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TitleHarmonic Analysis in Electric Power Systems with Independent Compon

ent Analysis

Author(s) Lian, Suo

Editor(s)

Citation

Issue Date 2013

URL http://hdl.handle.net/10466/13834

Rights

Harmonic Analysis in Electric Power Systems

with Independent Component Analysis

Suo Lian

February 2013

Doctoral Thesis at Osaka Prefecture University

Harmonic Analysis in Electric Power Systems

with Independent Component Analysis

This research was made in the Power System

Research Group, Department of Electrical and

Electronic Systems, Division of Electrical Engineering

and Information Science, Graduate School of

Engineering, Osaka Prefecture University, and

submitted for the doctoral thesis at Osaka Prefecture

University

Suo Lian

February 2013

Power System Research Group

Osaka Prefecture University

i

CONTENTS

CHAPTER 1 INTRODUCTION 1

1.1 Motivation .................................................................................................. 1 1.2 Organization of This Thesis ....................................................................... 6

CHAPTER 2 HARMONICS IN POWER SYSTEMS 7

2.1 Introduction ................................................................................................ 7 2.2 Fundamentals of Harmonics in Power Systems ......................................... 8

2.2.1 Examples of Harmonic Waveforms ................................................... 8 2.2.2 Representation of Harmonics in Power Systems ............................... 9 2.2.3 Power Quality Indices under Harmonic Distortion.......................... 10 2.2.4 Power Quantities under Nonsinusoidal Situations ........................... 11

2.3 Harmonic Sources .................................................................................... 12 2.3.1 Traditional Harmonic Sources ......................................................... 12

2.3.1.1 Transformer ............................................................................ 13 2.3.1.2 Rotating Machine ................................................................... 13 2.3.1.1 Arc Furnace ............................................................................ 14

2.3.2 Modern (Power-Electronic) Harmonic Sources ............................. 14 2.3.2.1 Fluorescent Lamp ................................................................... 15 2.3.2.2 Converter ................................................................................ 16

2.3.3 Future Sources of Harmonic ............................................................ 16 2.4 Effects of Harmonic Distortion in Power Systems .................................. 17

2.4.1 Thermal Losses on Transformer ...................................................... 17 2.4.2 Neutral Conductor Overloading ..................................................... 18 2.4.3 Miscellaneous Effects on Capacitor Banks ...................................... 19

2.4.3.1 Resonant Condition ................................................................ 19 2.4.3.2 Unexpected Fuse Operation ................................................... 20

2.4.4 Abnormal Operation of Electronic Relay ........................................ 20 2.5 Limits of Harmonic Distortion ................................................................. 21

2.5.1 IEEE Limits...................................................................................... 22 2.5.2 Limits of Japanese ............................................................................ 23

ii

2.6 Mitigation of Power Systems Harmonics ............................................... 23 2.6.1 Passive Harmonic Filters ................................................................. 24

2.6.1.1 Single Tuned Filters ............................................................... 24 2.6.1.2 High Pass Filters .................................................................... 25

2.6.2 Active Power Filters......................................................................... 26 2.7 Conclusion ................................................................................................ 29

CHAPTER 3 INDEPENDENT COMPONENT ANALYSIS 31

3.1 Introduction ............................................................................................ 31 3.2 Linear Model based ICA .......................................................................... 32 3.3 Restriction in ICA .................................................................................... 34 3.4 Ambiguities of ICA .................................................................................. 35 3.5 Centering and Whitening ......................................................................... 36

3.5.1 Centering the Variables .................................................................... 36 3.5.2 Whiting............................................................................................. 37

3.6 ICA by Maximization of Nongaussianity ................................................ 39 3.6.1 Nongaussian is Independent............................................................. 40 3.6.2 Fixed-Point Algorithm of Real Values ............................................ 41

3.6.2.1 Negentopy as Nongaussianity Measure ................................. 41 3.6.2.2 Approximating Negentopy ..................................................... 42 3.6.2.3 Fixed-Point Algorithm using Negentopy ............................... 43 3.6.2.4 Estimating Several Independent Components ....................... 44

3.6.3 Fixed-Point Algorithm of Complex Values ..................................... 45 3.6.3.1 Basic Concepts of Complex Random Variables .................... 45 3.6.3.2 Indeterminacy of Independent Components .......................... 47 3.6.3.3 Choice of the Nongaussianity Measure ................................. 47 3.6.3.4 Fixed-Point Algorithm of Complex Value ............................. 49

3.7 Conclusion ................................................................................................ 50

CHAPTER 4 EFFECT OF HARMONICS CAUSED BY LARGE SCALE PHOTOVOLTAIC

INSTALLATION IN POWER SYSTEMS 51

4.1 Introduction .............................................................................................. 51 4.2 Stochastic Aggregate Harmonic Load Model .......................................... 52 4.3 Modeling the PV Inverter ......................................................................... 55

iii

4.4 Numerical Simulation .............................................................................. 58 4.4.1 Mega-PV .......................................................................................... 58 4.4.2 Residential Type PV ........................................................................ 61

4.5 Conclusion ................................................................................................ 64

CHAPTER 5 ESTIMATION OF SYSTEM HARMONIC IMPEDANCE USING COMPLEX

ICA 65

5.1 Introduction .............................................................................................. 65 5.2 Current System Harmonic Impedance Estimate Method ......................... 66

5.2.1 General Principles ............................................................................ 66 5.2.2 Transients Based Methods (Invasive Method) ................................. 67

5.3 Estimation of System Harmonic Impedance using Complex ICA .......... 70 5.3.1 Norton Equivalent Circuit ................................................................ 70 5.3.2 Statistical Properties of Harmonic Current Sources ........................ 71 5.3.3 Estimation Algorithm ....................................................................... 73 5.3.4 Numerical Simulations ..................................................................... 76

5.3.4.1 Simulation Condition ............................................................. 76 5.3.4.2 When Customer Harmonic Impedance

Changing in Small Range .................................................... 79 5.3.4.3 When Customer Harmonic Impedance Changing Sharply .... 84 5.3.4.4 Changing the Kurtosis of the Harmonic Current Source ....... 87

5.4 Conclusion ................................................................................................ 93

CHAPTER 6 HARMONIC CONTRIBUTION EVALUATION OF INDUSTRY LOAD USING

COMPLEX ICA 95

6.1 Introduction .............................................................................................. 95 6.2 Power Direction Method .......................................................................... 96 6.3 Harmonic Contribution Evaluation of Industry Load

using Complex ICA ................................................................................ 98 6.3.1 Principle of Proposed Method.......................................................... 98 6.3.2 Characters of Industrial Load ......................................................... 100 6.3.3 Dealing to Ambiguities of ICA ...................................................... 101 6.3.4 Algorithm of Proposed Method ..................................................... 101

iv

6.4 Numerical Simulations ........................................................................... 102 6.4.1 Artificial Data ................................................................................ 102 6.4.2 Measurement Data ......................................................................... 107

6.5 Conclusion .............................................................................................. 110

CHAPTER 7 CONCLUSIONS 111

REFERENCES 115

LIST OF ACRONYMS 121

ACKNOWLEDGEMENTS 123

Chapter 1 Introduction

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CCHHAAPPTTEERR 11

Introduction

1.1. Motivation

Electric power systems are designed to operate with sinusoidal voltage and current.

However, nonlinear and electronically switching loads will distort steady state Alternate

Current (AC) voltage and current waveform [1]. The nonlinear loads and the Distributed

Generation (DG) power systems contribute to change the characteristics of voltage and

current waveforms in power systems, which differ from pure sinusoidal constant

amplitude signals. The impact of nonlinear loads in electrical power systems has been

increasing during the last decades. Such electrical loads, which produce non-sinusoidal

current consumption patterns (current harmonics), can be found in rectification

front-ends in motor drives, electronic ballasts for discharge lamps, personal computers

or electrical appliances. Harmonics in power systems mean the existence of signals,

superimposed on the fundamental signal, whose frequencies are integer numbers of the

fundamental frequency. The electric utility companies should supply their customers

with constant frequency equal to the fundamental frequency, which is 50/60 Hz, and

having a constant magnitude. The presence of harmonics in the voltage and current

waveform leads to a distorted signal for voltage and current, and the signal becomes

non-sinusoidal signal which it should not be. Thus the study of power system harmonics

is an important subject for power engineers [2]-[4].

Furthermore, the increasing of renewable energy sources are already became the main

source of harmonic current in power distribution systems. Nowadays, fossil fuel is the

main energy source in the worldwide, but the recognition of it as being a major cause of

environmental problems and increasing of generation cost makes the mankind to look

for alternative resources in power generation. Moreover, the increasing demand for

energy can create problems for the power distributors, like grid instability and even

outages. The necessity of producing more energy combined with the interest in clean


- 2 -

technologies yields in an increased development of power distribution systems using

renewable energy.

Among the renewable energy sources, a noticeable growth of grid-connected

Photovoltaic (PV) power plants connected to low-voltage distribution networks is

expected in the future. In Japan, the installing total capacity of PV will be about 53 GW

at 2030, which is ten times more than it in 2010 [5]. However, the large number of PV

could cause the voltage increase at the distribution system nodes, reduction of

distribution systems losses, and voltage and current waveform distortion [6]-[11]. In the

power distribution system which installing PV, the cause of current and voltage

waveform distortion are including the nonlinear loads of customers and the inverters of

PV.

For the increasing of harmonic distortion on power systems, the harmonic mitigation

method is necessary. The harmonic filters are designed for this purpose, which include

passive filter and active filter. The reliable design of a passive filter requires a correct

knowledge of the system harmonic impedance and its variations throughout the day to

avoid creating a resonance condition, which could destabilize a power system. Active

filters also require a good knowledge of the system harmonic impedance to ensure

stable controller operation and also can be used in the generation of the filter reference

currents. For harmonic analysis and mitigations, the identification and measurement of

harmonic impedance and harmonic sources have become an important issue in electric

power systems.

A common philosophy of harmonic analysis in power system is to conduct a

deterministic study based on the worst case in order to provide a safety margin in system

design and operation. However, this often leads to overdesign and excessive costs. Field

measurement data clearly indicates that voltage and current harmonics are time-variant

due to continual changes in load conditions. Consequently, statistical techniques for

harmonic analysis are more suitable, similar to other conventional studies like

probabilistic load flow and fault studies [12] [13]. Such an analysis would calculate

harmonic currents and voltages based not simply on the expected average or maximum

values, but would also obtain the complete spectrum of all probable values together with

their respective probabilities. As a statistical technique, Blind Source Separation (BSS) techniques have received

attention in applications where there is little or no information available on the


- 3 -

underlying physical environment and the sources. BSS algorithms estimate the source

signals from observed mixtures. The word ‘blind’ emphasizes that the source signals

and the way the sources are mixed, i.e. the mixing model parameters, are unknown. A

well-known example for the BSS is the cocktail party problem [14]. Assume that four

people are speaking simultaneously in a room, and there are four microphones located

in different parts of the room recording the sounds in the room. The recordings from the

microphones will be the mixture of the individuals’ speeches. In this problem neither

the individuals’ speeches nor the distances of individuals from the microphones are

known. The only known quantity is the recordings from the microphones. The object of

the BSS is to recover the individuals’ speeches from the recorded signals. The BSS is a

difficult problem to solve. However, some properties of the sources make the problem

solvable.

The important BSS technique is the Independent Component Analysis (ICA)

[15]-[18] which is based on the statistical independence and the sparsity of the source

signals. In this problem, source signals are the speeches of the individuals and

measurement signals are the recordings of the microphones. ICA was originally

developed to deal with problems that are closely related to the cocktail-party problem.

Since the recent increase of interest in ICA, it has become clear that this principle has a

lot of other interesting applications as well. Consider, for example, electrical recordings

of brain activity as given by an Electroencephalogram (EEG). The EEG data consists of

recordings of electrical potentials in many different locations on the scalp. These

potentials are presumably generated by mixing some underlying components of brain

and muscle activity. This situation is quite similar to the cocktail-party problem: we

would like to find the original components of brain activity, but we can only observe

mixtures of the components. ICA can reveal interesting information on brain activity by

giving access to its independent components. ICA has been an attractive technique for

different areas such as financial applications, audio separation, image separation,

telecommunications, and brain imaging applications [15].

Even though there are plenty of applications of ICA to the areas mentioned above, it

is limited application in power systems. The application of ICA for study on load profile

estimation is described in [19] and estimation of DG is present in [20]. In [21], ICA is

used to estimate the harmonic source. Results of system side harmonic impedance [22]

and the harmonic current contribution of industrial load [23]-[25] are presents in this


- 4 -

thesis.

To mitigate the current and voltage distortion of power system, the system harmonic

impedance estimation for filters and harmonic source estimation methods are very

important.

It is very desirable in many applications to directly measure the system harmonic

impedances. A number of impedance measurement methods have been developed for

this purpose. These methods can be classified into two types: 1) the transients- based

methods (invasive methods) and 2) the steady-state-based methods (noninvasive

methods) [26] [27]. The transients-based methods inject transient disturbances into the

system. The frequency-dependent system impedances are extracted from voltage and

current transients. Typical transient disturbances for this application are the capacitor

switching transients [28] [29] and controlled harmonic current injection method [30]

[31]. The main problems associated with these methods are the need for a high-speed

data acquisition system to measure pre and post disturbance of steady-state waveforms.

Typical disturbances are harmonic current injections produced by an external source or

switching of a network component. Also, these kinds of methods cannot get the system

impedance in real time, only provide instantaneous results which are valid for the

moment of the test. Sometimes, this may cause a negative effect on the normal

operation of the power system [26].

Identification of harmonic sources in power systems has been a challenging task for

many years. Many techniques have been applied to determine customer and systems

responsibility for harmonic distortion. As the method with synchronized measurements

in multiple points in the network [32] is a rather difficult and expensive task, more

practical approaches are based on measurement data at the Point of Common Coupling

(PCC) between the customer and the systems. Although there are a few indices dealing

with harmonic contribution determination at PCC, none are widely used in practice. The

most common tool to solve this problem is the harmonic power direction-based method

[33]. In this method, if harmonic active power flows from systems to customer, the

system is considered as the dominant harmonic generator. Unfortunately, [34] have

proven that this qualitative method is theoretically unreliable. Another group of

practical methods for harmonic source detection is to measure the system and customer

harmonic impedances and then calculate the harmonic sources behind the impedances.

There are a number of variations of this method [35]–[40]. These types of methods are


- 5 -

very difficult to implement. The main problem of these methods is that it need the

information of harmonic impedances which the customer impedances can only be

determined with the help of disturbances. Such disturbances are not readily available

from the system or are expensive to generate with intrusive means. Also, there have

other ways to estimate the harmonic current likes using the artificial neural networks

(ANNs) [41] [42]. Therefore, it is very desirable to estimate the harmonic contribution

of customers and systems without any information or estimated information of

network’s parameters just using the measurement data at PCC.

The system harmonic impedance estimation algorithm and the estimation of

harmonic current contribution of industry load at PCC are presented [22]-[25] in this

thesis. The both methods are based on the complex value Fast-ICA algorithm, which is

the most powerful algorithm in ICA algorithms. The main advantage of these methods is

that only harmonic voltage and current have to be measured for estimation without

knowing the systems information and disrupting the operation of any devices.

The system harmonic impedance estimation algorithm introduced the Norton

equivalent circuits to set up the linear mixing model of ICA model to estimate the

system harmonic impedance in condition that load harmonic impedance changing. Two

types of harmonic impedance models are introduced for simulation. The numerical

simulation of proposed method verified that the method is suitable to estimate the

system harmonic impedance.

The harmonic current contribution evaluation algorithm discussed the true harmonic

current contribution of an industry load at PCC, which is the main harmonic current

source in the power systems. The proposed method also needs the Norton equivalent

circuits to set up the linear mixing model of ICA model. The proposed method is

verified by both artificial data and measurement data. The artificial data simulation

proved that the propose method is suitable to evaluate the harmonic contribution at PCC.

The measurement data simulation shows that the customer side is mainly responsible for

the harmonic current distortion at PCC. However, the customer is not full responsible for

harmonic current at PCC. The system side also have responsible for harmonic current

distortion at PCC.


- 6 -

1.2. Organization of This Thesis

The remainder of this thesis is divided into seven chapters.

Chapter 2 describes the harmonic analysis in power systems. Fundamentals of

harmonic, harmonic sources, effect of harmonic distortion limits of harmonic distortion

and mitigation of harmonics in harmonic domain are presented.

Chapter 3 describes ICA, which main method for the estimation of system harmonic

impedance and harmonic contribution of industrial loads in this thesis. The linear model

of the ICA and the complex value Fast-ICA algorithm are presented.

In Chapter 4, the characterizations of the waveform distortion from grid-connected

PV plants under different installing capacity are discussed.

In Chapter 5, the technique that estimates system harmonic impedance at PCC is

presented. Harmonic impedances of a supply system characterize the frequency

response characteristics of the system at specific buses. It is very desirable in many

applications to directly measure the system harmonic impedances.

In Chapter 6, the estimation of harmonic contribution of industry load at PCC is

presented. The industry load can be departed to three part, which before the working,

under the working and after the working. And in each part, the loads are damping in the

same level. Thus, it can be assumed that the customer side harmonic impedance of each

part is changing small in mixing process.

Chapter 7 concludes the thesis with a summary, review of main contribution points.

Chapter 2 Harmonics in Power Systems

- 7 -

CCHHAAPPTTEERR 22

Harmonics in Power Systems

2.1. Introduction

The presence of non-linear loads and the increasing number of DG systems in power

systems contribute to change the characteristics of voltage and current waveforms in

power systems, which differ from pure sinusoidal constant amplitude signals. Under

these conditions advanced signal processing techniques are required for accurate

measurement of electrical power quantities. The impact of non-linear loads in electrical

power systems has been increasing during the last decades. Such electrical loads, which

introduce non-sinusoidal current consumption patterns (current harmonics), can be

found in rectification front-ends in motor drives, electronic ballasts for discharge lamps,

personal computers or electrical appliances. Harmonics in power systems mean the

existence of signals, superimposed on the fundamental signal, whose frequencies are

integer numbers of the fundamental frequency. The power system companies should

supply their customers with a supply having a constant frequency equal to the

fundamental frequency, 50/60 Hz, and having a constant magnitude. The presence of

harmonics in the voltage or current waveform leads to a distorted signal for voltage or

current, and the signal becomes non-sinusoidal signal which it should not be. Thus the

study of power system harmonics is an important subject for power engineers.

