system. Consequently, only active power is supplied by the source

thus power factor approaches unity. A fully functional Simulink model of Shunt

Active Power Filter has been designed based on ‘Instantaneous Power Theory’ or

‘p-q Theory’. The results of simulation comply with all the features described by

the theory, justifying employment of SAPF in the industry.

ii

ACKNOWLEDGEMENTS

We thank Almighty Allah who gave us the opportunity to complete this project and

to explore more knowledge in power systems. We learned so many new things and

concepts due to its research based nature.

Secondly, we are deeply indebted to our beloved parents for their unconditional

support and prayers throughout our lives.

And then, we offer our profound gratitude to our Internal Advisor Mr. Mirza

Muhammad Ali Baig for being calm and supportive through the whole course of

project, and for tolerating us. We are also magnanimous to our External Advisor

Dr. Abdul Qadir for his vision and guidance. Also we are grateful to our FYP

committee members namely, Sir Umer Sajid and Sir Muhammad Omar.

Lastly, we want to thank Miss Samiya Zafar for listening to our ideas and for her

altruistic suggestions. And Sir Fezan Rafique, for appreciating our project and hard

work.

CONTENTS

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PREFACE v

1. INTRODUCTION 1 1.1 Introduction 1 1.2 Types of Load 2 1.2.1 Linear Load 2 1.2.2 Non-Linear Load 2 1.3 Power Quality 3 1.4 Harmonic Injection Limit 3 1.5 Problems caused by Harmonics 4 1.5.1 Effect on Power System itself 4 1.5.2 Effect on Consumer itself 4 1.5.3 Effect on Communication System 4 1.5.4 Effect on Revenue Billing 5

2. BACKGROUND 6 2.1 Harmonic Mitigation Techniques 6 2.2 Passive Harmonic Filters 6 2.2.1 Promising features of Passive Harmonic Filters 7 2.2.2 Non-Promising features of Passive Harmonic Filters 8 2.3 Active Harmonic Filters (AHF) 8 2.3.1 Operation of AHF 8 2.3.2 Advantages of AHF 9

3. THE INSTANTANEOUS POWER THEORY 10 3.1 Historical Background of Power Theory 10 3.2 Basis of the p-q Theory 10 3.3 Background of p-q Theory 11 3.4 The Clarke Transformation 11 3.5 The p-q Theory 13 3.6 Use of p-q Theory in Shunt Active Filters 15 3.7 Symmetrical Components 16

4. SHUNT ACTIVE POWER FILTER 18 4.1 Shunt Active Filters 18 4.2 General description of Shunt Active Filters 18 4.3 The Active Filter Controller 19

4.4 Optimal Power Flow 20 4.5 Three Phase Three Wire Shunt Active Filter 20 4.6 The Simulink Model 22 4.6.1 3-Phase Source 22 4.6.2 Line Impedance (Zt) 22 4.6.3 V-I Measurement 22 4.6.4 Circuit Breaker 23 4.6.5 POWERGUI 23 4.6.6 Non-Linear Load 23 4.7 SHUNT APF 24 4.7.1 PQ and I-Compensation calculation 24 4.7.1.1 Clarke V 24 4.7.1.2 Clarke I 25 4.7.1.3 PQ Calculation 25 4.7.1.4 Low Pass Filter 25 4.7.1.5 Alpha-Beta Current 26 4.7.1.6 Compensation Currents 26 4.7.2 Hysteresis Controller 26 4.7.3 Universal Bridge 26 4.7.4 Capacitor 26 4.7.5 PI Controller 26 4.7.6 Coupling Inductor 26

5. SIMULATION RESULTS 5.1 Case I: Compensation of Non-Linear Load 27 5.2 Case II: Compensation of Non-Linear plus Unbalance Resistive Load 29 5.3 Case III: Compensation of Unbalance Resistive Load 30 5.4 Case IV: Compensation of Unbalance Inductive Load 32 5.5 Case V: Compensation of Unique Unbalance Resistive Load 33 5.6 Conclusion 34

A. SYMMETRICAL COMPONENTS 35

A.1 Simulink Model Of Sequence Power 36

B. HYSTERESIS CONTROLLER 39 B.1 Simulink Model Example 40 B.2 Hysteresis Band in Digital Hysteresis Current Controller 41

BIBLIOGRAPHY

v

PREFACE

___________________________________

The concept of harmonic compensation is the underlying idea of our project. The compensation

is not done by signal comparison, as it is done in active filtering, but on the basis of ‘power

selection’ in harmonic components and unbalanced component of fundamental current

component. The inspiration behind this idea comes from our curiosity of power properties under

semiconductor load (nonlinear load). We went through H. Akagi’s book “Instantaneous Power

Theory and Applications to Power Conditioning”. It helped us understanding ‘the p-q Theory’

proposed by H. Akagi et al, originally in 1982. Proponent of p-q’s rival theory; Lesczek S.

Czarnecki pointed out many limitations and flaws in the theory, and put forward his own

‘Current Physical Component Theory’. Czarnecki made a comprehensive discourse on ‘Voltages

and Currents in Non Sinusooidal Conditions’, his research publication, by comparing power

definitions of various power theories.

This report is divided into five chapters. In chapter 1, we have discussed basic power definitions

and idea of harmonics in power systems. Chapter 2 discusses the aged solutions of harmonic

problem; ‘Passive Harmonic Filters’ are discussed in detail. Chapter 3 puts forward H. Akagi and

A. Nabae’s “Instantaneous Power Theory”, and tries to explain its various power terms. Chapter

4 is titled as per our project name, ‘Shunt Active Power Filter’, describes working algorithm

based on the p-q Theory, and our Simulink model, its building blocks and specifications in detail.

Chapter 5 is dedicated to simulation results, five cases of nonlinear and unbalanced load were

considered to demonstrate the feats of Active Power Filters. Appendices on Symmetrical

Components and Hysteresis Controller are also attached to assist the reader.

Finally, we want to acknowledge that we are immensely grateful to above mentioned scholars

and our teachers, without whom it would not be possible.

