ghosh monday july 23 2003

7/29/2019 Ghosh Monday July 23 2003

1/81

Applications of PoApplications of Po

Electronics to PoElectronics to PoTransmission & DistrTransmission & Distr

SystemsSystems

Arindam GhoshArindam Ghosh

Dept. of Electrical EnginDept. of Electrical Engin

Indian Institute of TechnologIndian Institute of TechnologIndiaIndia

EE--mail:mail: aghoshaghosh@@iitkiitk.a.a


2/81

Lecture ScheduLecture Schedu

Day 1:Day 1: Problems with bulk power Problems with bulk power & shunt compensation of transmi& shunt compensation of transmisystems.systems.

Day2: Series compensation of trDay2: Series compensation of tr

systems & other FACTS controllsystems & other FACTS controll Day 3: Power quality, custom powDay 3: Power quality, custom pownetwork reconfiguring devices.network reconfiguring devices.

Day 4: Distribution STATCOM, DDay 4: Distribution STATCOM, D

Voltage Restorer (DVR) & UnifieVoltage Restorer (DVR) & UnifieQuality Conditioner (UPQC)Quality Conditioner (UPQC)


3/81

Power TransmissioPower Transmissio

CharacteristicCharacteristicsTransmission lines are represeTransmission lines are represe

Lumped parameter short lineLumped parameter short line

50 mi).50 mi). Medium lines represented byMedium lines represented by or nominalor nominal--T models.T models.

Long lines represented by dLong lines represented by diparameter models.parameter models.


4/81

LosslessLossless Line RepreseLine Represe

TotalVoltage,Midpoint~ Receivin

~

Voltage,Source

~

====XV

VV

M

RS


5/81

Power Flow OvPower Flow Ov

Transmission LTransmission L~and~Let == VVVV RS

( )

jX

jVV

IS si1cos~

Then

+

=

XjVVIVjQP SSSS sin~~

havethenWe

2

+==+


6/81

PowPow

TraTra

V

PPP RSesin

linover thepowerrealThe

2

===

X

jVVIVjQP SRRR

sin~~

Similarly2

==


7/81

P

TT

( )

X

VQQQ RSl

cos12babsorbedpowerreactiveThe

2

==

The power-angle curves are shown inslide. Note that we have assumed parameter representation of the

a similar pattern also occurs even is modeled using distributed param


8/81

PowerPower--Angle CurAngle Cur

It is assumed thatIt is assumed that2

max ==X

VP


9/81

Midpoint VoltageMidpoint Voltage

2cos~ =VVM


10/81

FerrantiFerranti EffectEffect

( )( )RV

XYYV ~

22~

=


11/81

In 500 km long line where per kIn 500 km long line where per kreactance and admittance are reactance and admittance are and 0.0000031734 mho, the rand 0.0000031734 mho, the revoltage rises 25.64% above thevoltage rises 25.64% above the

voltage under novoltage under no--load (or even ligload (or even ligcondition. Also since the paramecondition. Also since the paramedepend on the line length, the rdepend on the line length, the rvoltage will be different for dvoltage will be different for d

lengths. In fact the longer tlengths. In fact the longer tmore is the voltage rise.more is the voltage rise.


12/81

Voltage StabilitVoltage Stabilit

Voltage stabilityVoltage stabilityis the ability ofis the ability ofsystem to return to the nominal system to return to the nominal voltages of all buses following a dvoltages of all buses following a din the system.in the system.

In addition, the system shall alsoIn addition, the system shall alsomaintain the nominal voltage at bmaintain the nominal voltage at bsteady state.steady state.

Conversely, the voltage instabiConversely, the voltage instabi

when the voltages at different when the voltages at different continuously following a disturbacontinuously following a disturbachange.change.


13/81

VV

( )pu whe

11~

+=

RLratio

RZ

V


14/81

VVo

( { (ratioratioRratio

RZZ

ZP

++

+=cos1

cos122

2

The transmission line impedance is fThe transmission line impedance is fgiven line. Thereforegiven line. Therefore ZratioZratio decrdecreload impedance increases. The maload impedance increases. The mapower occurs when the load and lipower occurs when the load and li

impedances are same. The power impedances are same. The power after that and the voltage monotafter that and the voltage monotodecreases.decreases.


15/81

VolVol


16/81

VVo


17/81

Angle StabilityAngle Stability

PPmm = Mechanical powe= Mechanical powe

PPee = Electrical power= Electrical power


18/81

MultimachineMultimachine StabStab


19/81

MultimaMultimac

Modern power systems are intercoModern power systems are interco

operate close to their stability limoperate close to their stability liminterconnected systems it is comminterconnected systems it is commnatural response of a group of clonatural response of a group of clomachines oscillating against othemachines oscillating against othemachines.machines.

These oscillations have a frequeThese oscillations have a frequen0.1 Hz to 3 Hz.0.1 Hz to 3 Hz.

The lowest frequency mode The lowest frequency mode generators of the system. Thgenerators of the system. Thigroups the system into two groups the system into two generators in one part oscillatinggenerators in one part oscillatingother part (interother part (inter--area oscillation).area oscillation).


20/81

MultimaMultima

The higher frequency modes The higher frequency modes localized with small groups oscillalocalized with small groups oscillaeach other (local modes).each other (local modes).

Unfortunately, the interUnfortunately, the inter--area osarea osbe initiated by a small disturbancbe initiated by a small disturbancof the system.of the system.

These small frequency oscillationThese small frequency oscillationthe category of dynamic stabithe category of dynamic stabilanalyzed in linear domain tanalyzed in linear domain t

linearizationlinearization of the entire intof the entire intsystems model.systems model.