The power system harmonics problem is not a new problem; it has been noticed since

the establishment of the AC generators, where distorted voltage and current waveforms

were observed in the thirtieth of 20th century [1]. Concern for waveform distortion

should be shared by all electrical engineers in order to establish the right balance

between exercising control by distortion and keeping distortion under control. There is a

need for early co-ordination of decisions between the interested parties, in order to

achieve acceptable economical solutions and should be discussed between

manufacturers, power supply and communication authorities. Electricity supply


- 8 -

Figure 2-1. Example of harmonic waveforms.

authorities normally abrogate responsibility on harmonic matters by introducing

standards or recommendations for the limitation of voltage harmonic levels at the PCC

between consumers [2]-[4].

2.2. Fundamentals of Harmonics in Power Systems

2.2.1. Examples of Harmonic Waveforms

Harmonics are component of a distorted periodic waveform whose frequencies are

integer multiples of the fundamental frequency. The distorted waveform can be a sum of

sinusoidal signals. When the waveform is identical, it can be shown as a sum of pure sine

waves where the frequency of each sinusoid is an integer multiple of the fundamental

frequency of the distorted wave. This multiple is called a harmonic of fundamental. The

sum of the sinusoidal is called the Fourier series. Figure 2-1 shows examples of harmonic

waveforms. Here the fundamental frequency is the frequency of the power system. That is

60 Hz and the multiples that are 300Hz, 420Hz called fifth and seventh harmonics


- 9 -

respectively. The combine waveform shows the result of adding the harmonics on to the

fundamental.

2.2.2. Representation of Harmonics in Power Systems

The Fourier series represents an effective way to study and analysis harmonic

distortions. It allows inspecting the various constituents of a distorted waveform

through decomposition.

Generally, any periodic waveform can be expanded in the form of below:

1

000 sincos)(h

hh thBthAAtf ............................................. (2.1)

where )(tf is a periodic function of frequency 0f angular frequency 00 2 f and

period 00 /2/1 fT .

(2.1) can be further simplied, which yields:

1

00 sin)(h

hh thCCtf .............................................................. (2.2)

where

h

h

hhhhB

AandBACAC 122

00 tan,,

0h hth order harmonic of the periodic function

0C magnitude of the DC component

hC and h magnitude and phase angle of the hth harmonic component

(2.2) is known as a Fourier series and it describes a periodic function made up of the

contribution of sinusoidal functions of different frequencies.

The component with 1h is called the fundamental component. Magnitude and phase angle of each harmonic determine the resultant waveform )(tf . Generally, the

frequencies of interest for harmonic analysis include up to the 40th or so harmonics.

The Fourier coefficients are given as follows:


- 10 -

txwheredxtfdttfT

AT

0

2

000 ,)(21)(1

....................................... (2.3)

dxhxtfdtthtfT

AT

h

2

000)cos()(1)cos()(2 ..................................... (2.4)

dxhxtfdtthtfT

BT

h

2

000)sin()(1)sin()(2 ...................................... (2.5)

where ,,2,1 h .

2.2.3. Power Quality Indices under Harmonic Distortion

Total Harmonic Distortion (THD) is an important index widely used to describe

power quality issues in transmission and distribution systems. It considers the

contribution of every individual harmonic component on the signal. THD is defined for

voltage and current signals, respectively, as follows:

2

2

1

1h

hV VV

THD ................................................................................ (2.6)

2

2

1

1h

hI II

THD ................................................................................. (2.7)

where 1V , 1I represent the fundamental peak voltage and current, respectively. This

means that the ratio between Root Mean Square (RMS) values of signals including

harmonics and signals considering only the fundamental frequency define the total

harmonic distortion.

Total Demand Distortion (TDD): Harmonic distortion is most meaningful when

monitored at the PCC — usually the customer’s metering point — over a period that

can reflect maximum customer demand, typically 15 to 30 minutes as suggested in

Standard IEEE-519.7 Weak sources with a large demand current relative to their rated

current will tend to show greater waveform distortion. Conversely, stiff sources


- 11 -

characterized for operating at low demand currents will show decreased waveform distortion. The total demand distortion is based on the demand current LI , over the

monitoring period:

2

21h

h

L

II

TDD .................................................................................. (2.8)

2.2.4. Power Quantities under Nonsinusoidal Situations

It is noteworthy to emphasize that all quantities referred to in this section are based

on the trigonometric Fourier series. These quantities are expressed in a way that they

account for the contribution of individual harmonic frequency components.

A distorted periodic current or voltage waveform expanded in to a Fourier series is

expressed as follows

1

01

)cos()()(h

hh

h

h thItiti ......................................................... (2.9)

1

01

)cos()()(h

hh

h

h thVtvtv ...................................................... (2.10)

where

hI is the thh harmonic peak current

hV is the thh harmonic peak voltage

The instantaneous power is shown in follows with refer to (2.9) and (2.10)

)()()( titvtP ...................................................................................... (2.11)

Active power: Every harmonic provides a contribution to the average power that can

be positive or negative. However, the resultant harmonic power is very small relative to

the fundamental frequency active power.


- 12 -

1

0)cos(

21)(1

h

hhhh

T

IVdttpT

P ................................................. (2.12)

Reactive power is defined as

1

0)sin(

21)(1

h

hhhh

T

IVdttqT

Q ................................................... (2.13)

2.3. Harmonic Sources

Harmonic in power systems have been known since the adoption of alternating

current as a means for electric energy transmission. It is important to stress that

harmonic waveform distortion is just one of many different disturbances that perturb the

operation of electrical systems. It is also a unique problem in light of an increasing use

of power electronics that basically operate through electronic switching. Fortunately, the

sources of harmonic currents seem to be sufficiently well identified, so industrial,

commercial, and residential facilities are exposed to well known patterns of waveform

distortion. Different nonlinear loads produce different but identifiable harmonic spectra.

This makes the task of pinpointing possible culprits of harmonic distortion more

tangible. Systems and customers of electric power have to become familiar with the

signatures of different waveform distortions produced by specific harmonic sources.

This will facilitate the establishment of better methods to confine and remove them at

the sites where they are produced. In doing this, their penetration in the electrical system

affecting adjacent installations will be reduced.

2.3.1. Traditional Harmonic Sources

Prior to the development of power electronic switching devices, harmonic current

propagation was looked at from the perspective of design and operation of power

apparatus devices with magnetic iron cores, like electric machines and transformers. In

fact, at that time the main source of harmonics must have involved substation and

customer transformers operating in the saturation region. Harmonic distortion produced

under transformer saturation probably at peak demand or under elevated voltage during


- 13 -

very light load conditions is only one of numerous situations that generate harmonic

waveform distortion. Possibly, electric furnaces should be regarded as the second most

important cause of concern in high-power applications in industry, second to power

converter utilization.

The traditional type of harmonic source consist of

Transformers

Rotating machines

Arc Furnaces

2.3.1.1. Transformer

Power transformers are sources of harmonics since they use magnetic materials that

are operated very close to and often in the non-linear region for economic purposes.

This result in the transformer magnetizing current being non-sinusoidal and containing

harmonics (mainly third) even if the applied voltage is sinusoidal.

A transformer operating on the saturation region will show a nonlinear magnetizing

current which contains a variety of odd harmonics, with the third dominant. The effect

will become more evident with increasing loading. In an ideal lossless core, no

hysteresis losses are produced. The magnetic flux and the current needed to produce

them are related through the magnetizing current of the steel sheet material used in the

core construction. When the hysteresis effect is considered, this nonsinusoidal

magnetizing current is not symmetrical with respect to its maximum value. The

distortion is typically due to triplen harmonics (odd multiples of three, namely, the 3th,

9th, 15th, etc.), but mainly due to the third harmonic. This spectral component can be

confined within the transformer using delta transformer connections. This will help

maintain a supply voltage with a reasonable sinusoidal waveform.

2.3.1.2. Rotating Machine

As a result of small asymmetries on the machine stator or rotor slots or slight

irregularities in the winding patterns of a three-phase winding of a rotating machine,

harmonic currents can develop. These harmonics induce an electromotive force on the

stator windings at a frequency equal to the ratio of speed/wavelength. The resultant


- 14 -

distribution of magneto motive forces in the machine produces harmonics that are a

function of speed. Additional harmonic currents can be created upon magnetic core

saturation. However, these harmonic currents are usually smaller than those developed

when the machines are fed through variable frequency drives.

2.3.1.3. Arc Furnace

The melting process in industrial electric furnaces is known to produce substantial

amounts of harmonic distortion. The introduction of fundamental frequency harmonics

develops from a combination of the delay in the ignition of the electric arc along with

its highly nonlinear voltage-current character. Additionally, voltage changes caused by

the random variations of the arc give rise to a series of frequency variations in the range

0.1 to 30 kHz; each has its associated harmonics. This effect is more evident in the

melting phase during the interaction of the electromagnetic forces among the arcs.

2.3.2. Modern (Power-Electronic) Harmonic Sources

Nowadays, the sources of waveform distortion in power systems are multiple and, in

industrial installations, they can be found from small (less than 1 kVA) to several tens

of megavoltamperes. However, as mentioned earlier, commercial and residential

facilities can also become significant sources of harmonics. This is particularly true

when the combined effects of all individual loads served by the same feeder are taken

into account. The use of electricity involving loads that require some form of power

conditioning like rectification and/or inversion is on the rise.

The greatest majority of industrial nonlinear loads are related to solid-state switching

devices used in power converters that change electric power from one form to another.

This includes, among others, AC to DC energy conversion for DC motor speed control,

and AC to DC and back to AC at variable frequencies for processes involving speed

control of induction motors. Most bulk energy conversion processes take place in the oil,

mining, steel mill, pulp and paper, textile, and automobile industries. Other applications

include manufacturing assembly lines and electrolytic coating processes, which can

produce significant amounts of harmonic current generation.

The modern (Power Electronic) type of harmonic source consist of


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Fluorescent lamps

Converter (Rectifiers)

Inverters

Static VAR compensation

Cycloconverters

HVDC transmission.

Static inductive thyristors

DC motor drives

Regulated dc power supplies

Battery chargers etc.

2.3.2.1. Fluorescent Lamp

Fluorescent tubes are highly nonlinear in their operation and give rise to odd

harmonic currents of important magnitude. As a brief portrayal of the fluorescent lamp

operation, it can be state that magnetic core inductors or chokes contained inside the

start ballasts function to limit the current to the tube. Likewise, they use a capacitor to

increase the efficiency of the ballast by increasing its power factor. Electronic ballasts

operate at higher frequency, which permits the use of smaller reactors and capacitors.

The use of higher frequencies allows them to create more light for the same power input.

This is advantageously used to reduce the input power. In a four-wire, three-phase load,

the dominant phase current harmonics of fluorescent lighting are the third, fifth, and

seventh if they use magnetic ballast and the fifth with electronic ballast.

Furthermore, lighting circuits frequently involve long distances and combine with a

poorly diversified load. With individual power factor correction capacitors, the complex

LC circuit can approach a resonant condition around the third harmonic. Therefore,

these are significant enough reasons to oversize neutral wire lead connections in

transformers that feed installations with substantial amounts of fluorescent lighting.

Capacitor banks may be located adjacent to other loads and not necessarily as individual

power factor compensators at every lamp.


- 16 -

2.3.2.2. Converter

The increasing use of the power conditioners in which parameters like voltage and

frequency are varied to adapt to specific industrial and commercial processes has made

power converters the most widespread source of harmonics in distribution systems.

Electronic switching helps the task to rectify 50-/60-Hz AC into DC power. In DC

applications, the voltage is varied through adjusting the firing angle of the electronic

switching device. Basically, in the rectifying process, current is allowed to pass through

semiconductor devices during only a fraction of the fundamental frequency cycle, for

which power converters are often regarded as energy-saving devices. If energy is to be

used as AC but at a different frequency, the DC output from the converter is passed

through an electronic switching inverter that brings the DC power back to AC.

Converters can be grouped into the following categories:

Large power converters like those used in the metal smelter industry and in HVDC

transmission systems

Medium-size power converters like those used in the manufacturing industry for

motor speed control and in the railway industry

Small power rectifiers used in residential entertaining devices, including TV sets

and personal computers. Battery chargers are another example of small power

converters.

2.3.3. Future Sources of Harmonic

The challenge for electrical system designers in systems and industry is to design the

new systems and adapt the present systems to operate in environments with escalating

harmonic levels. The sources of harmonics in the electrical system of the future will be

diverse and more numerous. The problem grows complicated with the increased use of

sensitive electronics in industrial automated processes, personal computers, digital

communications, and multimedia.

Systems, who generally are not regarded as large generators of harmonics, may be

lining up to join current harmonic producers with the integration of distributed

resources in the rise. Photovoltaic, wind, natural gas, carbonate full cells, and even

hydrogen are expected to play increasingly important roles in managing the electricity


- 17 -

needs of the future. Distributed generators (DG) that presently provide support to

systems, especially during peak demand hours, will be joined by numerous harmonic

producing units, fueled by natural gas or even wind, called microturbines.

2.4. Effects of Harmonic Distortion in Power Systems

By the turn of the century, electronic loads have been increased continually. This is a

clear indication that residential customers are joining industrial and commercial

customers as harmonic current generators at a rapid pace. Considering the limited

awareness of residential customers on harmonics created by household equipment

(multiple TV sets, computers and entertaining devices, fluorescent lighting, etc.), power

systems may find it increasingly difficulty to set up rules for implementing remedial

actions at this user level. Even in commerce and industry, the concept of harmonic

filters is far from adequately well known. This is often due to the lack of information on

the effect that harmonics producing nonlinear loads can impose on sensitive industrial

processes and equipment and commercial applications.

As a result, a considerable number of electricity users are left exposed to the effects

of harmonic distortion on industrial, commercial, and residential loads. In a broad

manner, these can be described as the following.

2.4.1. Thermal Losses on Transformer

Modern industrial and commercial networks are increasingly influenced by

significant amounts of harmonic currents produced by a variety of nonlinear loads like

variable speed drives, electric and induction furnaces, and fluorescent lighting. Add to

the list uninterruptible power supplies and massive numbers of home entertaining

devices including personal computers.

All of these currents are sourced through service transformers. A particular aspect of

transformers is that, under saturation conditions, they become a source of harmonics.

Delta–wye or delta–delta-connected transformers trap zero sequence currents that would

otherwise overheat neutral conductors. The circulating currents in the delta increase the

RMS value of the current and produce additional heat. This is an important aspect to

watch. Currents measured on the high-voltage side of a delta-connected transformer will

not reflect the zero sequence currents but their effect in producing heat losses is there.


- 18 -

In general, harmonics losses occur from increased heat dissipation in the windings and

skin effect; both are a function of the square of the RMS current, as well as from eddy

currents and core losses. This extra heat can have a significant impact in reducing the

operating life of the transformer insulation. Transformers are a particular case of power

equipment that has experienced an evolution that allows them to operate in electrical

environments with considerable harmonic distortion. In industry applications in which

transformers are primarily loaded with nonlinear loads, continuous operation at or

above rated power can impose a high operating temperature, which can have a

significant impact on their lifetime.

2.4.2. Neutral Conductor Overloading

In single-phase circuits, return currents carrying significant amounts of harmonic

components flow through transformer neutral connections increasing the RMS current.

Furthermore, zero sequence currents (odd integer multiples of 3) add in phase in the

neutral. Therefore, the operation of transformers in harmonic environments demands

that neutral currents be evaluated in grounded-wye connected transformers to avoid the

possibility of missing the grounding connection as a consequence of overloading. In

balanced three-phase, four-wire systems, there is no current on the neutral, for which the

presence of neutral currents under these conditions should be attributed to the

circulation of zero sequence harmonics, which are mostly produced by single-phase

power supplies. In systems that are not entirely balanced, the unbalanced current

circulates on the return (neutral) conductor. Because this conductor is usually sized the

same as the phase conductors for being able to handle unbalanced currents comfortably,

it may experience overheating if those currents are subsequently amplified by zero

sequence currents. Large numbers of computers in office buildings make a formidable

source of harmonic currents produced by their electronic switched power supplies. A

common practice is to size neutral conductors to carry as much as two times the RMS

current that phase conductors can take. Monitoring temperature increase on the neutral

conductor of transformers might be a good start to detect whether zero sequence

harmonic currents are not overstressing neutral connections. This is true as long as the

system does not incur increased levels of current unbalance that would produce a

temperature rise in neutral conductor temperature.


- 19 -

2.4.3. Miscellaneous Effects on Capacitor Banks

Increased voltage can overstress and shorten the life of capacitor banks. Voltage,

temperature, and current stresses are the drivers of capacitor bank conditions that lead to

dielectric breakdown. Operating voltage can increase in distribution systems under light

load conditions or when fuse links operate to isolate a failed capacitor unit, leaving the

remaining units exposed to an overvoltage condition. For example, a 5% increase in the

nominal voltage of a capacitor unit would cause it to deliver.

Harmonic distortion is definitively another factor that contributes to impose voltage

stresses on capacitor banks. This is a serious condition in industrial facilities with

unfiltered large power converters. These operating limits are for continuous operation.

Thus, it will be important to take into account these limits also in the design of

harmonic filters because capacitor banks in single–tuned filters are meant to act as a

sink for the entire amount of harmonic currents of the corresponding tuned frequency.

2.4.3.1. Resonant Condition

The resonant conditions involve the reactance of a capacitor bank that at some point

in frequency equals the inductive reactance of the distribution system, which has an

opposite polarity. These two elements combine to produce series or parallel resonance.

In the case of series resonance, the total impedance at the resonance frequency is

reduced exclusively to the resistive circuit component. If this component is small, large

values of current at such frequency will be developed. In the case of parallel resonance,

the total impedance at the resonant frequency is very large (theoretically tending to

infinite). This condition may produce a large overvoltage between the

parallel-connected elements, even under small harmonic currents. Therefore, resonant

conditions may represent a hazard for solid insulation in cables and transformer

windings and for the capacitor bank and their protective devices as well. Resonant

frequencies can be anticipated if the short-circuit current level at the point where the

capacitor bank is installed is known.