JAWAD, WALEED, DANISH,

FURQAN AND FARHAN

1

Chapter 1

INTRODUCTION

___________________________________

1.1 INTRODUCTION

Electric power is the rate of electrical energy flow at any point. In other words it can also be

defined as the product of the instantaneous voltage and current. If i(t) and v(t) represent the

instantaneous sinusoidal current and voltage respectively then the instantaneous power P(t) can

be given as;

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( )

( )

( )

( )

( )

( )

( ) ( )

If we consider θv as reference i.e. θv = 0o and θi = ø then the above equation will become:

( )

( )

( )

( ) ( ) ( )

Where pact(t) is the real or active power absorbed by the electrical components and preactive(t) is

the power oscillating in the power system due to the charging components like inductance and

capacitance. This reactive power is in the form of increasing current which causes higher line

losses, voltage drops, etc.

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Figure 1.1 - Conventional concept of active and reactive power

But according to modern definitions pact(t) is not the total useful power and preactive(t) is also

cause by power electronic components or non-linear loads (Further discussion of 3-phase power

and various terms involved in it are explained in Chapter No. 3). Moreover, non-linear loads like

switching circuits, speed control circuits, welding plants, thermal loads, variable frequency

drives; motor controls, power invertors, cyclo convertors, high frequency induction furnaces, etc

tends to produce harmonics distortion in power lines. These high frequency harmonic

components cause greater magnetization and heating losses in the cores of electrical machines

and transmission lines.

The effects of harmonic distortion are hard to measure but the end results are easy to understand,

higher operating costs and lower reliability.

1.2 TYPES OF LOAD

Loads can be characterized into many types according to their nature, function etc. The type of

load we are interested in are

1. Linear load

2. Non-linear load

1.2.1 LINEAR LOAD

Electrical loads whose current wave has a linear relation with the voltage wave are termed as

linear loads. These loads do not cause any harmonic in the electrical system .

1.2.2 NON-LINEAR LOAD

The nonlinear loads are referred to as the loads that distort the current waveform shape due to

the switching action and the current and voltage waveforms are not identical in shape, e.g.

fluorescent lamp, PC and TV etc. Figure 1.2 shows how harmonics injected by non-linear

loads distort the current waveform.

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Figure 1.2 – Distortion in current waveforms due to harmonics

1.3 POWER QUALITY

Power quality is an issue of great concern for electrical power industry (utility and consumer

both) as the power quality determines the fitness of electrical power to consumer devices and the

efficiency of consumer devices. There are many ways in which electric power can be of poor

quality. The presence of harmonic content in AC power is an important factor that describes

quality of electrical power. Power quality may be defined as;

“The deviation of voltage from the ideal, continuous single-frequency sine wave with a rated

constant frequency and amplitude. Also current deviation from the ideal, single-frequency sine

wave with a constant frequency and amplitude that is in phase with the supply voltage.”

1.4 HARMONIC INJECTION LIMIT

The increasing use of power electronic devices has result in greater control over electric power

and larger production but power systems are often subjected to harmonic distortion due to

increasing applications of these nonlinear loads. Non-linear loads inject harmonics into the

power system which travel through the system and create problems even for those consumers

who do not actually use non-linear loads, by making the supplied voltage non-linear at PCC

(Point of Common Coupling).

Therefore, there is a limit for every individual customer for harmonic injection depending upon

full load current and short-circuit capacity. Table 10.3 of IEEE Std. 519 prescribes limits for

TDD (Total Demand Distortion) for individual customers.

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Table 1.1 – Harmonic Limits

1.5 PROBLEMS CAUSED BY HARMONICS

Following are the problems that are caused by the presence of harmonics in power system.

1.5.1 EFFECT ON POWER SYSTEM ITSELF

The major effect of power system harmonics is to increase the current in the system. This is

particularly the case for the third harmonic, which causes a sharp increase in the zero

sequence current, and therefore increases the current in the neutral conductor.

1.5.2 EFFECT ON CONSUMER ITSELF

Non-linear loads also causes harmonics/distortions in utility supplied voltages due to which

even the linear loads draw non -linear current. Harmonics can also cause thyristor firing errors

in converter. The performance of consumer equipment, such as motor drives and computer

power supplies, can be adversely affected by harmonics.

1.5.3 EFFECT ON COMMUNICATION SYSTEM

Harmonic currents flowing on the utility distribution system or within an end-user facility

can create interference in communication circuits sharing a common path. Voltages included

in parallel conductors by the common harmonic currents often fall within the bandwidth of

neutral voice communications. Harmonic currents on the power system are coupled into

communication system by either induction or direct conduction.

Table 10.3: Current Distortion Limits for General Distribution System (120V through 69,000V)

Maximum Harmonic Current Distortion in Percentage of IL

Individual Harmonic Order (Odd harmonics)

ISC/IL h<11 11 ≤ h <17 17≤ h <23 23≤ h <35 35< h TDD

<20*

4.0 2.0 1.5 0.6 0.3 5.0

20<50 7.0 3.5 2.5 1.0 0.5 8.0

50<100 10.0 4.5 4.0 1.5 0.7 12.0

100<1000 12.0 5.5 5.0 2.0 1.0 15.0

>1000 15.0 7.0 6.0 2.5 1.4 20.1

Even harmonics are limited to 25% of the odd harmonic limits above

Current distortions are result in a DC offset (e.g. half wave converters) are not allowed *All power generation equipments are limited to these values of current distortion regardless of

actual ISC/IL

Where,

ISC = maximum short circuit current at PCC

IL = maximum Demand load current (fundamental frequency component) at PCC

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1.5.4 EFFECT OF REVENUE BILLING

Electrical utility companies usually measure energy consumption in two quantities energy

consumed and the maximum power used for given period. Both energy and demand are

measured using the so-called watt -hour and demand meters.

Harmonic currents from non -linear loads can impact the accuracy of watt-hour and demand

meter adversely. Traditional watt -hour meters are based on the induction motor principle.

Conventional magnetic disk watt -hour meters tend to have a negative error at harmonic

frequencies. That is, they register low for power at harmonic frequencies if they are properly

calibrated for fundamental frequency. This error increases with increasing frequency.

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Chapter 2

BACKGROUND

__________________________________

2.1 HARMONIC MITIGATION TECHNIQUES

The growing concern of good power quality and awareness with the harms to power system due

to harmonics along with the penalties imposed by utility companies and the standards to the limit

of THD made by IEEE and other organizations are the driving factors for invention and adoption

of various methods, devices and equipment for harmonic mitigation. These equipment include:

1. Line reactors

2. Isolation transformers

3. K-Factor transformers

4. Phase shifting transformer

5. Harmonic filters

The most effective and useful of the list are current harmonic filters. There are two types of

harmonic filters: 1) passive harmonic filter and 2) active harmonic filters.

2.2 PASSIVE HARMONIC FILTERS

Conventional solutions to the harmonic distortion problems have existed for a long time. The

passive filtering is the simplest conventional solution to mitigate the harmonic distortion. A

passive harmonic filter is built using an array of capacitors, inductors and/or resistors. They

restrict the harmonic currents to flow into power system and divert them by providing a low

resistance path. They remove distortion due to harmonic currents and hence remove distortions

in voltages caused by non-linear voltage drop across the line impedance. They are connected as

shunt branch or parallel with load. Passive filters are designed to be capacitive at fundamental

frequency in order to correct displacement power factor and provide reactive Volt-Amperes.

Passive filters provide a low impedance path compared to the system impedance and hence the

harmonic current flows proportional to the impedance of parallel paths, therefore, performance

of passive harmonic filter largely depends upon the system impedance and its topologies and this

factor is not accurately known and is subjected to changes very frequently. Usually a separate

tuned harmonic filter is required to mitigate a single harmonic.

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Figure 2.1 – Typical Passive Harmonic Filters

Single tuned, double tuned, high pass and c-type are four types of passive harmonic filter but

most common of them is single tuned filters. Tuned filter is the most common type of passive

filters. It is a series combination of inductor and capacitor which resonates at tuning frequency,

therefore, its impedance is very low for tuned harmonic. Because of low impedance at tuned

frequency the filter now becomes the source of harmonic current rather than the utility. Another

popular type of passive filter is the high-pass filter (HPF). A HPF will allow a large percentage

of all harmonics above its corner frequency to pass through.

The following figure shows impedance vs. frequency curve of a tuned passive harmonic filter.

The response is highly capacitive (with high impedance) at fundamental frequency, therefore

provides reactive power at fundamental frequency as well. At tuned frequency the response is

resistive while above tuned frequency response is inductive.

Figure 2.2 – Frequency vs. Impedance curve of Single Tuned Filter

2.2.1 PROMISING FEATURES OF PASSIVE HARMONIC FILTERS

Passive filters are the most common and economical solution to current harmonic distortion

problems because of their simple design and low initial cost. The most promising features of

passive harmonic filters are:

1. They are simple to design

2. Low initial cost (compared to Active Harmonic Filter)

Single-Tuned High-Pass

Double-Tuned C-Type

Capacitive

Response Inductive

Response

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3. Shunt passive filters have extra advantage of providing reactive power. Hence they

correct distortion power factor and displacement power factor as well.

4. Brings down the THD in line current to the allowable limits.

2.2.2 NON-PROMISING FEATURE OF PASSIVE HARMONIC FILTERS

Some non-trivial issues with passive current harmonic filters are:

1. Filtering characteristics depend upon the source impedance (i.e. impedance of system and

its topology) which may not be well defined or may be subjected to variations.

2. Resonance problem may occur with loads and network

3. Usually a series of passive harmonic filters are required to compensate various

harmonics

4. Works good when power factor is not very low and the magnitude of specified harmonic

remains constant

5. The response is not dynamic i.e. if the requirement of certain harmonic increases or with

the change of load or if certain other harmonics are required the filters have to be

redesigned

6. Doesn’t solve the problem of load unbalancing or neutral shifting

2.3 ACTIVE HARMONIC FILTER (AHF)

Static response of passive harmonic filter and other problems have led to a power electronic

solution of harmonic distortion i.e. Active Harmonic Filters (AHF); a modern solution to old

harmonic current problems. Nowadays, passive filters are used to cancel the switching frequency

of active filters and high frequencies. Tuned filters are used besides the active filters to cancel

specific frequencies and decrease the power of active filters. Active filters have been designed,

improved, and commercialized in the past three decades. They are applicable to compensate

current-based distortions such as current harmonics, reactive power, and neutral current. They

are also used for voltage-based distortions such as voltage harmonics, voltage flickers, voltage

sags and swells, and voltage imbalances and load unbalancing and neutral shifting. Moreover,

unlike passive filters, they do not cause harmful resonances with the power distribution systems.

Consequently, the AHFs performances are independent of the power distribution system

properties.

2.3.1 OPERATION OF AHF

The main aim of the APF is to compensate for the harmonics and reactive power dynamically.

The APF overcomes the drawbacks of passive filters by using the switching mode power

converter to perform the harmonic current elimination. AHF continuously monitors the load

current, filter out the harmonic content and generates compensation current signals using any of

the algorithms like P-Q Theory, dq transform, sliding mode control, unity power factor method,

algorithm based on DSP, etc. compensation current signals are fed to hysteresis controller or

PWM converter as reference signals to generate gating signals for fast switching IGBT inverter.

The inverter generates harmonic currents required by the load through charging and discharging

of capacitor. These currents are injected into the system near the load through an interfacing

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inductor or a coupling transformer. The performance of AHF is independent of system

impedance as it compares the injected currents with reference signals and tries to minimize the

error.

There are three topologies of AHF: i) Series AHF, ii) Shunt AHF and iii) Hybrid AHF. We have

selected Shunt AHF for this project which is ideal for current harmonic compensation. A

generalize block diagram of SAHF is given below;

Figure 2.3 – Generalized Block Diagram of Active Filter

2.3.2 ADVANTAGES OF ACTIVE HARMONIC FILTER

1. Widely compensated harmonic spectrum

2. Only one filter needed to eliminate all the unwanted harmonics

3. Improved stability of the power system due to the lack of parallel resonance

4. Its performance is dynamic and take into account the changes in load

5. It may also be programmed to eliminate specific number of harmonics

6. They may also be programmed to eliminate harmonics with or without compensation of

reactive power

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Chapter 3

THE INSTANTANEOUS POWER THEORY

___________________________________

3.1 HISTORICAL BACKGROUND OF POWER THEORY

As to our best knowledge, the term power theory has occurred in literature for the first time in

Fryze’s paper in 1931. It is now used as a classifier of various power concepts developed by

scientists who studied power properties of electrical circuits. In such a meaning it is used in

phrases such as, Budeanu’s power theory, Fryze’s power theory, Shepherd and Zakikhani’s

(S&Z) power theory, Kusters and Moore’s (K&M) power theory, Depenbrock’s power theory

also known as FDB method, Nabae and Akagi’s Instantaneous Reactive Power Theory,

Czarnecki’s CPC power theory, Tenti and Tedeschi (T&T) conservative power theory and so on.