Note how changing any of these parameters can shift the resonant frequency. This is

a practice actually used sometimes in certain applications involving excessive heating in

transformers connected to non-linear loads. If this frequency coincides with a


- 20 -

characteristic harmonic present at the site, that current will see large upstream

impedance and the existing voltage harmonic distortion will be amplified. Capacitor

banks can be applied without concern for resonance conditions as long as the nonlinear

load and capacitor bank are less than 30% and 20%, respectively, the rated of the

transformer, assuming a typical transformer impedance around 5% to 6%. Otherwise,

the capacitors should be used as a harmonic filter, with a series reactor that tunes them

to one of the characteristic harmonics of the load. Generally, fifth and seventh

harmonics are the most commonly found and account for the largest harmonic currents.

2.4.3.2. Unexpected Fuse Operation

As mentioned earlier, RMS voltage and current values may increase under harmonic

distortion. This can produce undesired operation of fuses in capacitor banks or in

laterals feeding industrial facilities that operate large nonlinear loads. Capacitor banks

can be further stressed under the operation of a fuse on one of the phases, which leaves

the remaining units connected across the other phases. They are thus left subject to an

unbalanced voltage condition that can produce over voltages and detune passive

harmonic filters if they are not provided with an unbalance detection feature.

2.4.4. Abnormal Operation of Electronic Relay

Variable Frequency Drive (VFD) operation leading to shut-down conditions is often

experienced in applications involving oil fields in which solid material (sand) abruptly

demand higher thrust power, mining works in which sudden increases in lifting power

occur, and high inertia loads, among others. In all these cases, the protective relays trip

as a response to over currents exceeding the established settings. Similar effects can be

experienced under the swift appearance of harmonic distortion on current or voltage

waveforms exceeding peak or RMS preset thresholds. Therefore, when protective relays

trigger during the operation of a nonlinear load, harmonic distortion should be assessed.

It might well be that an unpredicted overloading condition is the cause of the

unexpected operation, but often increased harmonic levels following nonlinear load

growth are the reason for similar behavior. On the other hand, third harmonic currents

produced by severe line current unbalance may cause nuisance relay tripping in VFD


- 21 -

applications. Therefore, nuisance and missed relay tripping in installations with

nonlinear loads should be assessed by checking the harmonic distortion levels and by

inspecting the relays for possible threshold-setting fine-tuning. The onset of this type of

occurrence in industrial installations may be used as an warning to start considering

harmonic filtering actions.

2.5. Limits of Harmonic Distortion

The most widespread standards for harmonic control worldwide are due to Institute

of Electrical and Electronics Engineers (IEEE) in the U.S. In 1981, the IEEE issued

Standard 519-1981 which aimed to provide guidelines and recommended practices for

commutation notching, voltage distortion, telephone influence, and flicker limits

produced by power converters. The standard contended with cumulative effects but did

little to consider the strong interaction between harmonic producers and power system

operation. The main focus of the revised IEEE-519 standard in 1992 was a more

suitable stance in which limitations on customers regarding maximum amount of

harmonic currents at the connection point with the power utility did not pose a threat for

excessive voltage distortion. This revision also implied a commitment by power

companies to verify that any remedial measures taken by customers to reduce harmonic

injection into the distribution system would reduce the voltage distortion to tolerable

limits. The interrelation of these criteria shows that the harmonic problem is a system,

and not a site problem. Compliance with this standard requires verification of harmonic

limits at the interface between systems and customers, more commonly known as PCC.


- 22 -

2.5.1. IEEE Limits

Limits of allowable voltage distortion set by IEEE 519 are provided in Table 2-1.

Table 2-1. IEEE 519 voltage distortion limits.

Bus voltage at PCC Individual %,h

V Voltage THD, %

kV69V 3.0 5.0

kV161kV69 V 1.5 2.5

kV161V 1.0 1.5

Given in Table 2-2, IEEE 519 provides for harmonic current distortion in general

distribution, transmission systems based on the compared to the system’s short –circuit capacity (the ratio

LscII / ).

Table 2-2. IEEE 519 current distortion limits.

LscII / Lh

II / , % - General distribution systems ( kV69120V V ) TDD

(%) 11h 1711 h 2317 h 3523 h 35h

<20 4.0 2.0 1.5 0.6 0.3 5

20-50 7.0 3.5 2.5 1.0 0.5 8

50-100 10 4.5 4.0 1.5 0.7 12

100-1000 12 5.5 5.0 2.0 1.0 15

>1000 15 7.0 6.0 2.5 1.4 20

LscII / Lh

II / , % - General distribution systems ( kV161kV69 V ) TDD

(%) 11h 1711 h 2317 h 3523 h 35h

Limits are half those for general distribution systems.

LscII / Lh

II / , % - General distribution systems ( VkV161 ) TDD

(%) 11h 1711 h 2317 h 3523 h 35h

<50 2.0 1.0 0.75 0.3 0.15 2.5

50 3.0 1.5 1.15 0.45 0.22 3.75


- 23 -

2.5.2. Limits of Japanese

Japan also has the standard of harmonic distortion, where the voltage limits are same

as IEEE 519 voltage limits. The current limits are provided in Table 2-3.

Table 2-3. Japan current distortion limits ( kW/mA ).

Bus voltage at

PCC 5h 7h 11h 13h 17h 19h 23h 23h

kV6.6 3.5 2.5 1.6 1.3 1.0 0.90 0.76 0.70

kV22 1.8 1.3 0.82 0.69 0.53 0.47 0.39 0.36

kV33 1.2 0.86 0.55 0.46 0.35 0.32 0.26 0.24

kV66 0.59 0.42 0.27 0.23 0.17 0.16 0.13 0.12

kV77 0.5 0.36 0.23 0.19 0.15 0.13 0.11 0.10

kV110 0.35 0.25 0.16 0.13 0.10 0.09 0.07 0.07

kV154 0.25 0.18 0.11 0.09 0.07 0.06 0.05 0.05

kV220 0.17 0.12 0.08 0.06 0.05 0.04 0.03 0.03

kV275 0.14 0.10 0.06 0.05 0.04 0.03 0.03 0.02

2.6. Mitigation of Power Systems Harmonics

Todays, various harmonic mitigation techniques are available to solve harmonic

problems in power systems. One of the most common methods for control of harmonic

distortion in industry is the use of passive filtering techniques that make use of

single-tuned or band-pass filters. Passive harmonic filters can be designed as

single-tuned elements that provide a low impedance path to harmonic currents at a

punctual frequency or as band-pass devices that can filter harmonics over a certain

frequency bandwidth. The more sophisticated active filtering concepts operate in a wide

frequency range, adjusting their operation to the resultant harmonic spectrum. In this

way, they are designed to inject harmonic currents to counterbalance existing harmonic

components as they show up in the distribution system. Active filters comprise CD, CA,

series, and parallel configurations.


- 24 -

Figure 2-2. Typical single tuned filters

2.6.1. Passive Harmonic Filters

2.6.1.1. Single Tuned Filter

Harmonics can be efficiently reduced through the use of a passive filter which

consists, basically, of a series combination of a capacitor and a reactor tuned to specific

harmonic frequency. The most common harmonic filter in industrial applications is the

passive single tuned filters that shown in Figure 2-2. The passive single tuned filters

present very low impedance at the tuning frequency. Thus, passive filter design must

take into account expected growth in harmonic current sources or load reconfiguration

because it can otherwise be exposed to overloading, which can rapidly develop into

extreme overheating and thermal breakdown. The design of a passive filter requires a

precise knowledge of the harmonic-producing load and of the power system. A great

deal of simulation work is often required to test its performance under varying load

conditions or changes in the topology of the systems.

Because passive filters always provide reactive compensation to a degree dictated by

the voltampere size and voltage of the capacitor bank used, they can in fact be designed

for the double purpose of providing the filtering action and compensating power factor

to the desired level. If more than one filter is used — for example, sets of 5th and 7th or

11th and 13th branches — it will be important to remember that all of them will provide

a certain amount of reactive compensation.

As discussed earlier, this filter is a series combination of an inductance and a

capacitance. In reality, in the absence of a physically designed resistor, there will always

be a series resistance, which is the intrinsic resistance of the series reactor sometimes

used as a means to avoid filter overheating. All harmonic currents whose frequency


- 25 -

coincides with that of the tuned filter will find a low impedance path through the filter.

The resonant frequency of this filter can be expressed by the following expression:

LCf r

21

....................................................................................... (2.14)

where:

rf : resonant frequency in hertz

L : filter inductance in henrys

C : filter capacitance in farads

The process of designing a filter is a compromise among several factors: low

maintenance, economy, and reliability. The design of the simplest filter that does the

desired job is what will be sought in the majority of cases. The steps to set up a

harmonic filter using basic relationships to allow for a reliable operation can be

summarized as follows:

1. Calculate the value of the capacitance needed to improve the power factor and to

eliminate any penalty by the electric power company. Power factor compensation

is generally applied to raise power factor to around 0.95 or higher.

2. Choose a reactor to tune the series capacitor to the desired harmonic frequency.

For example, in a six-pulse converter, this would start at the fifth harmonic and it

would involve lower frequencies in an arc furnace application.

3. Calculate the peak voltage at the capacitor terminals and the rms reactor current.

4. Choose standard components for the filter and verify filter performance to assure

that capacitor components will operate within recommended limits. This may

require a number of iterations until desired reduction of harmonic levels is

achieved.

2.6.1.2. High Pass Filters

Band-pass filters, high-pass in particular, are known by their small impedance value

above the corner frequency. Typical frequency response of a high-band pass filter is

shown in Figure 2-3. This filter draws a considerable percentage of frequency harmonic

currents above the corner frequency. Therefore, this frequency must be placed below all


- 26 -

Figure 2-3. Typical high pass filter

harmonic currents that have an important presence in the installation. In planning to

adopt a high-pass filter as a harmonic mitigating measure, the following aspects should

be considered:

The impedance-frequency characteristic of a high-pass filter will entail a very

different filtering action as compared with that provided by a single tuned filter.

Harmonic current elimination using a high-pass filter may require a quite different

sizing of filter elements, particularly of the capacitor bank, compared with a

single-tuned filter. For example, a 3-MVAR bank used in a fifth harmonic filter in a

60-Hz application may fall short in size when used as part of a high-pass filter with a

corner frequency of 300 Hz. Obviously, this will very much depend on the additional

harmonic currents that the high-pass filter will be draining off. First-order high-pass

filters are characterized by large power losses at fundamental frequency, for which they

are less common. The second-order high-pass filter is the simplest to apply; it provides

a fairly good filtering action and reduces energy losses at fundamental frequency. The

third-order high-pass filter presents greater operating losses than the one, second-order

high-pass filter and is less effective in its filtering action.

2.6.2. Active Power Filters

An Active Power Filter (APF) generates a harmonic current spectrum that is opposite

in phase to the distorted harmonic current it measures. The typical APF is shown in

Figure 2-4. Harmonics are thus cancelled and the result is a non-distorted sinusoidal

current. An APF is a high performance power electronics converter and can operate in

different modes: harmonics elimination, power factor correction, voltage regulation and

load unbalance compensation. Different control approaches are possible but they all

share a common objective: imposing sinusoidal currents in the grid, eventually with


- 27 -

Figure 2-4. Typical active power filters.

unity power factor, even in the case of highly distorted mains voltage. As a

demonstration of the capabilities of the APF, one control approach has been selected. It

uses a simple and robust power circuit interface without load current sensors and an

efficient signal processing without heavy or complex computations.

Different approaches such as notch filter, scalar control, instantaneous reactive power

theory, synchronous detection method, synchronous d–q frame method, can be used to

improve the active filter performance.

Usually, the voltage-source is preferred over the current-source to implement the

parallel APF since it has some advantages. In this Chapter it is used the voltage-source

parallel topology, schematically shown in Figure 2-4. The filter generates currents in the

connection point in order to:

Cancel/minimize the harmonic content in the AC system.

Correct the power factor at fundamental frequency.

Regulate the voltage magnitude.

Balance loads.

So, the AC distribution system only carries the active fundamental component of the

load current. Very different current control algorithms can be applied to the active filter.

The current reference for the active filter connection node usually satisfies one of the

two following strategies:

Power factor correction, harmonic elimination, and load unbalance compensation.

Voltage regulation, harmonic elimination, and load unbalance compensation.

The voltage regulation strategy is a concurrent objective faced to the power factor


- 28 -

compensation because the two depend on the reactive current. However, any control

algorithm has enough flexibility to be configured, in real-time, to either objectives or for

the two, in a weighted form.

The filter performance should be evaluated in a typical distribution system with

different loads, linear and non-linear. The relevant performance indexes will be

characterized by the THD of the mains current, with and without filter, in the following

two indexes basis: filter effectiveness index and filter capacity index. The Filtering

Effectiveness index (FE) is the relation between the total harmonic distortion of the

current supplied by the mains with and without filter in a pre-defined frequency range.

THD

THDFE APF ...................................................................................... (2.15)

The Filtering Capacity index (FC) is the relation between the total apparent power

supplied by the filter and the total mains or load apparent power:

Load

APF

S

SFC .......................................................................................... (2.16)

These two indexes are the basis for evaluating the filter performance in static

operation. In transient operation, only special conditions can be evaluated and they

usually are not under the restrictions of power quality measurements.


- 29 -

2.7. Conclusion

The wide spread utilization of power electronic devices has significantly increased the

number of harmonic generating apparatus in the power systems. The harmonics

distortions of the voltage and current have adverse effects on electrical equipment. The

harmonics effect on power systems can be summarized as increase losses of devices,

equipment heating and loss of life, and interference with protection, control and

communication circuits as well as customer loads. Harmonics have already became one

of the major power quality concerns. The estimation of harmonic from nonlinear loads is

the first step in a harmonic analysis and this may not be straightforward task. To eliminate

this situation, the harmonic analysis becomes an important and necessary task for

engineers in power systems.


- 30 -

Chapter 3 Independent Component Analysis

- 31 -

CCHHAAPPTTEERR 33

Independent Component Analysis

3.1. Introduction

BSS techniques have received attention in applications where there is little or no

information available on the underlying physical environment and the sources. BSS

algorithms estimate the source signals from observed mixtures. The word ‘blind’

emphasizes that the source signals and the way the sources are mixed, i.e. the mixing

model parameters, are unknown. A well-known example for the BSS is the cocktail

party problem [14]. Assume that four people are speaking simultaneously in a room,

and there are four microphones located in different parts of the room recording the

sounds in the room. The recordings from the microphones will be the mixture of the

individuals’ speeches. In this problem neither the individuals’ speeches nor the

distances of individuals from the microphones are known. The only known quantity is

the recordings from the microphones. The object of the BSS is to recover the

individuals’ speeches from the recorded signals. The BSS is a difficult problem to solve.

However, some properties of the sources make the problem solvable.

The important BSS technique is the ICA [14]-[18] which is based on the statistical

independence and the sparsity of the source signal. In this problem source signals are

the speeches of the individuals and measurement signals are the recordings of the

microphones. Independent component analysis was originally developed to deal with

problems that are closely related to the cocktail-party problem. Since the recent increase

of interest in ICA, it has become clear that this principle has a lot of other interesting

applications as well. Consider, for example, electrical recordings of brain activity as

given by an EEG. These potentials are presumably generated by mixing some

underlying components of brain and muscle activity. This situation is quite similar to

the cocktail-party problem: we would like to find the original components of brain


- 32 -

activity, but we can only observe mixtures of the components. ICA can reveal

interesting information on brain activity by giving access to its independent

components.

3.2. Linear Model based ICA

ICA is a BSS algorithm which transforms the observed signals into mutually,

statistically independent signals. It thus exploits the statistical independence between

the sources. Statistical properties of signals are a key factor in estimation by ICA, since

there is almost no other information available. Assuming there are N sources and M measurements, neglecting the noise term,

then the linear mixing model of ICA can be written as:

)()()()(

)()()()()()()()(

2211

22221212

12121111

tsatsatsatx

tsatsatsatx

tsatsatsatx

NMNMMM

NN

NN

.............................................. (3.1)

In a compact form (3.1) is

)()( tt Asx ........................................................................................... (3.2)

where T

mxxx ],,,[ 21 x : M-dimensional vector of observed signals T

nsss ],,,[ 21 s : N -dimensional vector of unknown source signals ][ mnaA : M×N unknown matrix called mixing matrix

t : Time or sample index with .,,2,1 Tt

The mna are some parameters that depend on the mixing condition. It would be very

useful if you could now estimate the original speech signals s , only using the recorded

signals x .

The mixing model given above is an instantaneous mixing model which means that

there is no time delay in the mixing. Neglecting the noise term, the matrix


- 33 -

representation of (3.2) is given as:

Asx .................................................................................................. (3.3)

Here x and s are TM and TN matrices, A is an NM full column

rank matrix. There if we knew the mixing parameters mna , we could solve the linear equation in

(3.1) simply by inverting the linear system. The point is, however, that here we know neither the mna nor the s , so the problem is considerably more difficult. One approach

to solving this problem would be to use some information on the statistical properties of the signals s to estimate both the mna and the s .

Actually, and perhaps surprisingly, it turns out that it is enough to assume that s are,

at each time instant t , statistically independent. This is not an unrealistic assumption in

many cases, and it need not be exactly true in practice. Independent component analysis can be used to estimate the mna based on the information of their independence, and

this allows us to separate the original signals s from their mixtures x .

The objective of the ICA is to find estimates of the s and A from the available

observations x . The recovery model can be written as

)()()()(

)()()()()()()()(

2211

22221212

12121111

txwtxwtxwty

txwtxwtxwty

txwtxwtxwty

MNMNNN

MM

MM

............................................ (3.4)

where T

nxxx ],,,[ 21 x : M-dimensional vector of observed signals T

nyyy ],,,[ 21 y : N-dimensional vector of separated signals ][ ijwW : N×M estimated matrix called separating matrix.

Rewriting (3.4) in matrix form as:

Wxy ................................................................................................. (3.5)


- 34 -

Where y is an estimate of the sources s with the dimension of TN and W is an

MN matrix called separating matrix which is the pseudo-inverse of matrix A . The

mixing matrix A and the separating matrix W are assumed to be constant over the

mixing process.

3.3. Restrictions in ICA

To make sure that the basic ICA model just given can be estimated, source and mixed

signals should satisfy the following properties.

1. The independent components are assumed statistically independent.

This is the principle on which ICA rests. Surprisingly, not much more than this

assumption is needed to ascertain that the model can be estimated. This is why ICA is

such a powerful method with applications in many different areas. Basically, random variables

nyyy ,,, 21 are said to be independent if information

on the value of iy does not give any information on the value of jy for ji .