When a phrase like Budeanu’s power theory is used, we can have an idea, of what the phrase

refers to, and can find its details in literature. Used in such a way, the term theory can be

regarded as a system of terms defined in this theory, relations between these terms, and their

interpretations. The term power theory is also used sometimes in a different meaning, namely, as

a kind of a database of what is known on power properties of electrical circuits, thus as a

collection of true statements on power related phenomena, and mathematical expressions related

to these proper, ties and interpretations. Individual statements belong to this database, meaning,

to power theory, as long as it is not proven that these statements do not hold truth. Regarded as

such a database, the power theory is not a system of terms and relationships between them, i.e., it

is not a theory in the previous meaning. At the same time, however, anyone who reveals even a

single, but a new power related property, contributes to the development of the power theory

regarded as a database.

3.2 BASIS OF THE p-q THEORY

The p-q Theory is primarily based on a set of instantaneous calculation of powers time domain.

Voltage and current are sampled instantaneously which means there is no restriction on the shape

of their waveforms, and it can be applied to three phase with or without a neutral wire. It is based

in time domain rather than frequency domain. Thus, it is valid in the steady state and also in the

transient state. This shows the theory is very flexible and efficient in designing controllers for

active filters and power conditioners based on power electronics devices.

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Traditionally a three-phase system is considered as superposition of three single-phase circuits.

The p-q Theory first transforms and map linearly the voltages and currents from the abc

coordinates to αβ0 axes, and then defines instantaneous powers in these coordinates. Hence, this

theory always considers the three-phase system as single unit, not a superposition or sum of three

single-phase circuits.

3.3 BACKGROUND OF THE p-q THEORY

Originally the p-q Theory was born in Japan and its first version was published in the Japanese in

July 1982, in a regional conference and then in IEE Japan. After sometime, it was republished in

1983 in an international conference, and, in the year of 1984 finally it was published in IEEE

Transactions, and came into eyes of international audience. The development of the original p-q

theory was primariy base on the works and visions of previous power electronics specialists.

Many were working in the domain of compensation of nonactive power generated by non linear

loads. In the 1960’s scientists started to doubt the power definitions of Constantin Budeanu, that

its fails to define electrical power in presence of non linear elements. And in the beginning of

1970’s some papers were published which can be considered as a underlying idea of nonactive

power compensation. Many authors said, like “ compensation of distortive power is unknown to

date”, distortive was defined by Budeanu in 1927 . They also presented that “a nonlinear resistor

behaves like a reactive-power generator while having no energy-storing elements,” and showed

the very first practical and complete way to true-power-factor correction. Fukao et al said, “by

connecting a reactive-power source in parallel with the load and by controlling it in such a way

as to supply reactive power to the load, the power network will only supply active power.

Therefore, an ideal power transmission would be possible.”

Gyugyi and Pelly in proposed the point that nonactive power can be compensated by a naturally-

commutated cyclo-converter without any use of energy storage elements or conventional

methods of reactive power compensation. presence of nonactive power without any energy

storage elements was also a subjective and highly debated topic. All of the points presented

before were only the ideas of different scholars, and no tangible proof were given. In 1976,

Harashima et al used perhaps for the first time, the term “instantaneous reactive power”, but only

for single phase context. After sometime, Gyugyi and Strycula used the term “active ac power

filters”. And In 1981, Takahashi et al, put forward what can be said as basis of the p-q Theory.

The equations they gave are in fact considered a subset of the p-q Theory. But yet no physical

meaning of the terms introduced in their publications, was given. The p-q Theory will be

discussed in detail in the coming topics.

The p-q Theory uses the Clarke transformation, also known as the transformation, which

consists of a real matrix that linearly maps three phase voltages and currents into the frame,

which is a stationary reference frame.

3.4 THE CLARKES TRANSFORMATION

The transformation or the Clarke transformation maps the three-phase instantaneous

voltages in the abc phases, va, vb, and vc, into the instantaneous voltages on the αβ0-axes vα, vβ,

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and v0. The Clarke Transformation and its inverse transformation of three-phase generic voltages

are given by,

(3.1) and

(3.2) One advantage of applying the αβ0 transformation is to separate zero-sequence components from

the abc-phase components. The α and β axes contains only positive and negative sequence

components. Since there is no neutral in a three- phase, three-wire system and therefore on zero

sequence currents, so i0 can be eliminated from the above equations, thus resulting in matrix

simplification. If the three-phase voltages are exactly balanced in a four wire system, so no zero-

sequence voltage is present, therefore v0 can be eliminated. However, when zero-sequence

voltage and current components are present, the complete transformation should be considered.

If v0 is eliminated from the transformation matrices, the Clarke transformation and inverse

Clarke transformation becomes,

(3.3)

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And inverse Clarke transformation,

Similarly, we can transform line currents using above matrices. The matrices in (3.3) and (3.4) is

nothing but an axes-transformation from abc coordinates to αβ0 frame, as illustrated Fig. 3-1.

αβ0 frame is a stationary frame, it should not be confused with the dq0-tranformation or rotation

phase vectors on phasor domain. Note that these all value are instantaneous value of real life

sample taken from line voltage and load current. The a, b and c axes are spatially shifted by 120o

from each other, while the and axis are 90 degrees apart fom each other, and the axis is

parallel to the a axis. This graphical representation explains the above mentioned points,

Figure 3-1 - Graphical representations. (a) The abc to αβ0 transformation (Clarke transformation).

(b) Inverse αβ0 to abc transformation (inverse Clarke transformation).

3.5 THE p-q THEORY

The p-q Theory is defined in three-phase three wire systems or three phase four wire systems

with a neutral conductor. Three instantaneous powers are defined namely,- the instantaneous

(3.4)

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zero-sequence power p0, the instantaneous real power p, and the instantaneous imaginary power

q, are defined from the instantaneous phase voltages and line currents on the αβ0 axes as,

(3.5) If there are no zero-sequence current components in three-phase, three-wire systems, that is, i0 =

0. In this case, only the instantaneous powers defined on the axes exist, because the product v0 i0

is always zero. Similarly, if ther is no zero-sequence voltage components in three phase four wire

systems, that is v0 = 0.Hence, in three-phase, three-wire systems, the instantaneous real power p

represents the total energy flow per unit time in terms of components. The power components p

and q are related to the same α-β voltages and currents, and can be written together:

(3.6) Both of these powers have constant values and a superposition of oscillating components.

Therefore, it is interesting to separate p and q into two parts:

(3.7)

Similarly, if we have three phase four wire system zero sequence power can be expresses as the

sum of an average and oscillating components.

(3.8)

These quantities are illustrated in the following figure for an electrical system represented in a-b-

c coordinates and have the following physical meaning:

Figure 3.2 - Physical meaning of the various terms defined in the α-β-0 reference frame.

15

= mean value of the instantaneous zero-sequence power in αβ0 frame corresponds to the

energy per time unit which is transferred from the power supply to the load through the zero-

sequence components of voltage and current.

O = oscillating value of the instantaneous zero-sequence power in αβ0 frame it means the

energy per time unit that is exchanged between the power supply and the load through the zero-

sequence components. The zero-sequence power only exists in three-phase systems with neutral

wire. Furthermore, the systems must have unbalanced voltages and currents and/or 3rd

harmonics in both voltage and current of at least one phase.

= mean value of the instantaneous real power in αβ0 frame corresponds to the energy per time

unit which is transferred from the power supply to the load, through the a-b-c coordinates, in a

balanced way (it is the desired power component).

= oscillating value of the instantaneous real power in αβ0 frame It is the energy per time

unit that is exchanged between the power supply and the load, through the a-b-c coordinates.

= instantaneous imaginary power in αβ0 frame corresponds to the power that is exchanged

between the phases of the load. This component does not imply any transference or exchange of

energy between the power supply and the load, but is responsible for the existence of undesirable

currents, which circulate between the system phases. In the case of a balanced sinusoidal voltage

supply and a balanced load, with or without harmonics, its value is equal to the conventional

reactive power.

= oscillating value of the instantaneous imaginary power in αβ0 frame.

3.6 USE OF THE p-q THEORY IN SHUNT ACTIVE FILTER

The original concept of active filtering was introduced by Strycula and Gyugyi in 1976. Now a

shunt active filter can be implemented practically, and many shunt active filters are working all

over the world. Their controllers determine in real time the compensating current reference, and

source a power converter to synthesize the compensating current reference with high fidelity.

Figure 3.3 illustrates the basic idea behind the shunt current compensation. It shows a source

supplying power to a nonlinear load that is being compensated by a shunt active filter. Shunt

active filter is in actual is a shunt compensator. We assumed that the shunt active filter behaves

as a three phase controlled current source that can generate harmonics in phase opposition

depending upon current references i*ca, i*cb, and i*cc.

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The calculated real power p of the load can be separated into its average (p) and oscillating (~p)

parts. Likewise, the load imaginary power q can be separated into its average (q) and oscillating

(~q) parts. Then, undesired portions of the real and imaginary powers of the load that should be

compensated are selected.

The reason for incorporating a minus sign in the compensating powers is to emphasize that the

compensator must inject harmonics in perfect phase opposition. Remember that convention of

current direction is selected so in figure 3.3, that source current is the sum of load current and

filter current. Inverse Clarke transformation from αβ0 to abc-coordinates is applied then to

calculate the compensating current references i*ca, i*cb, and i*cc, instantaneously.

Ideally compensating current can be calculated by subtracting line current from a pure sine wave

of the same peak value as the fundamental component of current drawn by non linear load.

3.7 SYMMETRICAL COMPONENTS

Symmetrical components are useful tool in power system analysis. They are discussed in detail

in Appendix A. The following conclusions can be written from the equations for the real and

imaginary powers of the symmetrical components:

1. The positive and negative sequence components in voltages and currents may contribute to the

average real and imaginary powers.

2. The instantaneous active and nonactive powers consists of an oscillating components due to

the vector product of the positive sequence voltage and the negative sequence current, and the

Figure 3.3 - Basic principle of shunt current compensation.

17

negative sequence voltage and the positive sequence current. Therefore circuits without any

harmonic generating load can have oscillating active or nonactive power.

In Appendix A, we have analyzed the instantaneous real and imaginary powers of the sequence

components of symmetrical components.

18

Chapter 4

SHUNT ACTIVE POWER FILTER

_______________________________________

4.1 SHUNT ACTIVE FILTERS

An idea on shunt active power filter was provided by gyugyi and strycula in 1976. Shunt active

filter is being widely used commercially. The main advantage of using shunt active filter is

finding of compensating current reference and then forcing a power converter to take

synthesizing action accordingly. So we can also conclude that this method is also adaptive. In

other words shunt active filter can efficiently and continuously compensate for harmonics in a

selected non-linear load.

4.2 GENERAL DESCRIPTION OF SHUNT ACTIVE FILTERS

Shunt active filters is comprises consist of two main blocks:

1. The PWM converter (power processing)

2. The active filter controller (signal processing)

Figure 4.1 – Basic configuration of shunt active filter

The PWM converter comprises of a 3 arm bridge inverter and a PWM controller that generates

the gating signals for the bridge inverter. The active filter controller is responsible for calculating

19

the compensatory reference currents which the bridge inverter tries to accurately track and inject

into the system at PCC. The generalized diagram of shunt active filter is shown in figure 4.1

4.3 THE ACTIVE FILTER CONTROLLER

The block diagram of active filter controller based on the pq-Theory is shown in figure

Figure 4.2 – Control block for constant instantaneous power control strategy

Clarke transformation matrix transforms source voltage and load current from abc-coordinates to

αβ-axes. Through these voltage and current, real and imaginary powers are calculated

(instantaneous). According to the power requirements, compensatory current is calculated. And

after calculating, these currents are inversely transformed back to abc-coordinates.