Technically, independence can be defined by the probability densities. Denote by ),,,( 21 n

yyyp the joint Probability Density Function (PDF) of the iy , and by )( iyp

the marginal PDF of iy , i.e., the PDF of iy when it is considered alone. Then the iy

are independent if and only if the joint PDF is factorable in the following way:

)()()(),,,( 2121 nn ypypypyyyp .............................. (3.6)

2. The independent components must have nongaussian distributions.

Sources should have nongaussian distributions. Objective functions used in ICA

estimation are based on the higher order statistics. However, for Gaussian distributions

higher-order statistics are either zero or contain redundant information. Also, the joint

probability distribution of Gaussian variables is rotationally symmetric under

orthogonal transformation assuming that the data is whitened. This is the result of the

property that the uncorrelated jointly Gaussian variables are necessarily independent.

Therefore only a single Gaussian variable can be estimated. Several Gaussian

distributed random variables can not be distinguished and thus not estimated by ICA.

3. For simplicity, we assume that the unknown mixing matrix is square.


- 35 -

In other words, the number of independent components is equal to the number of

observed mixtures. It simplifies the estimation very much. Then, after estimating the

matrix A , we can compute its inverse 1A , and obtain the independent components simply by

xAs 1 ........................................................................................... (3.7)

It is also assumed here that the mixing matrix is invertible. If this is not the case,

there are redundant mixtures that could be omitted, in which case the matrix would not

be square; then find again the case where the number of mixtures is not equal to the

number of Independent Components (ICs). Thus, under the preceding three assumptions

(or at the minimum, the two first ones), the ICA model is identifiable, meaning that the

mixing matrix and the ICs can be estimated up to some trivial indeterminacies.

3.4. Ambiguities of ICA

In the ICA model in (3.3), it is easy to see that the following ambiguities or

indeterminacies will necessarily hold:

1. Cannot determine the variances (energies) of the independent components.

The reason is that, both s and A being unknown, any scalar multiplier in one of the sources is could always be canceled by dividing the corresponding column ia of

A by the same scalar i as below:

i

ii

i

i sx

ia1 .................................................................... (3.8)

As a consequence, we may quite as well fix the magnitudes of the independent

components. Since they are random variables, the most natural way to do this is to assume that each has unit variance: 12 isE . Then the matrix A will be adapted in the

ICA solution methods to take into account this restriction. Note that this still leaves the

ambiguity of the sign: we could multiply an independent component by 1 without

affecting the model.


- 36 -

2. Cannot determine the order of the independent components.

The reason is that, again both s and A being unknown, it can be freely change the

order of the terms of estimated ICs, and call any of the independent components the first

one. A non-singular diagonal matrix K and its inverse 1K can be multiplied by the mixing matrix A without changing the measurement matrix x as below:

sAsAKKx 1 ................................................................................ (3.9)

Ordering indeterminacy can be eliminated by prior information about the sources.

Scaling and ordering indeterminacy can be combined into one equation

KWA ........................................................................................... (3.10)

In the ICA problem, sources s , measurements x , mixing matrix A and the

separating matrix W can be real or complex valued. In this thesis sources,

measurements and the mixing matrix are complex valued.

3.5. Centering and Whitening

3.5.1. Centering the Variables

Without loss of generality, it can be assume that both the mixture variables and the

independent components have zero mean. This assumption simplifies the theory and

algorithms quite a lot. If the assumption of zero mean is not true, we can do some

preprocessing to make it hold. This is possible by centering the observable variables.

This means that the original mixtures x are preprocessed by

xxx E ...................................................................................... (3.11)

before doing ICA. Thus the independent components are made zero mean as well, since

xAs EE 1 ................................................................................ (3.12)


- 37 -

The mixing matrix, on the other hand, remains the same after this preprocessing, so we

can always do this without affecting the estimation of the mixing matrix. After

estimating the mixing matrix and the independent components for the zero-mean data, the subtracted mean can be simply reconstructed by adding xA E1 to the zero-mean

independent components.

3.5.2. Whiting

As already discussed in above, the ICA problem is greatly simplified if the observed

mixture vectors are first whitened or sphered. A zero-mean random vector Tnzz 1z is said to be white if its elements iz are uncorrelated and have unit

variances

ijji zz E ....................................................................................... (3.13)

In terms of the covariance matrix, this obviously means that ,IzzE T with I the unit matrix. The best-known example is white noise; then the elements iz would be

the intensities of noise at consequent time points ,2,1i and there are no temporal

correlations in the noise process. The term “white” comes from the fact that the power

spectrum of white noise is constant over all frequencies, somewhat like the spectrum of

white light contains all colors.

Because whitening is essentially decorrelation followed by scaling, the technique of

PCA can be used. This implies that whitening can be done with a linear operation. The

problem of whitening is now: given a random vector x with n elements, find a linear

transformation V into another vector z such that

Vxz ............................................................................................... (3.14)

is white.

The problem has a straightforward solution in terms of the PCA expansion. Let neeeE ,, 21 be the matrix whose columns are the unit-norm eigenvectors of the

covariance matrix TxxECx . These can be computed from a sample of the vectors x either directly or by one of the on-line PCA learning rules. Let ndddiag 1D


- 38 -

be the diagonal matrix of the eigenvalues of xC .Then a linear whitening transform is

given by

TΕDV 21

........................................................................................ (3.15)

This matrix always exists when the eigenvalues 1d are positive; this is not a

restriction that xC is positive semidefinite, in practice positive definite for almost any

natural data, so its eigenvalues will be positive.

It is easy to show that the matrix V of (3.15) is indeed a whitening transformation. Recalling that xC can be written in terms of its eigenvector and eigenvalue matrices

E and D as TEDECx , with E an orthogonal matrix satisfying IEEEE TT ,

it holds:

IEDEDEEDVxxVEzzE 2/12/1 TTTTT ............................ (3.16)

The covariance of z is the unit matrix, hence z is white.

The linear operator V of (3.15) is by no means the only unique whitening matrix. It

is easy to see that any matrix UV , with U an orthogonal matrix, is also a whitening

matrix. This is because for UVxz it holds:

IUIUUVxxUVEzzE TTTTT ............................................. (3.17)

An important instance is the matrix TΕΕD 2/1 . This is a whitening matrix because it is obtained by multiplying V of (3.15) from the left by the orthogonal matrix Ε . This

matrix is called the inverse square root of xC , and denoted by 2/1xC , because it comes

from the standard extension of square roots to matrices.


- 39 -

Figure 3-1. The process of ICA estimation.

3.6. ICA by Maximization of Nongaussianity

Nongaussianity is actually of paramount importance in ICA estimation. Without

nongaussianity the estimation is not possible at all. Therefore, it is not surprising that

nongaussianity could be used as a leading principle in ICA estimation. This is at the

same time probably the main reason for the rather late resurgence of ICA research: In

most of classic statistical theory, random variables are assumed to have Gaussian

distributions, thus precluding methods related to ICA.

The simple process of ICA is shown in Figure 3-1. Estimation of the separating

matrix W is an iterative process based on the optimization of an objective function. The

methods based on finding the columns of the mixing matrix from the edges of the joint

distribution of mixtures can be used when the problem is simple. The state-of-the-art

ICA algorithms are based on the statistical properties of the signals. The main properties

that the ICA exploits are the statistical independence and the higher order statistics

which are discussed in the following sections.


- 40 -

3.6.1. Nongaussian is Independent

The central limit theorem is a classic result in probability theory that the distribution

of a sum of independent random variables tends toward a Gaussian distribution, under

certain conditions. Loosely speaking, a sum of two independent random variables

usually has a distribution that is closer to Gaussian than any of the two original random

variables.

Assume that the data vector x is distributed according to the ICA data model (3.2),

which is a mixture of independent components. For pedagogical purposes, assume in

this motivating section that all the independent components have identical distributions.

Estimating the independent components can be accomplished by finding the right linear

combinations of the mixture variables based on (3.5). Thus, to estimate one of the independent components, consider a linear combination of the ix , which denote by

(3.18)

i

ii

T xby xb ....................................................................... (3.18)

where b is a vector to be determined. Then, take (3.2) into (3.18) can get the follow

equation:

sAbAsb )()( TTy ................................................................ (3.19)

Thus, y is a certain linear combination of the is , with coefficients given by AbT . Denote this vector by q . Then we have

i

ii

TT sqy sqxb ............................................................. (3.20)

If b were one of the rows of the inverse of A , this linear combination xbT would actually equal one of the independent components. In that case, the corresponding q

would be such that just one of its elements is 1 and all the others are zero.


- 41 -

Next, vary the coefficients in q to see how the distribution of sqTy changes in

(3.20). The fundamental idea is that since a sum of even two independent random variables is more Gaussian than the original variables, sqTy is usually more Gaussian than any of the is and becomes least Gaussian when it in fact equals one of

the is . In this case, obviously only one of the elements iq of q is nonzero.

Therefore, it is could take as b a vector that maximizes the nongaussianity of xbT . Such a vector would necessarily correspond to a bAq T , which has only one nonzero

component. This means that sqxb TTy equals one of the independent components.

Maximizing the nongaussianity of xbT thus gives us one of the independent components.

3.6.2. Fixed-Point Algorithm of Real Values

Several ICA algorithms have been derived from the statistical measures explained in

the passed years. This section presents the well known ICA algorithms: Fixed point

algorithms, so called Fast-ICA algorithms are based on the fixed point iteration scheme.

Fast-ICA algorithms are developed based on the measure of the kurtosis in [16] and

negentropy in [17]. Here the Fast-ICA algorithm is presented based on the negentropy.

3.6.2.1. Negentropy as Nongaussianty Measure

Negentropy is based on the information-theoretic quantity of differential entropy.

Entropy is the basic concept of information theory. The entropy of a random variable is

related to the information that the observation of the variable gives. The more “random”,

(unpredictable and unstructured the variable) is the larger its entropy. The (differential)

entropy H of a random vector X defined as

i

ii aXPaXPXH )(log)()( .............................. (3.21)

A fundamental result of information theory is that a Gaussian variable has the largest

entropy among all random variables of equal variance. This means that entropy could be

used as a measure of nongaussianity. Entropy is small for distributions that are clearly

concentrated on certain values when the variable is clearly clustered, or has a PDF that

is very “spiky”.


- 42 -

To obtain a measure of nongaussianity that is zero for a Gaussian variable and always

nonnegative, one often uses a normalized version of differential entropy, called negentropy. Negentropy J of random vector y is defined as follows

)()()( yHyHyJ gauss .......................................................... (3.22)

where gaussy is a Gaussian random vector of the same correlation (and covariance)

matrix as y . Due to the above-mentioned properties, negentropy is always nonnegative,

and it is zero if and only if y has a Gaussian distribution. Negentropy has the additional

interesting property that it is invariant for invertible linear transformations.

The advantage of using negentropy as a measure of nongaussianity is that it is well

justified by statistical theory. In fact, negentropy is in some sense the optimal estimator

of nongaussianity, as far as the statistical performance is concerned. The disadvantage

of using negentropy is computationally very difficult. Estimating negentropy using the

definition would require an estimate (possibly nonparametric) of the PDF. Therefore,

simpler approximations of negentropy are very useful, as will be discussed next. These

will be used to derive an efficient method for ICA.

3.6.2.2. Approximating Negentropy

The classic method of approximating negentropy is using higher-order cumulants,

using the polynomial density. This gives the approximation as:

223

481

121)( ykurtyEyJ ............................................. (3.23)

The random variable y is assumed to be of zero mean and unit variance.

In the case where use only one nonquadratic function G , the approximation becomes

2])()([)( GEyGEkyJ .............................................. (3.24)

where


- 43 -

k is positive constant

is a gaussian variable of zero mean and unit variance

G is two nonquadratic function.

This is a generalization of the momentbased approximation in (3.22) if y has a

symmetric distribution, in which case the first term in (3.22) vanishes. Indeed, taking 4)( yyG , one then obtains a kurtosis-based approximation. In particular, choosing a

G that does not grow too fast, one obtains more robust estimators. The following

choices of G have proved very useful:

yayg 11 tanh)( ....................................................................... (3.25)

2/exp)( 22 yyyg .............................................................. (3.26)

33 )( yyg ..................................................................................... (3.27)

where 21 1 a is some suitable constant, often taken equal to 1.

3.6.2.3. Fixed-Point Algorithm using Negentropy

Fast-ICA is based on a fixed-point iteration scheme for finding a maximum of the

nongaussianity of as measured in (3.2). More rigorously, it can be derived as an

approximative Newton iteration. The Fast-ICA algorithm using negentropy combines

the superior algorithmic properties resulting from the fixed-point iteration with the

preferable statistical properties due to negentropy.

The following fixed-point iteration can be used in Fast-ICA algorithm.

wzwzwzw TT gEgE .......................................... (3.28)


- 44 -

3.6.2.4. Estimating Several Independent Components

The key to extending the method of maximum nongaussianity to estimate more independent component is based on the following property: The vectors iw

corresponding to different independent components are orthogonal in the whitened

space. To recapitulate, the independence of the components requires that they are uncorrelated, and in the whitened space we have j

T

i

T

j

T

iE wwzwzw , and therefore

uncorrelatedness in equivalent to orthogonality. This property is a direct consequence of

the fact that after whitening, the mixing matrix can be taken to be orthogonal. The

iw are in fact by definition the rows of the inverse of the mixing matrix, and these are

equal to the columns of the mixing matrix, because by orthogonality TAA 1 . Thus, to estimate several independent components, we need to run any of the one unit

algorithms several times with vectors nww ,,1 , and to prevent different vectors from

converging to the same maxima we must orthogonalize the vectors nww ,,1 after

every iteration. A simple way of orthogonalization is deflationary orthogonalization

using the Gram-Schmidt method. This means that it estimate the independent

components one by one. The orthogonalization is shown below:

j

p

j

j

T

ppp wwwww

1

1 ........................................................ (3.29)

Then give a detailed version of the Fast-ICA algorithm that uses the symmetric

orthogonalization as below:

1. Center the data to make its mean zero.

2. Whiten the data to give z . 3. Choose m , the number of ICs to estimate. Set counter 1p . 4. Choose an initial value of unit norm for pw randomly.

5. Let wzwzwzw TT gEgE , where g is defined as in

(3.25)–(3.27). 6. Normalize pw by dividing it by its norm.


- 45 -

7. Do the following orthogonalization: j

p

j

j

T

ppp wwwww

1

1

8. Let ppp www .

9. If pw has not converged, go back to step 5.

10. Set 1 pp . If mp , go back to step 4.

3.6.3. Fixed-Point Algorithm of Complex Values

Sometimes in ICA, the ICs and/or the mixing matrix are complex-valued. For

example, in signal processing in some cases frequency (Fourier) domain representations

of signals have advantages over time-domain representations. Especially in the

separation of convolutive mixtures, it is quite common to Fourier transform the signals,

which results in complex-valued signals. Furthermore, in the system impedance

estimation and harmonic current contribution problem which discussing in this thesis

are used the complex value Fast-ICA, where the measurement data (harmonic current,

harmonic voltage) are complex values.

In this chapter, we show how the Fast-ICA algorithm can be extended to complex

valued signals [18]. Both the independent component s and the observed mixtures x

assume complex values. For simplicity, we assume that the number of independent

component variables is the same as the number of observed linear mixtures. The mixing

matrix A is of full rank and it may be complex as well. In addition to the assumption of the independence of the components is , an assumption on the dependence of the real

and complex parts of a single IC is made here. Assume that every is is white in the

sense that the real and imaginary parts of is are uncorrelated and their variances are

equal; this is quite realistic in practical problems.

3.6.3.1. Basic Concepts of Complex Random Variables

A complex random variable y can be represented as ivuy where u and v

are real-valued random variables. The density of y is Rvufyf , . The

expectation of y is viEuEyE . Two complex random variables 1y and 2y

are uncorrelated if *21

*21 yEyEyyE , where ivuy * designates the complex

conjugate of y . The covariance matrix of a zero-mean complex random vector


- 46 -

nyy ,1y is

nnn

n

H

CC

CC

E

1

111

yy ....................................................... (3.30)

where *kjjk yyEC and Hy stands for the Hermitian of y , that is, y transposed

and conjugated. The data can be whitened in the usual way. In this complex ICA model, all is have zero mean and unit variance. Moreover, it is

required that they have uncorrelated real and imaginary parts of equal variances. This can be equivalently expressed as ISS HE and 0TE SS . In the latter, the

expectation of the outer product of a complex random vector without the conjugate is a null matrix. These assumptions imply that is must be strictly complex; that is, the

imaginary part of is may not in general vanish.

For a zero-mean complex random variable, the definition of kurtosis can be easily

generalized as:

yyEyyEyyEyyE

yyEyyEyEykurt

****

**4)(

........................... (3.31)

but the definitions vary with respect to the placement of conjugates * actually, there

are 42 ways to define the kurtosis. We choose the definition as:

2

2)(4

22224

yE

yEyEyEykurt ................. (3.32)

where the last equality holds if y is white, i.e., the real and imaginary parts of y are

uncorrelated and their variances are equal to 21 . This definition of kurtosis is intuitive since it vanishes if y is Gaussian.


- 47 -

3.6.3.2. Indeterminacy of Independent Components

The independent components S in the ICA model are found by searching for a

matrix B such that Bxs . However, as in basic ICA, there are some indeterminacies. In the real case, a scalar factor i can be exchanged between is and a column ia of

A without changing the distribution of x : iiiiii ss 1 aa . In other words, the

order, the signs and the scaling of the independent components cannot be determined. Usually one defines the absolute scaling by defining 12 isE ; thus only the signs of

the independent components are indetermined. Similarly in the complex case there is an unknown phase jv for each js . Let us

write the decomposition

iiiiii svvs 1 aa .............................................................................. (3.33)

where the modulus of jv is equal to one. If is has a spherically symmetric

distribution, themultiplication by a variable iv does not change the distribution of is .

Thus the distribution of x remains unchanged as well. From this indeterminacy it follows that it is impossible to retain the phases of is , and BA is a matrix where in

each row and each column there is one nonzero element that is of unit modulus.