20

4.4 OPTIMAL POWER FLOW

For fulfilling optimal power flow condition, constant instantaneous power control strategy is

used. In this method, only active average real power from the load is used and remaining power

i.e. from harmonics is exchanged with shunt active filter

Figure 4.3 – Optimal power flow provided by shunt current compensation

The PWM converter takes synthesizing action on compensating current. The filter control block

has the job perform signal analysis in real time to calculate the instantaneous compensatory

reference current signals. The figure 4.3 shows the most common topology of an active filter for

harmonic compensation (maximum compensation) of a specific non linear load. It consists of a

voltage source inverter with a PWM current control, here hysteresis current control, and an

active filter controller that performs an instantaneously working control-algorithm. The shunt

active filter controller works in a closed-loop manner, continuously acquiring the samples of the

load current and calculating the instantaneous values of the compensating current reference i*c

for the PWM converter. In an ideal case, the PWM converter may be considered as a linear

power amplifier, where the compensating current ic tracks correctly its reference.

4.5 THREE PHASE THREE WIRE SHUNT ACTIVE FILTER

A distinguishing feature of three phase three wire system is that it does not have a common or

neutral wire, and so there is no zero sequence current components. Therefore there is no need to

calculate zero sequence power is for three phase three wire system. The figure 4.4 depicts the

most significant part of a three phase three wire shunt active power filter for harmonic current

compensation. The control block that calculates the instantaneous power takes phase-voltages of

point of common coupling (PCC) as the inputs and the line currents of the nonlinear load that

should be compensated. This means that the shunt active filter has a selective compensation

characteristic. In other words we can set it to compensate a particular non-linear load even if

there are other non-linear loads connected to PCC.

21

The filter control consists of following control block:

1. Instantaneous powers calculation

2. Powers-compensation select

3. dc-voltage regulation

4. Determine reference current

The Instantaneous powers calculation block calculate the instantaneous power of the non linear

load. According to p-q Theory, only the active and reactive power exist, because there is no

neutral, zero-sequence power is not present. The Powers-compensation select block determine

the working output of the shunt active filter. In other words, it selects the parts of the active (real)

and reactive (imaginary) power of the non linear load, that should be compensated by the filter.

Moreover, the dc-voltage regulation block determines the extra amount of power ploss, that

causes an additional flow of energy to (from) the dc link to keep its voltage fixed around

reference voltage Vref. This active power ploss is summed up to the compensating active power

pc, and together with the compensating reactive power qc, are passed to the calculation of

reference current block. It determine the instantaneous compensatory current reference from the

udesired powers. The power injector of the shunt active filter consists of a three-phase voltage

source inverter made up of IGBT or power MOSFETs and anti parallel diodes. The PWM

current controller force the inverter to simulate a controlled current source. In order to avoid high

inductive kick, the coupling of a inverter to the system must be made through a coupling

inductor, commonly known as a commutation inductor or a series coupling inductor. The leakage

inductive reactance of a coupling transformer can also serve this purpose, that is to limit di/dt, so

the coupling inductor can be eliminated ftom the power circuit. Figure also shows a small RC

pssive filter, it filters the higher order harmonics generated due to the switching of power

converter.

Figure 4.4 – Three-phase, three-wire shunt active filter

22

4.6 SIMULINK MODEL DESCRIPTION

Figure 4.5 shows the complete system design of Shunt Active Power Filter in a three-phase

three-wire system. Harmonic contents in Line and load currents, voltage at PCC and unbalancing

in load is observed in presence and in absence of SAPF.

4.6.1 3-PHASE SOURCE

3-Phase Source block of SimPower System is used as 3Ф Voltage source with following ratings:

Phase-to-phase rms voltage (V) 400V

Frequency (Hz) 50Hz

Internal connection Yg

Source resistance (Ohms) 0.001

Source inductance 1e-8

4.6.2 LINE IMPEDANCE (Zt)

Line impedance (Zt) is a three phase RLC branch used to represent line impedance having

following parameters.

Resistance R (Ohms) 0.01

Inductance L (H) 1e-6

4.6.3 V-I MEASUREMENT

Three phase VI Measurement block is used to measure line to ground voltages and line currents.

23

Figure 4.5 - Complete System

4.6.4 CIRCUIT BREAKER

A three phase Circuit Breaker is connected in series with line and SAPF. Breaker timing is

defined such that it connects SAPF with system after some time simulation has started in order to

have a better look at line current harmonics and the effect of SAPF.

4.6.5 POWERGUI

This block is needed to run any SimPower System model. It provides option for configuration of

simulation and analysis of system.

4.6.6 NON-LINEAR LOAD

The non-linear load has two components connected in parallel as highlighted in figure 4.6. One

is three phase rectifier and the other is an unbalance resistive load. The two loads are switched on

one after another to have a better look at compensation of harmonics and balancing of system

with SAPF.

Figure 4.6 –Non-linear and unbalance resistive load

24

4.7 SHUNT APF

The design and simulation of Shunt APF, shown in figure 4.7, is the main objective of our

project. We shall look under mask of this object to find components and blocks integrated to

design SAPF.

Figure 4.7 - Shunt APF

4.7.1 PQ & I-COMPENSATION CALCULATION

This is the heart of APF. PQ Theory algorithm for calculation of active and reactive power and

compensation currents (currents that are to be injected in order to compensate harmonic

distortion) is implemented in this block. Embedded MATLAB Function block is used to

implement all the mathematical operations involved in the algorithm. Figure 4.8 shows all the

blocks required for calculation of compensation currents.

4.7.1.1 CLARKE V

This block takes phase voltages Va, Vb and Vc as input and transforms them to Vα and Vβ using

Clarke transformation through the following functions:

function [x,y] = VCT(u,v,w)

x = sqrt(2/3)*(u-(0.5*v)-(0.5*w));

y = sqrt(2)*(0+(0.5*v)-(0.5*w));

25

Figure 4.8-I-compensation calculation

4.7.1.2 CLARKE I

This block takes line currents Ia, Ib and Ic as input and transforms them to Iα and Iβ using Clarke

transformation through the following functions:

function [x,y] = ICT(u,v,w)

x = sqrt(2/3)*(u-(0.5*v)-(0.5*w));

y = sqrt(2)*(0+(0.5*v)-(0.5*w));

4.7.1.3 PQ CALCULATION

This block calculates active and reactive powers of three phase system using Vα, Vβ and Iα, Iβ.

The functions used in this block are:

function [P,Q] = PQ(x1,x2,y1,y2)

P = (x1*y1)+(x2*y2);

Q = (x2*y1)-(x1*y2);

4.7.1.4 LOW PASS FILTER

Analog Filter Design block is used to implement 5th

order Butterworth Low pass filter with cut

of frequency of 2*pi*50 rad/sec. this is used to filter out the component of active power

transferred only due to the fundamental current component.