3.6.3.3. Choice of the Nongaussianity Measure

In the complex values, the distributions for the complex variables are often

spherically symmetric, so only the modulus is interesting. Thus it could be used a

nongaussianity measure that is based on the modulus only. Based on the measure of

nongaussianity as in (3.24), we use the following:

2

ZWWHGEJG .......................................................... (3.34)

where G is a smooth even function, w is an n-dimensional complex vector where


- 48 -

12

wzwHGE ......................................................... (3.35)

Maximizing GJ it is just estimate one IC. Estimating n independent components

is possible, just as in the real case, by using a sum of n measures of nongaussianity,

and a constraint of orthogonality. Thus one obtains the following optimization problem:

jkjHk

E

ww ....................................................................... (3.36)

under constraint

n

j

Gj

J1

W .................................................................................... (3.37)

where 1jk for kj and 0jk otherwise.

It is highly preferable that the estimator given by the contrast function is robust

against outliers. Themore slowly G grows as its argument increases, the more robust

is the estimator. For the choice of G we propose now three different functions, the derivatives g of which are also given:

ya

ygyayG

1

111 21, ................................. (3.38)

ya

ygyayG

2

2221,log ................................ (3.39)

yygyyG 32

3 ,21

....................................................... (3.40)


- 49 -

where 21,aa are some arbitrary constants (for example, 1,1 21 aa seem to be suitable). Of the preceding functions, 1G and 2G grow more slowly than 3G and thus

they give more robust estimators. 3G is motivated by kurtosis (3.32).

3.6.3.4. Fixed-Point Algorithm of Complex Value

Based on the discussion above, now give the fixed-point algorithm for complex

signals under the complex ICA model as:

1. Center the data to make its mean zero.

2. Whiten the data to give z .

3. Choose m , the number of ICs to estimate. Set p to 1p .

4. Select a normalized random vector pw as initial value.

5. Renew the separating matrix pw as

p

H

p

H

p

H

p

H

p

H

pp

ggE

gE

wzwzwzw

zwzwzw

222

2*)( ........ (3.41)

where, )(yg and )(yg defined as (3.29)-(3.31).

6. Normalize pw by dividing it by its norm.

7. Orthogonalization as:

1

1

p

j

p

H

jjpp wwwww ............................................................... (3.42)

8. Let ppp www .

9. If pw has not converged, go back to step 5.

10. Set 1 pp . If mp , go back to step 4.


- 50 -

3.7. Conclusion

In this chapter, the ICA algorithm, known as one of BSS techniques, is introduced.

ICA has been an attractive technique for different areas such as financial applications,

audio separation, image separation, telecommunications, and brain imaging

applications.

Even though there are plenty of applications of ICA to the areas mentioned above,

there is limited application in power systems. There is a study on load profile estimation

[19] using ICA. The application of ICA for estimation of DG is present in [20]. In [21],

ICA is used to estimate the harmonic source. Results of system side harmonic

impedance [22] and the harmonic current contribution of industrial load [23]-[24] are

presents in this thesis.

Chapter 4 Effect of Harmonics Caused by Large Scale Photovoltaic Installation in Power Systems

- 51 -

CCHHAAPPTTEERR 44

Effect of Harmonics Caused by Large Scale

Photovoltaic Installation in Power Systems

4.1. Introduction

Nowadays, the electric power is mainly generated by the fossil fuel. However, the

recognition of it as being a major cause of environmental problems and increasing of

generation cost make the mankind to look for alternative resources in power generation.

Moreover, the increasing demand for energy can create problems for the power systems,

like grid instability and even outage. The necessity of producing more energy combined

with the interest in clean technologies make an increasing development of power

distribution systems using renewable energy.

Among the all renewable energy sources, grid-connected PV power plants connected

to Low Voltage (LV) and Medium Voltage (MV) distribution systems is expected to a

noticeable growth in the future. In Japan, the installing total capacity of PV will be

about 53GW in 2030, which is ten times more than it in 2010 [5]. The large number of

PVs could cause the voltage increase at the distribution system nodes, reduction of

distribution systems losses, and voltage and current waveform distortion [6]-[10].

The effect of the harmonic distortion at PCC which installing the large number of

PVs is studied in [11]. The study aims to find out how the increasing of PV influences

the power quality of connecting point. The numerical simulations consist of two parts:

Mega-PV and residential type PV. In the Mega-PV simulation, the capacity of 10% is

discussed. The Mega-PV commonly connected to the system in LV or HV bus, which

includes huge number of load, thus, the assumed capacity is practical enough. However,

residential type PV, which connected to LV distribution systems, is expected to a

noticeable growth in the future. Therefore, in the simulation of residential type PV, the

capacity of 10% and 30% are presented to discuss the harmonic influence of it. In the


- 52 -

power distribution system which installing PV, the causes of current and voltage

waveform distortion are the upper systems, the nonlinear loads of customers and the

inverters of PV. The study confirmed that in the particular case, the harmonic current at

PCC may exceed the limits of harmonic current which is shown in Table 2-3.

In this Chapter, the characterization of the waveform distortion from grid-connected

PV plants under different installing capacity is discussed. First, we use the stochastic

aggregate harmonic load model to represent the load harmonic current source which not

including PV’s harmonic current. Next, we model the inverter of PV with measurement

data [6]. Finally, we change the installing capacity of PV in power distribution systems

to discuss the effect of harmonic current cause of PV.

4.2. Stochastic Aggregate Harmonic Load Model

As the biggest artificial system in the word, the power systems include huge number

of elements like generators, transmission lines and loads etc. Especially, the loads are

changing with the demand of customers and it is difficult and unmeaning to model the

each part of loads. The most common way is to model the loads at the MV and LV bus

as aggregate load model and the practice shown that it is very powerful in voltage

stability and sensitivity analysis [43]-[45]. Thus the aggregate harmonic load models

also very useful for harmonic study.

Power system include a lot type of nonlinear loads as shown in Chapter 2, and at the

MV and LV bus, it is common that the nonlinear loads are represented by the aggregate

effect of individual loads. For harmonic propagation studies based on the current

injection method, it is generally required that the aggregate harmonic load be represented

by a harmonic current source in parallel with some linear components such as resistance,

inductance, and capacitance.

The generic approach in developing a stochastic Aggregate Harmonic Load (AHL)

model based on harmonic field measurements. Stochastic models of AHLs were

established and classified according to consumers’ sectors/activities (commercial,

residential, and industrial). The application of the model is primarily in stochastic

harmonic propagation studies in MV distribution systems. Typically, a large number of

linear and nonlinear loads connected at the MV bus of a distribution transformer,

commonly known as the PCC, form an aggregate load. Linear loads do not produce

harmonic currents, but they are a significant component of the aggregate load as they


- 53 -

draw fundamental current, therefore, affect the current THD at the PCC. On the other

hand, nonlinear loads produce harmonic currents according to their individual harmonic

current spectrum. Harmonic current produced by AHLs is usually significantly smaller

than the algebraic sum of the harmonic currents produced by the individual nonlinear

load, mainly due to phase cancellation.

The harmonic current spectra of AHLs are primarily influenced by their harmonic

load composition and types which, in general, varies according to the class of customers.

For example, electronic home appliances, such as television sets, video players, and

fluorescent lamps form the majority of nonlinear loads of residential consumers,

whereas fluorescent lamps and computers are typical nonlinear loads of commercial

consumers. As a result, the composite harmonic current spectrum of a residential AHL

is likely to be different from that of the commercial AHL.

Field measurements indicate that harmonic current distortions at the PCC vary

randomly with a trend component closely correlating with the power demand of the

aggregate load. The random variation is primarily due to the combined effect of

continuous changes in operating conditions and usage pattern of linear and nonlinear

loads. At the same time, there is a need to account for uncertainties in harmonic current

distortions of the respective composite harmonic loads due to various factors. For

example, the harmonic current spectrum of composite harmonic loads is expected to

deviate from sample measured results within a range due to the different types of

electronic equipment.

In this chapter, we use the stochastic AHL model which is proposed in [46]. Hence, random variables are used to represent AHL parameters i

hiEIaK ，， associated with

the production of harmonic current distortions at the PCC. (4.1) is modified and written

in its normalized form as follows to represent random characteristics of harmonic

current.

21

)2(2

)1(1 ][][][][

aa

IaRIaRKRIR hhE

h

................................................ (4.1)

where ][h

IR denotes random variables corresponding to the Probability Density

Function (PDF) that describes harmonic current spectrum at the PCC, ][E

KR denotes

random variables that correspond to the PDF that describes a fraction of the AHL


- 54 -

Figure 4-1. Harmonic current load model.

participating into the total demand of the aggregate load, ][i

aR denotes random

variables corresponding to the PDF that describes a weighted coefficient representing

the fraction of the respective composite harmonic loads (type 1 and type 2) into the total demand of AHL, )(i

hI denotes random variables corresponding to the PDF that

describes the magnitude and phase, of the hth harmonic current distortion of the i

type composite harmonic loads.

The load model at the PCC [46] used in this thesis is showed in Figure 4-1. The

harmonic load model consists of a single stochastic harmonic current source that

represents harmonic current spectra of the AHLs, and a single stochastic current which

represents the linear loads consist of R, L, and C (i.e., induction motors, resistive loads

etc.). In Figure 4-1, I denotes the total fundamental current of load at PCC which are used by the nonlinear loads and linear loads.

hI denotes the total harmonic currents

that flows to the system side which produced by the nonlinear load. E

K denotes the

participation of nonlinear load of total load.

For more particular, it is necessary to depart the nonlinear load to more part based on

the production of harmonic currents as (4.1). In this simulation, the nonlinear load is

departed to two parts. Specific of two type individual harmonic current are showed [11] in Figure 4-2 where (1)

hI denotes the electric device like lamps and (2)

hI denotes the

electric device like TV and PC that produce harmonic current more than (1)h

I . In

common ways, only up to 25th harmonics are need to discuss, therefore, the harmonic


- 55 -

Figure 4-2. Individual harmonic current (1)

hI and (2)

hI .

Array(DC)

PCU

DC/AC converter

Transformer Grid

Figure 4-3. Block structure of the grid-connected PV Plant.

current emission by the PV inverters are characterized as high order. So we introduced

the harmonic current up to 50th order to simulate in this chapter.

4.3. Modeling the PV Inverter

The block structure of the grid-connected PV plant is shown in Figure 4-3. It is

composed of arrays of PV modules, interfaced to the external grid through a Power

Conditioning Unit (PCU). The PCU is composed of a number of stages. At the Direct

Current (DC) site, it is possible to find a DC/AC converter (inverter). At the AC side,

need the coupling transformer to connect to the power grid.

The harmonic distortion, originated from PV systems at the PCC with the low voltage

or medium voltage external grid, depends on the combined harmonic characteristics of


- 56 -

Figure 4-4. THDI and generated power of the PV inverter.

the distribution system, where the PV system is installed, and on the PCU performance.

In practice, there are substantial contributions at low-order harmonics (mainly for h=3,

h=5 and h=7) in the harmonic currents injected into the power systems.

In principle, the analysis of the harmonic distortion is similar to the one that could be

carried out for any inverter interfaced system. However, some peculiarities of PV

systems are the dependence of their operating conditions on climatic variables (i.e.,

temperature, irradiance and shading effect etc), which limit the time interval of

operation during a day and significantly impact on the shape of voltage and current

waveforms. The climatic variables may provide significant correlation among the

operational characteristics of closely located PV modules, resulting in strong similarities

among the waveforms at the inverter output. Yet, particular cases, like the operation of

some PV modules under shaded conditions, may result in strong unbalance of the phase

currents provided by the PV systems with different string or module integrated inverters

connected to the grid.

The contributions to harmonic distortion are increasingly important at low active

power level. However, at high levels of generated power, the harmonic distortion is

relatively low. Therefore, in these conditions the type of inverter control may have very

little impact on the harmonic distortion. Several field measurements carried out on PV

inverters have shown a similar dependence of the harmonic current emission on the

generated power. Similar harmonic behavior appeared both for PV systems interfaced to

the grid through a transformer and for PV systems without coupling transformer [6] [9].


- 57 -

Figure 4-4 shows the Total Harmonic Distortion (THD) of the current (THDI) and the

generated power of a 5 kW inverter without coupling transformer, measured over 1 day

in the summer season [6].

We use the measurement data of Figure 4-4 to model the harmonic emission of PV

inverters and assumed that all type of PV inverter have same character. In Figure 4-4, in

the morning and evening hours, the THDI is up to five times higher than during the rest

of the day. The shape of the THDI is characterized by a comparative high value under

low power generation conditions, with a sharp decay for increased generation; the THDI

remains below 10% when the inverter loading exceeds approximately 18% - 20% of the

rated power. It means that the inverters of PV have more nonlinear character in low rate

condition than high rate working condition. The individual harmonic currents show a

similar character, especially in low-order harmonics.

In the distribution system that include PV and nonlinear loads, the harmonic current

source have two as shown in Figure 4-5 without considering the upper system’s

harmonic. Where I denotes the fundamental current with supplying by the upper system, and PVI is the PV’s output, LI is the load’s fundamental current. LhI and PVhI are

harmonic current of load and PV respectively.

Figure 4-5. Harmonic source of distribution system with PV.


- 58 -

Figure 4-6. Power distribution systems with Mega PV.

4.4. Numerical Simulation

The purpose of the numerical simulation is to show that with the increasing of

installed PV in the power distribution system, the harmonic distortion at PCC also

increasing, and when the installing capacity of PV are increasing continually, the limit

of harmonic current at PCC will exceed the standard limit level of harmonic current

which is showed in Table 2-3.

4.4.1 Mega-PV

Mega-PV is often established by the company for getting the profits. A power

distribution system with Mega-PV, usually named mega-solar system is shown in Figure

4-6 where PCC1 is the connecting point of load and PCC2 is the connecting point of

mega-PV. Assumed the upper system’s capacity is 10 MVA, the system’s short circuit

capacity is about 150MVA, thus the system harmonic impedance is about 6.7%. The PV

is connecting at PCC2 and the capacity is about 1MVA, which including four 250KVA

PC system. All the transformers assumed to be same, and the connecting transformer’s

voltage ratio is 6.6KV/210V. The loads assumed to typically residential load. In this

condition, the ratio of PV is 10% of total load which is common in modern distribution

systems. The numerical simulation results are shows in Figure 4-7-4-9.

Figure 4-7 is THDI at PCC2 for one day. The line of PV 0% is the harmonic current


- 59 -

distortion of load without PV where the harmonic are emission by the load totally. We

assumed the loads are typically residential load model which are small in night and

large in day. The simulation shows that from 0 h to 6 h in the morning, the distortion

level of harmonic current is small than other times. The THDI is under the standard

limit level for full day. The line of PV 10% is THDI which including the load and 10%

PV of full load. The simulation result shown the harmonic current distortion is not

increasing so much in this condition. It is still satisfy the standard of harmonic distortion

levels.

Figure 4-7. Current total harmonic distortion at PCC2.

Figure 4-8. Voltage total harmonic distortion at PCC2.

Figure 4-8 shows the voltage Total Harmonic Distortion (THDV). The line of PV 0%


- 60 -

Figure 4-9. The ratio of 5th harmonic current and fundamental current at PCC2.

is the THDV without the PV and the line of PV 10% is the THDV with include the 10%

of PV. Most maximum values of the voltage THDV recorded at the observatory occurred

during the day. However, Figure. 4.8 shows only a small increase in the average THDV

value occurred when the PV has generated power. It must be noted that most generating

periods were normally coincident with the times when most customer loads would be

operating.

Figure 4-9 is the ratio of 5th harmonic current and fundamental current. In the all

harmonic current order, the 5th harmonic current is usually more than others. So it is very important to analysis the 5th harmonic current. The line of PV 0% is the 15 II

without the PV and the line of PV 10% is the 15 II with include the 10% of PV. The

result shows the 15 II increasing in the day with the PV have worked times. It is also

know the 5th harmonic current is under the standard limits in this simulation.

Compare Figure 4-7 to Figure 4-8, it can get a conclusion that the influence of

harmonic current distortion is larger than voltage distortion. It is because the nonlinear

loads consume the energy (fundamental current) when it emits harmonic current,

however, PV inverter is output the energy when it emits harmonic currents. The result is

that the influence of PV inverter is bigger in harmonic current distortion and smaller in

voltage distortion. The simulation result shows that, when the PV’s installing capacity is

10% of load capacity or smaller than it, the harmonic current distortion do not enough

to anxious the power quality. But it is clear that installing the PV bring the potential

concern about the power quality.


- 61 -

Figure 4-10. Power distribution systems with small scales PV.

4.4.2 Residential Type PV

In recent years, the install of residential type PV increase continually and will became

main harmonic current source in the future. The investigation on the harmonic distortion

due to the operation of multiple PV inverters connected to the distribution system has

been carried out with reference to inverter model of Figure 4-4.

A power distribution system with residential type PV is shown in Figure 4-10, where

PCC1 is the connecting point of individual load bus and PCC2 is the connecting point

of total load to the upper system. Assumed the upper system’s capacity is 10 MVA, the

system’s short circuit capacity is 150 MVA, the system harmonic impedance is about

6.7%. All the transformers assumed to be same, and the ratio is 6.6KV/210V. The load

is typically residential load and some loads installed the small scales of PV.

The simulation is assumed that the installing capacity of PV is 0%, 10% and 30% and

trying to verify the influence of harmonic distortion at PCC2 in these conditions. The

simulation results are shown in Figure 4-11-4-13. Figure 4-11 is THDI at PCC2 for one

day. The line of PV 0% is the harmonic current distortion at PCC2 without PV. We

assumed the loads are typically residential load model which are small in night and

large in day. The simulation shows that from 0 h to 6 h , the distortion level of

harmonic current is small than other times. The THDI is under the standard limit level

for full day. The line of PV 10% is THDI which including the load and 10% PV of full


- 62 -

Figure 4-11. Current total harmonic distortion at PCC2.

load. The simulation result shows that harmonic current distortion is not increasing so

much in this condition. The line of PV 30% is THDI which including the load and 30%

PV of full load. As the capacity of PV up to 30% of full loads, the harmonic

contribution of PV is increasing rapidly. The THDI is already exceed the standard limit

in the middle of day, which PV produce more power.

Figure 4-12 shows the THDV. The line of PV 0% is the THDV without the PV and

the line of PV 10% is the THDV with include the 10% of PV. Most maximum values of

the voltage THDV recorded at the observatory occurred during the day. However, Figure.

4-12 shows only a small increase in the average THDV value occurred when the PV has

generated power. It must be noted that most generating periods were normally coincident

with the times when most customer loads would be operating. The line of PV 30% is

THDV which including the load and 30% PV of full load, which the THDV in increased.