26

4.7.1.5 ALPHA BETA CURRENT

Now oscillating component of active power, reactive power, Vα and Vβ are used to find

harmonic currents in alpha-beta co-ordinates.

function [Ic1,Ic2] = ICOM(Posc,q,V1,V2)

Ic1 = (-1/(V1^2 + V2^2))*((Posc*V1)+(q*V2));

Ic2 = (-1/(V1^2 + V2^2))*((Posc*V2)-(q*V1));

4.7.1.6 COMPENSATION CURRENTS

Compensation currents in terms of abc phases are calculated by taking inverse Clarke transform

of alpha-beta compensation currents.

function [ICa,ICb,ICc] = Ical(Ic1,Ic2)

ICa = sqrt(2/3)*(Ic1);

ICb = sqrt(2/3)*((-0.5*Ic1)+((sqrt(3)/2)*Ic2));

ICc = sqrt(2/3)*((-0.5*Ic1)-((sqrt(3)/2)*Ic2));

4.7.2 HYSTERESIS CONTROLLER

Hysteresis Current Controller is one of the technique available for the generation of PWM

signals that controls the gates of inverter’s transistors. A detailed description is given in

4.7.3 UNIVERSAL BRIDGE

Gating signals generated by hysteresis current controller are fed to Universal Bridge three-arm

IGBT fast switching inverter. The inverter generates exactly the required harmonic currents.

4.7.4 CAPACITORS

Capacitors are discharged through the inverter to generate compensation currents. These

capacitors then become the source of harmonics rather than the main source.

4.7.5 PI CONTROLLER

PI controller is used to remove steady sate error. Here we want it to maintain Vdc by comparing it

with a constant value of Vref. If Vdc is lesser than Vref then it would create a positive ploss signal

and if Vdc is greater than Vref it would create negative ploss signal.

4.7.6 COUPLING INDUCTOR

An inductor is used to couple power inverter with point of common coupling (PCC). Its job is to

limit L.di/dt effects. Leakage inductance of a coupling transformer can also be used.

Appendix B.

27

Chapter 5

SIMULATION RESULTS

___________________________________

After having a lot of theoretical background, lets have some simulation results that helps in better

understanding of theory and dynamic and versatile behaviour of Active Power Filter (APF).

Following are the cases differ by nature of load connected across the source. All the simulations

are performed in the SIMULINK environment, a proprietary software from Mathworks Inc. A

detailed description of simulation model was covered in previous chapter.

5.1 CASE I: COMPENSATION OF NON-LINEAR LOAD

Figure 5.1 - Load of case 1 that is to be compensated using APF

Figure 5.1 shows a three phase rectifier with a resistor connected on DC side as a non-linear

load, its powers are shown in Figure 5.2 The source, load and compensation currents are shown

in Figure 5.3, it can be observed from Figure that APF is switched on at 0.02 seconds. After

compensation the source current has become a single frequency pure sinusoid having THD equal

to 0.70% which is within limits as standardized by IEEE in Std. 519.

28

Figure 5.2 - Powers of load shown in Figure 5.1

Figure 5.3 - Currents of the load shown in Figure 5.1

29

5.2 CASE II: COMPENSATION OF NON-LINEAR PLUS UNBALANCE

RESISTIVE LOAD



30


This case is simulated to show the dynamic behaviour of the APF. The APF is switched on at

0.02 seconds with the non-linear load only. The circuit breaker shown in Figure 5.4 is closed at

0.05 seconds and now the load has been changed. And the APF has only taken about 15

milliseconds to respond to the changing behaviour of the load. These switching instants and the

dynamic behaviour of the APF can easily be observed from the Figure 5.6

5.3 CASE III: COMPENSATION OF UNBALANCE RESISTIVE LOAD


This case is simulated to show the capability of APF to make the load balance, that is

compensating negative sequence currents so that the source will provide positive sequence only.

31



One thing that is worth noticing in Figure 5.8 is the presence of q (vai) in case of resistive but

unbalance load. And this is one of the points used by L.S. Czarnecki (author of the CPC Theory)

to criticize the Akagi’s theory.

32

5.4 CASE IV: COMPENSATION OF UNBALANCE INDUCTIVE LOAD



This is also a very important case, as this case shows the presence of active power, even if the

load is the purely reactive. The oscillations in both the powers is due to the unbalanced load

condition due to which a negative sequence current is also present. On the basis of this and the

previous case following conclusions regarding power due to symmetrical components can be

drawn assuming the source voltages are balanced and having positive phase sequence.

1. The presence of constant instantaneous active power p(W) is due to the positive sequence

current and only in case of purely resistive load but may be unbalance.

2. The presence of constant instantaneous imaginary power q(VAI) is due to the positive

sequence current and only in case of purely inductive/capacitive load but may be

unbalance.

3. The presence of oscillations in both of the instantaneous powers is due to the negative

sequence current whether a load is purely resistive or inductive/capacitive.

33


5.5 CASE V: COMPENSATION OF UNIQUE UNBALANCE RESISTIVE

LOAD


This is the unique case, simulated to show that even the load is connected between the single

phase and the ground, but after compensation the balanced three phase current is supplied by the

source i.e. the APF is acting as a resistive load for the remaining two open phases .

34



5.6 CONCLUSION

All the five cases that are discussed, helps in better understanding of power theory presented by

Akagi and his co-workers and some insights which conventional power theory fails to provide in

its phasor domain. Also the five cases shows the dynamic and versatile behavior of APF.

35

Appendix A

SYMMETRICAL COMPONENTS

___________________________________

Symmetrical components (also known as Fortescue’s components) are basically used for the

analysis of three phase electrical power systems. It’s ubiquitous in power system analysis and an

essential tool in fault analysis. If the phase quantities are expressed in frequency domain and, a

vector can be formed for the three phase quantities or three phase voltages phasors. A vector for

three phase voltages could be written as,

where the subscripts 0, 1, and 2 refer to the zero, positive, and negative sequence components.

The sequence components, only differ by their phase angles, and are symmetrical i.e. all

individual components are equal and are 2/3π radians or 120° apart. Operator α displaces any

phasor by an angle of 120°, if multiplied.