As the same reason discussed in Chapter 4.4.1, the influence of harmonic voltage

distortion is smaller than harmonic current when installing the PV.

Figure 4-13 is the ratio of 5th harmonic current and fundamental current. The line of PV 0% is the 15 II without the PV and the line of PV 10% is the 15 II with include

the 10% of PV. The line of PV 10% is the 15 II increasing in the day with the PV

worked times. It is also known the 5th harmonic current is under the standard limits in this simulation. The line of PV 30% is 15 II which including the load and 30% PV of

full load. This time, the 15 II already exceed the standard limits.


- 63 -

Figure 4-12. Voltage total harmonic distortion at PCC2.

Figure 4-13. The ratio of 5th harmonic current and fundamental current at PCC2.

The simulation shows that with the increasing of installed PV, the harmonic current

and harmonic voltage also increased. When the capacity is up to 30%, the harmonic

current distortion will exceed the standard limit.


- 64 -

4.5. Conclusion

This Chapter has presented the results of numerical simulation of the harmonic

distortion at PCC where the load include nonlinear load and PV are present. The

stochastic aggregate harmonic load model is introduced to represent the harmonic

current of loads and used the inverter model that is modeled by measurement data to

represent the harmonic current of PV. The numerical simulations consist of two parts:

Mega-PV and residential type PV. In the Mega-PV simulation, the capacity of 10% is

discussed. The Mega-PV commonly connected to the system in LV or HV bus, which

includes huge number of load, thus, the assumed capacity is practical enough. However,

residential type PV, which connected to LV distribution systems, is expected to a

noticeable growth in the future. Therefore, in the simulation of residential type PV, the

capacity of 10% and 30% are presented to discuss the harmonic influence of it.

The simulation shows that with the increasing of installed PV, the harmonic current

and harmonic voltage also increased. When the capacity of Mega-PV and residential

type PV is 10%, the harmonic current distortion increased significantly. However, it is

still under the standard current limit. When the capacity of residential type PV is up to

30%, the harmonic current distortion at the connected point will exceed the standard

limit. The simulation shows that increasing of PV will make a serious power quality

problem where it is connected. In the particular areas of Japan, the capacity of PV is

already more than the 30% of full loads. The customers of these areas will experience

the large harmonic distortion, which effect customer’s electric devices like shorter the

life of it. For power company, it is necessary to decrease the harmonic distortion of

these areas using harmonic filter etc.

Chapter 5 Estimation of System Harmonic Impedance using Complex ICA

- 65 -

CCHHAAPPTTEERR 55

Estimation of System Harmonic Impedance using

Complex ICA

5.1. Introduction

Harmonic impedances of a supply system characterize the frequency response

characteristics of the system at specific buses. To decrease harmonic distortion, the

method such as passive filtering and active filtering can be used to sink local harmonic

currents and prevent them from penetrating the main supply system. The reliable design

of a passive filter requires a correct knowledge of the system harmonic impedance and

it’s variations throughout the day to avoid creating a resonance condition, which could

destabilize a power system. Active filters also require a good knowledge of the system

harmonic impedance to ensure stable controller operation and also can be used in the

generation of the filter reference currents.

It is very desirable in many applications to directly measure the system harmonic

impedances. A number of impedance measurement methods have been developed for

this purpose. These methods can be classified into two types: 1) the transients- based

methods (invasive methods) and 2) the steady-state-based methods (noninvasive

methods) [26] [27]. The transients-based methods inject transient disturbances into the

system. The frequency-dependent system impedances are extracted from voltage and

current transients. Typical transient disturbances for this application are the capacitor

switching transients [28] [29] and controlled harmonic current injection method [30]

[31]. The main problems associated with these methods are the need for a high-speed

data acquisition system and for the methods use pre- and post-disturbance steady-state

waveforms. Typical disturbances are harmonic current injections produced by an

external source or switching of a network component. Also, these kinds of methods

cannot get the system impedance in real time, only provide instantaneous results which


- 66 -

are valid for the moment of the test. Sometimes, this may cause a negative effect on the

normal operation of the power system [27].

The steady-state-based methods use the existing harmonic sources and the

measurable parameters to calculate the impedance. The advantage of this kind of

methods is that they use the natural harmonic variations and can be applied anywhere,

and the disadvantage is that good precision is difficult to achieve in the absence of some

predominant disturbing loads mainly at higher frequencies.

In this chapter, the technique that estimates system harmonic impedance at PCC using

complex value ICA is presented. ICA algorithm is a linear transformation method,

which transforms the observed signals into mutually statistically independent signals.

Since the observed signals are linear combinations of unknown independent source

components, ICA’s objective is to invert the unknown mixing matrix, thus estimating

source signals blindly without prior system knowledge.

The method introduced the Norton equivalent circuits to set up the linear mixing

model of ICA model to estimate the system harmonic impedance in condition that load

harmonic impedance changing. The method does not need to know the specific of

related systems, it just needs the one point measurement data at PCC to estimate. The

method is verified by the numerical simulation where the result shown our method can

estimate the system harmonic impedance correctly.

5.2. Current System Harmonic Impedance Estimate Method

5.2.1. General Principles

The basic principle of harmonic impedance is to make use of harmonic currents

injected at the point where it is to be measured. The simple block is illustrated in Figure 5-1. Where

hI denotes the harmonic current,

hV denotes the harmonic voltage and

hZ is the system harmonic impedance. Then using Ohm’s law can get the

hZ as:

h

h

hI

VZ

................................................................................................ (5.1)


- 67 -

hIh

Zh

V

Figure 5-1. Basic principal of system harmonic impedance estimate.

This is only useful when assumed that no harmonic voltage (background harmonic

voltage) was present in the network prior to the current injection. In cases where this assumption in not valid,

hV and

hI replace by

hV and

hI .

In practice, the power system is 3-phase and is not symmetrical. Furthermore, in most

cases, the injected harmonic currents are far from symmetrical. Thus, even if assume the

system is symmetrical, it is necessary to study the practicability of measuring the

positive sequence impedance using unsymmetrical current injection.

5.2.2. Transients Based Methods (Invasive Methods)

Switching of capacitor bank: Switching of capacitor bank is approximately

equivalent to causing an instantaneous short-circuit, resulting in a current in which the

Fast Fourier Transform (FFT) gives a very rich spectrum. Recording in the voltage and

current signals in a time window including the transient will then allow the assessment

of the system harmonic impedance, as seen from the connection point of the capacitor

bank. Generally, the results are not valid for fundamental and harmonic frequencies (at

least when pre-existing harmonic voltage is signification). It is however possible to take

account of the pre-existing harmonics by applying the same principle as already seen at

section 5.2.1 for steady state signals.

Advantages:

Rich spectrum with interharmonic frequencies.

Capacitor banks are widespread and their switching is a common operation.

Disadvantages:

Very short duration of the signals.


- 68 -

The presence of a capacitor bank is necessary.

The current are unsymmetrical and depend on the switching moment.

Switching of transformers: Immediately after the switching process, a transformer

will reach saturation depending on the switching moment. Transient inrush

currents-different on the three phases are then characterized by a high aperiodic

component, a rich spectrum content and a great value, during some seconds. It can be

used for harmonic impedance measurements.

Advantages:

Very high current levels compared to existing harmonics.

The switching current spectrum contains all harmonics up to about 1000 Hz.

Disadvantages:

Currents are highly unsymmetrical and depend on the switching moment.

Natural variations: Instead of switching a network element, it is possible to make

use of natural variations of currents and voltage in the system to calculate its

impedance.

Advantages:

This method is fully non-invasive.

Can be applied anywhere.

Disadvantages:

A good precision is difficult to achieve in the absence of some predominant

disturbing loads.

Direct injection of harmonic current: The technique involves a power electronic converter, which injects a voltage transient on to the energized network via an inductor.

The injected voltage transient has a frequency resolution of 6.25 Hz which provides

inter-harmonic values and thus allows for the interpolation of the system impedance at

the harmonic frequencies. The resulting transient current is correlated with the

disturbance voltage to determine the frequency dependent impedance. As the technique

employs controlled power electronic devices it may be used as a stand alone piece of

portable measurement equipment, or alternatively can be embedded into the functions of

an active shunt filter for improved harmonic control.

Advantages:

All harmonic currents are produced up to over 10001Hz.

Fairly strong currents can be produced during long duration..


- 69 -

Disadvantages:

A transformer with saturable zero sequence path in needed.

Large arrangements are needed, with a powerful DC source and blocking series

capacitors in the neutral connection of all neighboring saturable transformers.

Injected currents are strongly unsymmetrical although the connection is

symmetrical.

Pre-existing harmonics must be taken into account.

There is also a method that using the interharmonic current generation [31].

Interharmonic current generators have been specially designed for the measurement of

harmonic impedances. They have been successfully applied in LV, MV and even

(rarely) HV networks. In practice, such generators are commercially available for LV

systems only.

Advantages:

Nearly the whole spectrum (0-2.5 kHz) can be measured. Harmonic

frequencies are also obtained by interpolation.

Pre-existing harmonic has very little effect on the measurements at

interharmonic frequencies so that very low signal levels can be sufficient.

Disadvantages:

Fairly power signal generators are needed, especially for higher voltages levels.

Suitable connecting transformers with low reactance are needed.

Injected current is sometimes not symmetrical depending on generator.

As discussing above, the existing estimated method has a lot of disadvantages, like

disturbing the power system and, sometimes, it is also more expensive. There are still

do not have a appreciate way to estimate the system harmonic impedances.


- 70 -

5.3. Estimation of System Harmonic Impedance using Complex ICA

5.3.1. Norton Equivalent Circuit

There are a number of net models presently being used for power system harmonic

analysis [2], [3]. These techniques vary in terms of data requirements, modeling

complexity, problem formulation and solution algorithms. The harmonic producing

device is modeled as a supply voltage-dependent current source in common. The

program based on this method requires more knowledge of the harmonic producing

equipment than those based on the admittance matrix method.

The reference [47], [48] have proposed the one generator and one load equivalent

power system based on the power flow calculation which can simplify the power system

voltage stability and sensitivity analysis. For harmonic study, especially in the

frequency-domain methods, the Norton equivalent circuit [49] is commonly used to

analyze the condition where system-side and customer-side generate harmonics with

respect to each harmonic order [27] [34] [50] [51]. The single phase Norton equivalent circuit is shown in Figure. 5-2, where

uI anduZ are the system-side equivalent

harmonic current and impedance, and cI are

cZ the customer-side equivalent harmonic current and impedance, and

pccV and pccI are the harmonic voltage and current

measured at the PCC. Referring to Figure 5-2,

pccI and pccV can be expressed as:

uI

cI

PCC

uZ c

Zpcc

I

pccV

Figure 5-2. Single phase Norton equivalent circuit.


- 71 -

cu

ccuu

pccZZ

IZIZI

................................................................................ (5.2)

)(cu

cu

cu

pccII

ZZ

ZZV

........................................................................ (5.3)

5.3.2. Statistical Properties of Harmonic Current Sources

The ICA algorithm requires the statistical independence and nongaussianity of ICs

which are harmonic current sources in this thesis. For estimation the system harmonic

impedance, it is necessary to verify the statistical properties of harmonic current

sources.

The power demand of a customer is not constant throughout the day, week or year.

Considering the power demand of a load over a 24-hour period, variation of power

mostly depends on the type of the load. Similar load types have similar active/apparent

power profiles which can be categorized into classes including residential, industrial,

commercial load. Also parameters such as weather variables including temperature, the

day of week, time of day and others effect the variation of the power demand of the load.

These factors create a slow varying trend which illustrates hour-to-hour variations, i.e. a

peak in the afternoon and minimum at nighttime. At the same time there are fast

variations over this trend. These fast fluctuations represent load variations in seconds or

minutes and are considered random variations which are difficult to predict.

We can assume that when exclude the slow varying of load demand, the fast varying

of load demand is statistical independence and nongaussianity. We used the two ways to

exclude the slow varying of load demand. First, we measure the harmonic current and

harmonic voltage in minutes, which can get the fast variation more efficient. Second is

the one step of Fast-ICA algorithm where centralize the measurement data. After these

step, we assumed that the harmonic current sources are statistical independence and

nongaussianity.

To verify the assumption above, we use the measurement dada of industry load to

discuss its nongaussianity. Figure 5-3 shows the measurement data of one week at

industry load which include fundamental current, 5th harmonic current and 7th harmonic

current. Examining these harmonic current, they demonstrate similar slow varying


- 72 -

25 26 27 28 29 30 310

1

2

3

4

5

6

7

Time (day)

Har

mon

ic C

urre

nt (A

)

10

20

30

40

50

60Fundamental current

5th Harmonic current

7th Harmonic current

Fund

amen

tal

Cur

rent

(A)

Figure 5-3. Measurement data at industry for one week.

Figure 5-4. Measurement data of 5th harmonic current of a day.

components like the fundamental current. In the working times, the harmonic currents

are more than rest time. For the long times like one day, the harmonic current is hard to

say independently. It has same behavior day by day.

The measurement data of 5th harmonic current of a day is illustrated in Figure 5-4. In

Figure 5-4, it is easy to depart the one day data to three pars: before the working, under

the working and after the working. The kurtosis of 5th harmonic current in the ten

minutes is about -0.54 where the kurtosis of Gaussian distribution is 0. The kurtosis of

5th harmonic voltage in the ten minutes is about -0.9. Based on the central limit theorem,

it is clear that the harmonic current sources are statistical nongaussianity. We consider

that the residential and commercial load also independence and nongaussianity if the


- 73 -

measurement times are short enough. Moreover, the Fast-ICA algorithm needs to center

the measurement data, which can remove the slow varying of measurement to improve

the independence and nongaussianity.

5.3.3. Estimation Algorithm

The system harmonic impedance with ICA estimation algorithm describe in this

section. From (5.2) and (5.3), we can get the (5.4)

c

u

cu

cu

cu

cu

cu

c

cu

u

pcc

pcc

I

I

ZZ

ZZ

ZZ

ZZ

ZZ

Z

ZZ

Z

V

I

..................................................... (5.4)

Compared (5.4) to ICA model (3.2), we can assumed the equation (5.4) as a simple

linear ICA model where

measurement data is

pcc

pcc

V

I

X

original signal is

c

u

I

I

S

mixing matrix is

cu

cu

cu

cu

cu

c

cu

u

ZZ

ZZ

ZZ

ZZ

ZZ

Z

ZZ

Z

A .

The purpose of ICA method is usually to estimate the original signals s by

estimating the separating matrix W , which is the pseudo-inverse of mixing matrix A . But, our purpose is to estimate the system harmonic impedance

uZ which include in the

mixing matrix A . The separation matrix of (5.4) is shown below.


- 74 -

c

u

Z

Z

11

11W ..................................................................................... (5.5)

It is very clearly that the algorithm just needs to estimate the matrix W , then can understand the value of

uZ . The problem is what the W is not only include the

uZ ,

but also include the customer-side harmonic impedancec

Z . It is known that the

system-side harmonic impedance is normally determined by the short-circuit capacity of

the power system. It is because the system-side is huge system and rarely influenced by the sudden change of load variation. So we can assumed that the

uZ is not changing in

the mixing process of ICA model if the mixing time is short enough. Otherwise, the c

Z

is changing often even if the mixing time is short enough. Thus, the estimation of c

Z

will occur the errors easily. The Fast-ICA algorithm needs to orthogonalization that

allows us to estimate the independent component one by one. As discussed in Chapter

3.3, ICA cannot determine the order of the independent components, if a row of W which includes the

cZ estimated firstly, the other row of W which includes the

uZ

must occur the errors. These problems are needed to discuss more carefully in

estimation algorithm.

The harmonic load identification algorithm (Figure 5-5) described above can be

summarized as follows: ① Reading the measurement data at PCC, which include the harmonic current

pccI and harmonic voltage

pccV .

② Centralize and whiten the measurement data of ①.

③ Choose an initial value of unit norm to apply complex Fast-ICA algorithm to the measurement data. The purpose of this method is to estimate the system harmonic impedance

uZ , so we just need to estimate the first row of separating

matrix of equation (5.5). As we discussed above, the Fast-ICA algorithm cannot

determine the order of row of W . If the first row of W which includes the

uZ be estimated firstly, the

uZ can be estimate correctly. But, if the second row of

W which includes the c

Z be estimated firstly, and have estimated error because

of its change, it would influence the estimate result of second row of W which includes the

uZ , finally, the estimate result of

uZ may be occur the error that is

bigger than accepted region. So it is necessary to perform the Fast-ICA algorithm


- 75 -

Figure 5-5. Proposed system harmonic impedance estimate algorithm.


- 76 -

more one time.

④ The mixing matrix of proposed method is 22 column. It is possible to change the row of 1W as the initial value of second separation matrix 2W .

⑤ As result, can get the two separation matrix 1W and 2W , which the rows must be inverse. We just need to take attention to the first rows of two separation

matrix 1W and 2W which must include the first and second row of equation

(5.5) and more importantly, they do not influenced by the orthogonalization.

⑥ Recentralize and rewhiten the two separation matrix 1W and 2W .

⑦ For scaling problem of ICA, using the first column of separation matrix to normalize the 1W and 2W . And assumed the system harmonic impedance.

⑧ The magnitude of the system-side impedance is much smaller than that of the customer side at the fundamental frequency. Furthermore, since the system-side

impedance and customer-side impedance are inductive in most cases, which

mean that even at a higher harmonic order, the system-side harmonic impedance

is still much smaller than that of the customer side at non resonance frequencies. Using this rule to decide which one is the correct

uZ .

5.3.4. Numerical Simulations

5.3.4.1. Simulation Conditions

We are using the program simulation method to verifying the proposed method which

discussed in Chapter 5.3.3. In this simulation, the both system harmonic source and

customer harmonic source are assumed to be harmonic current source. Otherwise, the

initial simulation work was carried out using a single phase Norton equivalent circuit

(shown in Figure 5-2) approximation only for simplicity and to improve processing

speed. Therefore, the method is easy to intend for three phase power systems. As the

purpose of estimated, two type of experimental loads are used. Two system impedance

models have been illustrated in Figure 5-6 and Figure 5-7. The representation of Figure

5-6 (model 1) is a simplified system where only the supply inductance is estimated. In

Figure 5-7 (model 2), the representation is more detailed to include resonances caused

by added parallel capacitance. It should be noted that the work equivalent system

impedance considers distribution level components such as power transformers,


- 77 -

uI

PCC

1R

1L

Figure 5-6. System impedance model 1.

uI

PCC1R

1L

2R

1C

Figure 5-7. System impedance model 2.

transmission lines, feeder load etc. The parameters of system impedance at fundamental are ,4.01 R ,mΗ11 L ,μF1101 C 5.02R .

It is known that the harmonic current problems are mainly caused by lower order

harmonic current like 3th, 5th, 7th etc. Thus, in this thesis, we just discuss the lower order system harmonic impedance that is up to 25th. Aim to assume the system impedance

uZ

is not changing in the mixing process of system harmonic current source uI and

customer harmonic current source c

I , it is necessary to shorter the measurement period.

In this numerical simulation, assumed that the measurement period is 10 minutes and

sample the data per second. Then we have the 600 measurement data which is enough for ICA estimation and the sampling period is short enough to assume the

uZ is not

changing.

For harmonic current source, using the function pearsrnd of MATLAB program to represent the both system harmonic current

uI and system harmonic current c

I . The

function Pearson shown as:

k)s,,(m,RIIh ............................................................................... (5.6)


- 78 -

Table 5-1. Parameters of 5th harmonic current sources.

hI m s k

uI (real part) 8 1.5 0.4 0 -0.6

uI (imaginary part) -8 2 0.5 0 -0.6

cI (real part) 8 2 0.6 0 -0.6

cI (imaginary part) -8 2.5 0.7 0 -0.6

0 100 200 300 400 500 6000

10

20

0 100 200 300 400 500 600-30

-20

-10

0

Time(s)

Real part

Imaginary part

5thha

rmon

ic c

urre

nt(A

)

Figure 5-8. 5th system-side harmonic current

uI .

where, hI is the amplitude of h

th harmonic current, m is mean of random numbers,

is root-mean-square deviation of random numbers, s is stand deviation, k is kurtosis of

random numbers. It is impossible to show the all order of harmonic current in here, so

we just show the parameters of 5th harmonic in Table 5-1 as an example. We take the

same value of k in both sides to simple the simulation condition.

Based on the (5.6) where parameters are shown in Table 5-1, we can get the 5th

system harmonic current and 5th customer harmonic current. Figure 5-8 is 5th the system side harmonic current

uI , Figure 5-9 is 5th customer side harmonic current c

I , where

above line is real part and below line is imaginary part of current sources. As same way,

the harmonic current sources are represented in every order which have same character


- 79 -

0 100 200 300 400 500 6000

10

20

30

0 100 200 300 400 500 600-40

-20

0

Time(s)

5thha

rmon

ic c

urre

nt(A

) Real part

Imaginary part

Figure 5-9. 5th customer-side harmonic current

cI .

of Figure 5-8 and Figure 5-9.

5.3.4.2. When Customer Harmonic Impedance Changing in Small Range

The customer harmonic impedance is changing often with the change of load demand,

likes capacity install or uninstall etc. It is possible that the customer harmonic

impedance is changed when the harmonic current in mixing at PCC. It means the

mixing matrix of linear ICA model shown in (5.4) changed in the mixing process.

In this section, we discuss the load changed in small range firstly and discuss the

sharply change condition in next section. For numerical simulation, the measurement

data at PCC (harmonic current pccI and harmonic voltage

pccV ) can get based on the (5.4)

in every harmonic order. We already represented the harmonic current source as Figure

5-8 and Figure 5-9. Also we have two system harmonic impedance models (Figure 5-6

and Figure 5-7). The customer side 5th harmonic impedance is shown in Figure 5-10

where the customer harmonic impedance is changing randomly in small range. Others

order of customer harmonic impedances also have same form which the amplitudes are

different. Installing the system harmonic current source (Figure 5-8), system harmonic

impedance model 1(Figure 5-6) and customer harmonic current source (Figure 5-9),

customer harmonic impedance (Figure 5-10) to the (5.4), can get the measurement data

of PCC. For example, we just show the 5th measurement data in this chapter.


- 80 -

Figure 5-10. Customer side 5th harmonic impedance

cZ in small change range.

Figure 5-11. 5th harmonic current at PCC.

Figure 5-12. 5th harmonic voltage at PCC.


- 81 -

(a) Amplitude

(b) Phase

Figure 5-13. Estimated system harmonic impedance model 1.

Figure 5-11 is the 5th harmonic current and Figure 5-12 is the 5th harmonic voltage

measurement data at PCC. The above line is real part and the below line is imaginary

part of complex value. The Figure 5-11 shows the 5th harmonic current is flow from the

customer side to system side but not clear which side are more contribution to this

harmonic current. In harmonic analysis, the harmonic source estimation is the mainly

one and in some case, it is necessary to estimate the system side harmonic impedance.

For more information we can get in Chapter 6. The harmonic current and harmonic

voltage at every harmonic order can be calculated by the same way.

The proposed algorithm is applied to the measurement data of every harmonic orders,


- 82 -

thus get the system side harmonic impedance model 1 from order 2 to order 25. The

simulation results are shown in Figure 5-13. The Figure 5-13(a) is the amplitude of

system harmonic impedance and the Figure 5-13(b) is the phase of system harmonic

impedance. The theoretical values are calculated directly from the system harmonic load

model 1. The Estimated 1 is the estimated result using the ICA only once time and the

Estimated 2 is the estimated result by our proposed algorithm.

In the Figure 5-13, the Estimated 1 have large error in some harmonic orders like 5th,

9th, 13th etc where Estimated 2 have very well performance of the estimate of system

harmonic impedance. Especially, the estimate of phase in Figure 5-13 (b), the Estimated

1 cannot get the correct result in some orders. However, in other orders, Estimated 1 can

get good result where the first order of (5.5) estimated occasionally. The simulation

results are confirmed our proposed method that uses the Fast-ICA algorithm is suitable

to estimate the system-side harmonic impedance.

The simulation results of system harmonic impedance model 2 are shown in Figure

5-14. The Figure 5-14(a) is the amplitude of system harmonic impedance and the Figure

5-14(b) is the phase of system harmonic impedance. The theoretical values are

calculated directly from the system harmonic load model 1. The Estimated 1 is the

estimated result using the ICA only once time and the Estimated 2 is the estimated

result by our proposed algorithm.

In the Figure 5-14, the Estimated 1 also has large errors in some harmonic orders.

However, Estimated 2 still have very well performance of the estimate of system

harmonic impedance even if the system harmonic impedance have resonance condition

cause by the SC. In harmonic analysis in power systems, the calculation of harmonic

impedance resonant point is very important to the placement of the filters, which should

avoid constructing this condition. The simulation shows our proposed method can

estimate the resonant point of system harmonic impedance correctly.

The simulation result shows that the proposed method can estimate the system

harmonic impedance correctly in the condition where the customer harmonic

impedances are changed in small range.


- 83 -

(a) Amplitude

(b) Phase



- 84 -

Figure 5-15. Customer 5th harmonic impedance

cZ in change sharply.

Figure 5-16. Measurement Harmonic current at PCC.

5.3.4.3. When Customer Harmonic Impedance Changing Sharply

In this section, we are discussing the condition that the customer harmonic impedance

changed sharply in the mixing process. For instance, the customer’s 5th harmonic

impedance is shown in Figure 5-15. The above line of Figure 5-15 is phase of system harmonic impedance

cZ and the below line is amplitude of

cZ . We assumed that the

cZ is small in anterior half and doubled in last half. This kind of change is more often

in power distribution system likes the capacitor or large load installing or uninstalling.

The measurement data (5th harmonic current and 5th harmonic voltage) at PCC are


- 85 -

Figure 5-17. Measurement harmonic voltage at PCC.

illustrated in Figure 5-16 and Figure 5-17. The Figure 5-16 is the 5th harmonic current

that the above line is real part and the below line imaginary part of complex value. They

are changed sharply at sampling point 300 because customer harmonic impedance

changed. The Figure 5-17 is the 5th harmonic voltage that the above line is real part and

the below line is imaginary part of complex value. It is also change in middle point just

like to the harmonic current. They are calculate by the equation (5.4) where the system harmonic impedance

uZ is the model 1 which shown is Figure 5-6.

The simulation result is shown in Figure 5-18 where Figure 5-18(a) is amplitude and

Figure 5-18(b) is phase of system harmonic impedance. In Figure 5-18, the theoretical

values are calculated directly from the system harmonic load model 1. The Estimated 1

is the estimated result using the ICA only once time and the Estimated 2 is the estimated


It is clearly from the estimation result that the proposed method also can estimate the

system harmonic impedance correctly and certainly in this condition. The Estimated 1

has some error in some orders, especially in estimating of phase.


- 86 -

(a) Amplitude

(b) Phase

Figure 5-18. Estimated system impedance model 1 in c

Z change sharply.

For the system harmonic impedance model 2, the estimated results are illustrated in

Figure 5-19. Figure 5-19(a) is amplitude and Figure 5-19(b) is phase of system

harmonic impedance. In Figure 5-19, the theoretical values are calculated directly from

the system harmonic load model 1. The Estimated 1 is the estimated result using the

ICA only once time and the Estimated 2 is the estimated result by our proposed

algorithm.

Just like the estimated result in Chapter 5.3.4.2, the proposed method can estimate the

system-side harmonic impedance very well. In generic way to estimate system harmonic

impedance, like as harmonic current injection methods, the sudden change of customer


- 87 -

(a) Amplitude

(b) Phase Figure 5-19. Estimated system impedance model 2 in

cZ change sharply.

harmonic impedance will influence the estimate result.

5.3.4.4. Changing the Kurtosis of the Harmonic Current Source

As mention above, the proposed method is based on the complex Fast-ICA algorithm.

The ICA algorithm is needed to satisfy the three conditions which the independent

component (harmonic current source in this thesis) must be non-Gaussian. It is mean


- 88 -

Table 5-2. Parameters of 5th harmonic current sources.

hI m s k

uI (real part) 8 1.5 0.4 0 -0.4

uI (imaginary part) -8 2 0.5 0 -0.4

cI (real part) 8 2 0.6 0 -0.4

cI (imaginary part) -8 2.5 0.7 0 -0.4

Figure 5-20. The 5th system side harmonic current

uI .

that with changing the kurtosis of (5.6), the distribution of harmonic current source is

also change. As a result, the system harmonic impedance estimation is influenced. In

this section, we verify the proposed method in different condition where the distribution

of harmonic current sources is nearly to the Gaussian distribution. For simple, just change the k of (5.6) to represent the both system harmonic current

uI and customer

harmonic currentc

I . The parameters of 5th harmonic currents are shown in Table 5-2.

The 5th system harmonic current uI is illustrated in Figure 5-20. The Figure 5-21 is

the 5th customer harmonic currentc

I , where above line is real part and below line is

imaginary part of current sources. They are more Gaussian than harmonic source shown


- 89 -

Figure 5-21. The 5th customer side harmonic current

cI .

Figure 5-22. The 5th harmonic current at PCC.

in Figure 5-8 and Figure 5-9.

Installing the system side harmonic current source (Figure 5-20), harmonic

impedance model 1(Figure 5-6) and customer side harmonic current source (Figure

5-21) and harmonic impedance (Figure 5-10) to the (5.4), can get the measurement data

at PCC. For example, we just show the 5th measurement data in here.

Figure 5-22 is the 5th harmonic current and Figure 5-23 is the 5th harmonic voltage

measurement data at PCC, where above line is real part and below line is imaginary part

of current sources. Next step is perform the proposed algorithm to the measurement


- 90 -

Figure 5-23. The 5th harmonic voltage at PCC.

data.

The simulation result is shown in Figure 5-24 where Figure 5-24(a) is amplitude and

Figure 5-24(b) is phase of system harmonic impedance. In Figure 5-24, the theoretical

values are calculated directly from the system harmonic load model 1. The Estimated 1

is the estimated result using the ICA only once time and the Estimated 2 is the estimated


The estimation result shows that the both Estimated 1 and the Estimated 2 have

bigger error in higher orders. As a special, the Estimated 2 has biggest error in order 6.

It means that the proposed method could be failure when the distribution of harmonic

current sources near to the Gaussian distribution. However, the kurtosis of harmonic

current shown in Figure 5-4 is about -0.54 and kurtosis of the harmonic voltage is about

-0.9. It means the kurtosis of harmonic current sources are smaller than -0.5. It is satisfy

the condition of ICA algorithm that our proposed method is stronger enough to estimate

the system harmonic impedance.


- 91 -

(a) Amplitude

0 5 10 15 20 250

20

40

60

80

100

Harmonic order

Phas

e(de

gree

)

Estimated 1Theoretical

2Estimated 2

(b) Phase


We also discuss the Mean Absolute Error (MAE) of estimated system impedance with

changing the kurtosis of harmonic current sources from -0.8 to -0.3. The simulation was

performed 50 times to get the MAE. The simulation result is shown in Figure 5-25

where Figure 5-25(a) is amplitude and 5-25(b) is phase of estimated MAE of system

harmonic impedance. We discussed the some typical orders that include 5th, 7th, 15th and

25th. It is confirmed that the MAE of estimated amplitude and phase are became bigger

with the distribution of harmonic current sources near to the Gaussian distribution.

Especially, the biggest MAE occurs where the kurtosis of harmonic current sources is

-0.3. The simulation result also showed that the proposed method fail to estimate the


- 92 -

(a) Amplitude

(b) Phase

Figure 5-25. MAE of system impedance with change the kurtosis of harmonic current

source (model 1).

system side harmonic impedance when the kurtosis of harmonic current sources near to

the -0.3.


- 93 -

5.4. Conclusion

In this chapter, the technique that estimates system harmonic impedance at PCC is

presented. This method uses the complex value Fast-ICA algorithm. The method

introduced the Norton equivalent circuits to set up the linear mixing model of ICA

model to estimate the system harmonic impedance in condition that customer side

harmonic impedance changing. The proposed method just needs one point measurement

data at PCC which is not necessary the extra device. Furthermore, the method does not

require the any knowledge of system parameters. The estimate result of proposed

method is shown that the proposed method is suitable to estimate the system harmonic

impedance.

The main conclusions of this chapter have:

To model the system harmonic impedance as a blind source separation task and

to solve it using a statistical technique called ICA.

To propose a method to estimate the system harmonic impedance in the power

system without knowledge of network topology and parameters.

The proposed algorithm used the Fast-ICA twice to avoid the effect of

orthogonalization which is necessary in Fast-ICA algorithm.

The MAE of estimated system impedance is increased with the kurtosis of

harmonic current source when it changes from -0.8 to -0.3.


- 94 -

Chapter 6 Harmonic Contribution Evaluation of Industry Load using Complex ICA

- 95 -

CCHHAAPPTTEERR 66

Harmonic Contribution Evaluation of Industry Load

using Complex ICA

6.1. Introduction

Identification of harmonic sources in power systems has been a challenging task for

many years. Many techniques have been applied to determine customer and systems

responsibility for harmonic distortion. As the method with synchronized measurements

in multiple points in the network [32] is a rather difficult and expensive task, more

practical approaches are based on measurement data at the PCC between the customer

and the systems. Although there are a few indices dealing with harmonic contribution

determination at PCC, none are widely used in practice. The most common tool to solve

this problem is the harmonic power direction-based method [33]. In this method, if

harmonic active power flows from systems to customer, the system is considered as the

dominant harmonic generator. Unfortunately, [34] have proven that this qualitative

method is theoretically unreliable. Another group of practical methods for harmonic

source detection is to measure the system and customer harmonic impedances and then

calculate the harmonic sources behind the impedances. There are a number of variations

of this method [35]–[40]. These types of methods are very difficult to implement. The

main problem of these methods is that the customer impedances can only be determined

with the help of disturbances. Such disturbances are not readily available from the

system or are expensive to generate with intrusive means. Also, there have other ways

to estimate the harmonic current likes using the artificial neural networks (ANNs) [41]

[42].

Therefore, it is very desirable to estimate the harmonic contribution of customers and

system without any information or estimated information of network’s parameters just

using the measurement data at PCC.


- 96 -

Figure 6-1. Harmonic source detection at PCC.

In this chapter, the estimation of harmonic contribution of industry load at PCC using

complex value Fast-ICA is presented. The industry load can be departed to three part,

which before the working, under the working and after the working. And in each part,

the loads are damping in the same level. It is also same for nonlinear loads that are included in the full loads. Thus, It can be assumed that the customer impedance cZ is

not changing, more exactly, the cZ is changing small in mixing process. The main

advantage of this method is that only harmonic voltage and current have to be measured

to calculate harmonic contributions without knowing the systems information and

disrupting the operation of any loads. The artificial data simulation proved that the

proposed method is suitable to evaluate the harmonic contribution at PCC. The

measurement data simulation shows that the customer side is responsible to the harmonic

current distortion. The final goal of this method is to apply in a measurement device.

6.2. Power Direction Method

The power direction method can be explained with the help of Figure 6-1. In this figure, the disturbance sources are the customer harmonic source

cI and the system

harmonic sourceuI .

cZ anduZ are the harmonic impedances of the systems and

customer respectively. The circuit is applicable to different harmonic frequencies (the

values will be different). The task of harmonic source detection is to determine which

side contributes more to the harmonic distortion at the PCC, subject to the constraint


- 97 -

that measurements can only be taken at the PCC. To determine which side causes more

harmonic distortion at the harmonic order, the power direction method first measures

harmonic voltage and current at the PCC and then calculates the following harmonic

power index:

)(PCCPCC

IVRP ................................................................................... (6.1)

where PCC

V andPCC

I are the harmonic voltage and conjugation of harmonic current at

the PCC for a particular harmonic number. Since deals with one harmonic at a time in

this chapter, the subscript h that represents harmonic number h will be omitted

throughout the paper to avoid excessive subscripts. The direction of P is defined as

from system side to the customer side. Conclusion of the power direction method is the

following. If 0P , the system side causes more h th harmonic distortion.

If 0P , the customer side causes more h th harmonic distortion.

Reference [34] shows that the direction of active power is mainly affected by the

relative phase angle between the two harmonic sources. It has little bearing on the

relative magnitude of the sources. Note that it is the source magnitudes instead of phase

angles that are of main interest for the harmonic source detection problem. The

simulation results show that the power direction method is theoretically incorrect and

should not be used to determine harmonic source locations.


- 98 -

6.3. Harmonic Contribution Evaluation of Industry Load using Complex ICA

6.3.1. Principle of Proposed Method

In this section, the simple linear ICA model of system harmonic impedance is

proposed. The proposed method is only need the measurement of the harmonic current

and voltage at the PCC.