And α3 = 1, so that α

−1 = α

2.

The zero sequence components always have same phase; denote them as:

And, the positive and the negative phase sequences, respectively written as:

36

Thus,

where

Similarly, we can calculate sequence components by the equation,

A.1 SIMULINK MODEL OF SEQUENCE POWER

Our motive is to calculate Symmetrical components in time domain. We used transport delay

block in SIMULINK library to emulate αphasor vector in time domain. since there are 360

electrical degrees in one cycle of 20m seconds. ifαphasor represents 120 degrees shift in

electrical degrees it corresponds to 6.666m seconds.

Figure A.1 - SIMULINK block of Symmetrical components

37

Figure A.2 - SIMULINK model of Symmetrical components in time domain

Figure A.1 shows that sequence currents are fed into PQ measument block because we wish to

calculate power of each sequence and analyse them in the light of Akagi’spqTheory.The

conclusions from pq Theory can be written from the above equations for the real andImaginary

powers of the symmetrical components:

1. Average real power flows in system only due to the product of positive sequence currents and

source voltages.

2.The zero and negative sequence components in currents contribute to the oscillations in real

imaginary powers.

To test above points we coupled our block with APF. In order to have both seqeunces in line

current unbalanced resistive load was connected. APF was set to operate at adelay of 0.06secs,

in order to incorporate before and after compensation affects, in other words negative sequence

currents must be wiped out completely since all imaginary power would be compensated.

Whereas positive sequence current would remain unaltered since they are responsible for active

power transfer to the load.

Fig. A.3, evidently shows that power transfer by positive sequence currents it active, since the

currents are unaltered before and after the APF operation. Hence only positive sequence currents

are coming from the source. Note the transients after 0.06secs are due DC capacitor charging.

Similarly, Fig. A.4 shows that negative sequence currents are completely compensated by APF

and instantaneous powers of negative sequence also eliminated from the system.

38

Figure A.3 – Positive sequence powers and currents before and after compensation

Figure A.4 – Negative sequence powers and currents before and after compensation

39

Appendix B

HYSTERESIS CONTROLLER

__________________________________

Hysteresis current controller is one of the techniques available to control the voltage source

inverter (VSI) through the PWM in a manner that the output current of inverter then tries to

track the reference current fed to the hysteresis current controller which works in a close loop. Its

advantage is that it is very easy to implement while its disadvantage is that its switching

frequency is not constant. A diagrammatic working of hysteresis current controller is shown in

figure B.1

A hysteresis band is defined which sets the upper and lower limits for the output current to

deviate from its reference current signal. The thicker the band the more the deviations hence

more ripples in output current and vice versa.

Figure B.1 - Hysteresis Current Controller

An error signal e(t) which is the difference between the reference current signal iref(t) and the

actual output current signal iactual(t) of the inverter, is used to generate the gating signals for the

transistors of VSI. Depending on the upper and lower limits of the hysteresis band the range of

error signal can be determined. If e(t) is greater than the maximum value of error emax the

transistors are switched off and if e(t) is less than the minimum value of error emin then the

transistors are switched on. And if the error is within the range i.e. emin < e(t) < emax then the state

of the transistors remain unchanged.

The switching frequency of hysteresis current controller is not constant and depends upon the

following factors:

1. The width of the hysteresis band. Switching frequency is inversely proportional to the

width of the hysteresis band.

40

2. The voltage of the DC side. Switching frequency is directly proportional to the voltage of

the DC side.

B.1 SIMULINK MODEL EXAMPLE

To better understand the working of hysteresis controller, let us simulate a single phase inverter

model in SIMULINK Environment in which gating signals of transistors are controlled through

hysteresis controller. Figure B.2 shows a SIMULINK model in which ‘Relational Operator’ and

‘Logical NOT Operator’ blocks, together acting as a hysteresis controller. The output of

Relational Operator block is directly connected to one diagonal pair of transistors namely IGBT1

and IGBT2, and indirectly through Logical NOT Operator block, connected to another diagonal

pair of transistors namely IGBT3 and IGBT4. Relational Operator compares the reference and

load current values, if reference value is greater than load value than Relational Operator block

gives ‘1’ as output, and ‘0’ if reference value is less than load value.

Figure B.2 - SIMULINK model of single phase inverter

Let us simulate a model for a very short interval of time to see how these two block works

together as hysteresis controller. Refer to figure B.3 which shows a stem plot figure made using

MATLAB’s stem command by importing the SIMULINK data to the workspace. The red

coloured lines are representing the reference current values, the blue coloured lines are

representing the load (tracked) current values and the black coloured filled lines are representing

the decision of Relational Operator block taken by comparing the current values at every Ts

instant.

41

Figure B.3 - Comparison of reference and load current values sampled after every Ts

It can be seen in the figure B.3 that, when the red line (reference value) has higher value than the

blue line (tracked value) the black line (gating signal) is at value equal to ‘1’, that means the

transistor is switched 'ON' and when the opposite is true than the black line (gating signal) is at

value ‘0’, that means the transistor is 'OFF' and this process goes on controlling the gates of the

transistors.

B.2 HYSTERESIS BAND IN DIGITAL HYSTERESIS CURRENT CONTROLLER

When the hysteresis current controller is digitally implemented than the rate at which the

currents are sampled determines the width of the hysteresis band. The higher the sampling rate

the thinner the hysteresis band and vice versa. This thing can easily be understood by simulating

our SIMULINK model with different values of Ts.

Refer to figure B.4 (a) and (b), which shows a comparison of two cases which are simulated by

configuring the powergui block at different Ts values. It can be seen in the figure B.4 (a) and (b)

that, when the Ts value is higher (lower sampling rate) the hysteresis band is thicker. And when

the Ts value is lower (higher sampling rate) the hysteresis band is thinner.

42

(a) Model simulated at Ts = 5e-005 seconds

(b) Model simulated at Ts = 2e-005 seconds

Figure B.4 - Hysteresis Band at different sampling rate

Therefore, one should select a proper value of sampling time or sampling frequency such that the

maximum frequency content present in reference current signal is properly sampled and ripples

in the actual current should be as small as possible. However, higher the sampling rate higher

will be the switching frequency and hence more will be the switching losses .

BIBLIOGRAPHY

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Education