The problem of harmonic current source detection is to determine the contribution of

systems and customer sides in PCC. For this problem, it is common to assume that the

system and customer sides are represented by their respective Norton equivalent circuits as shown in Figure 6-1. In this figure, cI and

uI are the customer and system harmonic

current sources, cZ and uZ are the customer and system harmonic impedances,

respectively. Referring to Figure 6-1, harmonic current

pccI and harmonic voltage

pccV at PCC can

be expressed as:

cu

ccuu

pccZZ

IZIZI

................................................................................ (6.2)

)(cu

cu

cu

pccII

ZZ

ZZV

........................................................................ (6.3)

The current contribution of customer side pcccI and system side pccuI

can be determined

as

u

cu

u

pccuI

ZZ

ZI

................................................................................ (6.4)

c

cu

c

pcccI

ZZ

ZI

................................................................................. (6.5)

According to (6.2)-(6.5), the general representation of the linear equation can be

written as (6.6) which can be seen as a simple ICA model.


- 99 -

pccc

pccu

ucpcc

pcc

I

I

ZZV

I

11 .................................................................. (6.6)

where, pccI and pccV are corresponding to measurements x as:

pcc

pcc

V

I

X

pcccI and pccuI

are corresponding to independent components s as:

pccc

pccu

I

I

S

The mixing matrix as

uc ZZ 11

A

respectively. We have discussed above that the system harmonic impedance uZ is

assumed not changing in the mixing process, but the system harmonic impedance cZ

is changing in common load. That is why our purpose is to study the contribution of

industrial load. We will discuss it in next section for more details.

To estimate the harmonic current contribution of industry load, it is need to estimate

the following equation:

pcc

pcc

ucpccc

pccu

V

I

ZZI

I

111 .................................................................. (6.7)


- 100 -

where separation matrix 111

ucZZ W .

6.3.2. Characters of industrial load

Figure 6-2 shows the measurement data of four days which measurement at a factory

shown in Figure 6-3. The voltage of connecting point is 6.6kV. It is including several

factories as soon as a photovoltaic system where the ratio is 110kW. The measurement

data shown in Figure 6-2 is not a special one, it is more common in industrial load.

From the fundamental current, we can depart the one day data to three parts, which

before the working, under the working and after the working. And in each part, the

loads are swinging in the same level. It is also same for nonlinear loads that are included in the full loads. In the end, it is enough to assumed that the customer impedance cZ of

(6.6) is not changing, more exactly, the cZ is changing small in mixing process in each

parts.

Figure 6-2. Measurement data at industrial load include fundamental current,

5th harmonic current and 7th harmonic current.


- 101 -

Figure 6-3. Industrial model of measurement data.

Based on the discussion above, we assume that the mixing matrix A is not

changing in the mixing process when depart the one day to three parts, which is before

the working, under the working and after the working.

For the harmonic current source’s independent and non-Gaussian, we already have

conclusion in chapter 5 which based on measurement data.

6.3.3. Dealing to Ambiguities of ICA

The ICA method have two ambiguities which shown in Chapter 2. In proposed ICA

model (6.6), we have some prior knowledge on the mixing matrix and these can be used

to solve the ICA’s ambiguities problems. For variances problem, can use the first row of

mixing matrix in (6.6), to normalize the separation matrix. Also, permutation problem is solved considering the

cuZZ .

6.3.4. Algorithm of Proposed Method

The harmonic contribution algorithm with complex ICA described as follows.

1) Centralized and whiten the measurement data.

2) Applying the complex Fast-ICA algorithm to measurement data to obtain the

separation matrix HW . 3) Using the prior knowledge of mixing matrix to Reorder and scale the A , where

)( Hinver WA


- 102 -

Table 6-1. Parameters of harmonic current sources.

hI m s k

uI (Real part) 1 4 0.5 0 -0.8

uI (Imaginary part) -1 3.6 0.6 0 -0.8

(First period)

cI (Second period) (Real part) (Third period)

1.5 1.5 1.5

2.2+0.044sin(t) 5+0.1cos(t) 2+0.04sin(t)

0.4 0.6 0.35

0 0 0

-0.8 -0.8 -0.8

(First period) cI (Second period)

(Imaginary part) (Third period)

-2 -2 -2

2.2+0.044cos(t) 5+0.1sin(t) 2+0.04sin(t)

0.3 0.5 0.3

0 0 0

-0.8 -0.8 -0.8

4) Calculate the harmonic current contribution using (6.7).

5) Perform steps 1)-4) for each harmonic frequency of interest.

6.4. Numerical Simulations

6.4.1. Artificial Data

To calculate the harmonic currrent and harmonic voltage at PCC of Figure 6-1 based

on (6.6), we need the harmonic currrent sources and harmonic impedances of system

side and customer side.

For harmonic current source, using the function pearsrnd of MATLAB program to represent the both system harmonic current

uI and system harmonic current c

I . The

function pearsrnd is shown in (5.6). Parameters of system harmonic current source and

customer harmonic current source are shown in Table 6-1, where 036,,0362,036 Tt , T is sample data. We assumed that the parameters of

system harmonic current source uI do not change for full day and the customer

harmonic current source c

I change in three steps.

The harmonic currrent sources that based on (5.6) and Table 6-1 are shown in Figure 6-4 and Figure 6-5. Figure 6-4 is the system side harmonic current

uI , Figure 6-5 is


- 103 -

Figure 6-4. System harmonic current uI .

Figure 6-5. Customer harmonic current

cI .

customer side harmonic current c

I , where above line is real part and below line is

imaginary part of current sources. The system harmonic current are mainly damping

around the 4(A) for real part and -3.6(A) for imaginary part.

The customer side harmonic current is departed to three parts, where, the summit

harmonic current from 9h to 18h are more than other times because nonlinear load of

the work time are more than before the work and after the work.

For harmonic impedance, we assumed the system side harmonic impedance )(65.52.1

uZ , which is not changed in the mixing matrix. The customer side

harmonic impedance cZ are shown in Figure 6-6, where above line is real part and

below part is imaginary part of complex value. The cZ also departed to three parts


- 104 -

Figure 6-6. Customer side harmonic impedance.

where the impedance of work time is smaller than other times. Furthemore, we assumed the cZ is changed small region in each parts.

The measurement data at PCC are shown in Figure 6-7 which is calculated by (6.7). Figure 6-7(a) is harmonic current

pccI and Figure 6-7(b) is harmonic voltage

pccV . The

above line is real part and below line is imaginary part in both figures. Result of

assumed, the harmonic current is delay from the harmonic voltage which are measured

at same point. From the Figure 6-7(a) we can not decide the direction of harmonic

current and also can not decide which side is more contribute to this harmonic current.

That is way we need the proposed method to estimate the harmonic contribution of

industrial load at PCC.


- 105 -

(a) Harmonic current pcc

I

(b) Harmonic voltage pcc

V .

Figure 6-7. Measurement data at PCC.

Employ the proposed method to the above measurement data of Figure 6-7, we can get

the harmonic currrent contributin of system side and customer side withod disturbing the

systems. The result are shown in Figure 6-8 and Figure 6-9. The Figure 6-8 is the system side harmonic current

pccuI

which have theoretical value and estimated value. The


- 106 -

Figure 6-8. Separated system side harmonic current pccu

I

.

Figure 6-9. Separated customer side harmonic voltage pccc

I

.

pccuI

have a big change from the rest time to the work time even if the system side

harmonic current do not have this characters (Figure 6-4). It is theoretically right when

the customer side impedance of (6.4) is changed sharply to small, more harmonic

currents flow to the customer side from system side. There have bigger errors in the

work time than other times because the effect of customer side harmonic impedance is

bigger in this period.

The Figure 6-9 is the customer side harmonic current pccc

I

which also include

theoretical value and estimated value. The Figure 6-9 shows that the proposed method


- 107 -

can estimate the harmonic contribution of industrial load very well. The estimated value

is exactly represent the theoretical value of pccc

I

. From the simulation result , we also

know that the harmonic current at PCC is mainly caused by the customer side harmonic

source even if the the both system and customer have same level of harmonic current

source. It is because the system side harmonic impedance is more small than the

customer side impedance, thus, the harmonic currrnet are mainly follow to the system

harmonic impedance.

6.4.2. Measurement Data

In this sectin, we used the 5th harmonic measurement data which one day data of

Figure 6-2 to verify our proposed method. The measurement data are shown in Figure

6-10 and Figure 6-11. The Figure 6-10 is the 5th harmonic current at PCC where the

above line is real part and the below line is imaginary part of complex value. The both

real part and imaginary part are became bigger in the work time which from about 9 to

18:30. Also, it is changing in small regine in each parts. For a short measurement times,

the kurtosis of 5Th harmonic current is about to -5.4, which satisfy the one of ICA’

restriction.

Figure 6-10. The 5th Harmonic current at PCC.


- 108 -

Figure 6-11. the 5th Harmonic voltage at PCC.

The Figure 6-11 is the 5th harmonic voltage at PCC where the above line is real part

and the below line is imaginary part. The Figure 6-11 shows that the harmonic voltage

at PCC is hardly influenced by the customer harmonic source even if it is changing

sharply. In short measurement period, the kurtosis of 5th harmonic voltage is about to

-0.9. The 5th harmonic voltage also satisfies the ICA’s restriction.

The simulation results are shown in Figure 6-12 and Figure 6-13. Figure 6-12 is the separated system side harmonic current

pccuI

, where above line is real part and below

line is imaginary part. The simulation result shows that the system side harmonic

current are increasing in the work time just like the artificial data. It means that the

system side harmonic current are more flow to the customer side in the work time

because of the decreasing of customer side impedance. Figure 6-13 is the separated customer side harmonic current

pcccI

, where above line

is real part and below line is imaginary part. In figure 6-13, the pccc

I

has a big

amplitude in the work time shows that the harmonic current are mainly come from the

customer side when the factory working. The simulation result shows that the customer

is not full responsible for harmonic current at PCC. The system side also have

responsible for harmonic current distortion at PCC.


- 109 -

Figure 6-12. Separated system side harmonic current pccu

I

.

Figure 6-13. Separated customer side harmonic current pccc

I

.


- 110 -

6.5. Conclusion

In this Chapter, the technique which evaluates the system and customer harmonic

contribution at PCC is presented. The proposed method is based on the complex ICA

algorithm where transforms the observed signals into mutually statistically independent

signals. The main advantage of this method is that only harmonic voltage and current

have to be measured to calculate harmonic contributions without knowing the systems

information and disrupting the operation of any loads. The artificial data simulation

proved that the proposed method is suitable to evaluate the harmonic contribution at PCC.

The measurement data simulation shows that the customer side is mainly responsible for

the harmonic current distortion. However, the customer is not full responsible for

harmonic current at PCC. The system side also have responsible for harmonic current

distortion.

Chapter 7 Conclusions

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CCHHAAPPTTEERR 77

Conclusions

The main objective of this thesis is to estimate the system harmonic impedance and

the harmonic current contribution of industry load at PCC. The both methods are based

on the complex value Fast fixed point Independent Component Analysis (Fast-ICA)

algorithm, which is the most powerful algorithm in ICA algorithms. The main advantage

of these methods is that only harmonic voltage and current have to be measured for

estimation without knowing the systems information and disrupting the operation of any

devices. This thesis is organized in 7 chapters, described as follows:

In Chapter 1, as the introduction of the thesis, the background and motivation of these

researches are described. A common philosophy of harmonic analysis in power system is

to conduct a deterministic study based on the worst case in order to provide a safety

margin in system design and operation. However, this often leads to overdesign and

excessive costs. Field measurement data clearly indicates that voltage and current

harmonics are time-variant due to continual changes in load conditions. Consequently,

statistical techniques for harmonic analysis are more suitable, similar to other

conventional studies like probabilistic load flow and fault studies.

In Chapter 2, as the basic of our study, fundamentals of harmonic, harmonic sources,

effect of harmonic distortion, limits of harmonic distortion and mitigation techniques of

harmonics are presented. The wide spread utilization of power electronic devices has

significantly increased the number of harmonic generating apparatus in the power

systems. The harmonics distortions of the voltage and current have adverse effects on

electrical equipment such as increase losses of devices, equipment heating and loss of life.

To eliminate the harmonic current and voltage distortion, the harmonic analysis becomes

an important and necessary task for engineers in power systems.

In Chapter 3, the ICA algorithm, known as one of Blind Source Separation (BSS)

techniques, is introduced. BSS techniques have received attention in applications where


- 112 -

there is little or no information available on the underlying physical environment and

the sources. BSS algorithms estimate the source signals from observed mixtures. The

word ‘blind’ emphasizes that the source signals and the way the sources are mixed, i.e.

the mixing model parameters, are unknown. ICA transforms the observed signals into

mutually, statistically independent signals. It thus exploits the statistical independence

between the sources. Statistical properties of signals are a key factor in estimation by

ICA, since there is almost no other information available.

In Chapter 4, the characterizations of the waveform distortion from grid-connected

Photovoltaic (PV) plants under different installing capacity are described. The

stochastic aggregate harmonic load model is introduced to represent the harmonic

current of loads and used the inverter model that is modeled by measurement data to

represent the harmonic current of PV. The numerical simulation consists of two parts:

Mega-PV and residential type PV. As a result, with the increasing of installed PV, the

harmonic current and harmonic voltage at the connected point also increased. When the

installed capacity of PV is up to 30%, the harmonic current distortion at the connected

point will exceed the standard limit of Japan. The simulation shows that increasing of

PV will make a serious power quality problem where it is connected.

In Chapter 5, the technique that estimates system harmonic impedance at PCC is

described. Harmonic impedances of a supply system characterize the frequency

response characteristics of the system at specific buses. It is very desirable in many

applications to directly measure the system harmonic impedances. This method uses the

complex value Fast-ICA algorithm to estimate the system harmonic impedance. The

method introduced the Norton equivalent circuits to set up the linear mixing ICA model

to estimate the system harmonic impedance in condition that customer side harmonic

impedance changing. The method just needs one point measurement data at PCC which

is not necessary the extra device. Furthermore, the method does not require the any

knowledge of system parameters. The method used the Fast-ICA twice to avoid the

effect of orthogonalization which is necessary in Fast-ICA algorithm. As a result, the

method can estimate the system harmonic impedance correctly. However, when

changed the kurtosis of harmonic current source equation from -0.8 to -0.3, the MAE of

estimated system impedance is also increased

In Chapter 6, the estimation of harmonic contribution of industry load at PCC is

described. The industry load can be departed to three part easily, which before the


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working, under the working and after the working. And in each part, the loads are

swinging in the same level. Thus, it can be assumed that the customer side harmonic

impedance of each part is changing small in mixing process. As a result, the artificial

data simulation proved that the method is suitable to evaluate the harmonic contribution at

PCC. The measurement data simulation shows that the customer side is mainly

responsible for the harmonic current distortion. However, the customer is not full

responsible for harmonic current at PCC. The system side also have responsible for

harmonic current distortion.

The original contributions of this thesis can be summarized as follows:

With the increase of installed PV, the harmonic current and harmonic voltage of

power distribution systems also increased. When the capacity is up to 30%, the

harmonic current distortion will exceed the standard current limit.

System side harmonic impedance and harmonic contribution of industrial load

problems are formulated as a blind source separation task where statistical

properties of harmonic sources are considered by modeling them as random

variables. A statistical technique called independent component analysis is used

to perform the estimation.

The proposed methods just need one point measurement data (harmonic current

and harmonic voltage), which is easy to performed in distribution system without

the extra devices.

The proposed methods can estimate the system harmonic impedance and

harmonic contribution of industrial load without knowledge of network topology

and parameters.

In the system harmonic impedance estimation algorithm, the method uses the

Fast-ICA twice to avoid the effect of orthogonalization which is necessary in

Fast-ICA algorithm.

The numerical simulations have shown that the proposed method can estimate the

system harmonic impedance even if the customer harmonic impedance is

changing.

The estimated harmonic contribution of industrial load shows that the customer

side is mainly responsible to the harmonic current distortion at PCC. However, the

customer is not full responsible for harmonic current at PCC. The system side also

have responsible for harmonic current distortion.


- 114 -

There are many works to do in the future. First of all, it is necessary to verify the

proposed algorithms by plenty measurement data where it is mainly tested by

numerical simulation in this thesis. Others like improvement of estimation

performance, harmonic voltage contribution estimation are also needed to do.

The final goals of these algorithms are to apply in a measurement device or in a

harmonic filter to decrease the harmonic current and voltage distortion of the power

systems.

References

- 115 -

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References

- 120 -

List of Acronyms

- 121 -

List of Acronyms

AC

DG

PV

BSS

ICA

EEG

PCC

ANN

Fast-ICA

THD

TDD

RMS

VFD

IEEE

APF

FE

FC

ICs

LV

MV

AHL

PDF

PCU

DC

FFT

MAE

alternate current

distributed generation

photovoltaic

blind source separation

independent component analysis

electroencephalogram

point of common coupling

artificial neural network

fast fixed point independent component analysis

total harmonic distortion

total demand distortion

root mean square

variable frequency drive

institute of electrical and electronics engineers

active power filter

filtering effectiveness

filtering capacity

independent components

low voltage

medium voltage

aggregate harmonic load

probability density function

power conditioning unit

direct current

fast fourier transform

mean absolute error

List of Acronyms

- 122 -

Acknowledgements

- 123 -

Acknowledgements

I wish to express my great thanks to many people who have enabled and assisted in

this research. Without their warm helps, I can not finish this thesis.

First and foremost I would like to express my sincerely gratitude to my advisor

Professor Atsushi Ishigame, for his constant guidance, encouragement, and excitation;

without him, this thesis would have been impossible.

I also wish to thank Assistant Professor Satoshi Takayama, for his kindness, support,

concern and valuable suggestions.

I would like also to thank Professor Shigeo Morimoto and Professor Keiji Konishi,

for their precious time in serving my dissertation committee, and their detail comments

and valuable suggestions.

I would also like to thank fellow graduate students for creating a pleasant working

environment.

I have to save my final thoughts for my family. My father Tie Bao, mother Mei Hua,

sister Ying Chun and brother Siqintu. Of course, my wife Wurizhe gave me the most

energetic help. They all have always stood behind me, and have provided me with

immeasurable love, encouragement and support. They have been there through all the

trials and tribulations for my graduate education. They give me the strength and

determination to forget ahead through the most difficult times.

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