21/81

Power System StabilizPower System Stabiliz

An AVR regulates the generatAn AVR regulates the generatvoltage and also reduces the peakvoltage and also reduces the peakswing following any disturbance.swing following any disturbance.

However, its high gain contributeHowever, its high gain contributesdamping to the system and this rdamping to the system and this rlow frequency oscillations in the sylow frequency oscillations in the sy

These oscillations are the resThese oscillations are the resperiodic interchange of kinetic eneperiodic interchange of kinetic enedifferent generator rotors.different generator rotors.

A PSS provides positive damping tA PSS provides positive damping toscillations through negative feedoscillations through negative feedchanges in rotor kinetic energy.changes in rotor kinetic energy.


22/81

SubsynchronouSubsynchronou

Resonance (SSRResonance (SSR

SSR usually occurs in series capacitor comSSR usually occurs in series capacitor comtransmission systems. For a radial serietransmission systems. For a radial seriesystem, the naturalsystem, the natural undampedundamped frequencfrequenc

Cn

fX

Xff == 0

0

Complement

ff00 being the nominal system frequencybeing the nominal system frequency


23/81

SubsynchronousSubsynchronous ReRe

SSR occurs when the complementSSR occurs when the complementis close to one of theis close to one of the torsionaltorsional frfrof the turbineof the turbine--generator shaft sgenerator shaft s

A small voltage induced by rotor A small voltage induced by rotor can result in largecan result in large subsynchronousubsynchronouthat produce an oscillatory compothat produce an oscillatory comporotor torque whose phase is suchrotor torque whose phase is suchenhances the rotor oscillations.enhances the rotor oscillations.

If this torque overcomes the meIf this torque overcomes the me

damping, the oscillation in the shdamping, the oscillation in the shagrows and can reach damaging levgrows and can reach damaging lev


24/81

Shunt CompensatiShunt Compensati

Transmission SystTransmission Syst

A device that that is connected in paA device that that is connected in patransmission line is called atransmission line is called a shunt cshunt cIt is referred to as a compensatorIt is referred to as a compensator

compensates for the reactive powecompensates for the reactive powesystem. It cansystem. It can

Improve the voltage profileImprove the voltage profile

Improve the powerImprove the power--angle characteangle characte

Improve the stability marginImprove the stability margin

provide damping to power oscillatioprovide damping to power oscillatio


25/81

Ideal Shunt CompeIdeal Shunt Compe

The ideal shunt compensator is reprThe ideal shunt compensator is repran ideal current source that suppan ideal current source that suppreactive power and no real power.reactive power and no real power.assume that it is connected at thassume that it is connected at thaa losslesslossless line.line.


26/81

Voltage ProfilVoltage Profile

ImprovementImprovement

The figure shows the ideal voltageThe figure shows the ideal voltage--characteristics of an ideal shunt characteristics of an ideal shunt in which the midpoint voltage heldin which the midpoint voltage heldirrespective of the current injectirrespective of the current inject


27/81

InIn

We thus have to generate a currWe thus have to generate a curr

phase quadrature with the midphase quadrature with the midvoltage. As a consequence the voltage. As a consequence the injected by the compensator isinjected by the compensator is

( )

( ){ } ( 22cos14~

~~~Since

~,

2

2~

~,

~,

~For

=

=

=

=

===

X

VjI

III

VI

Xj

VVI

VVVVVV

Q

SRQ

RS

MRS


28/81

Real & Reactive PReal & Reactive Po

Midpoint shunt compensation improvMidpoint shunt compensation improvflow over a line. The real power floflow over a line. The real power floand the reactive power absorbed band the reactive power absorbed bareare

( ) [ cos18,2sin222

==XVQ

XVP le

The reactive power generated by thThe reactive power generated by thcompensator iscompensator is

( )[ ]2cos14 2

=X

VQQ


29/81

It is assumed thatIt is assumed that p12

=X

V

Power Angle CharacPower Angle Charac


30/81

ReaRea

ReRe

FoFortratrauniunipowpowrourou

uniunireqreqshushu

HoHoperper

an an perper

It is assumed thatIt is assumed that pu12

=X

V


31/81

RRe

RR

For For trtrpperere

ininlolololommvovo


32/81

Improvement of StImprovement of St

MarginMargin


33/81

Power Swing DamPower Swing Dam

The swing equation is given byThe swing equation is given by

( Mems

VPPdt

dH ~,

22

2

=

wherewhere PPmm is the mechanical power is the mechanical power that the electrical powerthat the electrical power PPee is dis dfunction of the load anglefunction of the load angle and tand tmagnitude of the midpoint voltagmagnitude of the midpoint voltag

because both these quantities cabecause both these quantities capower transmitted over a transmpower transmitted over a transmfor constant voltage at the two efor constant voltage at the two e


34/81

Power SPower S

TheThe linearizationlinearization of the swing equaof the swing equa

~~

22

2

+

+

e

M

M

e

s

PVV

P

dt

dH

The regulation of the midpoint voltagThe regulation of the midpoint voltag

the magnitude ofthe magnitude of VVmm is held constais held consta0

~= MV

Therefore we getTherefore we get

22

2

=

+

e

s

P

dt

dH


35/81

Power SPower S

The roots of the above equationThe roots of the above equation

on the imaginary axis of the son the imaginary axis of the s--plplimplies that the load angle will oimplies that the load angle will osa constant frequency ofa constant frequency of

es PH2

Obviously this solution is not accObviously this solution is not acce

must therefore add a derivative must therefore add a derivative angle in theangle in the linearizedlinearized swing equaswing equa


36/81

Power SPower S

Let us vary the midpoint voltage accLet us vary the midpoint voltage acc

dt

d

KV MM

=

~

We therefore getWe therefore get

~2

2

2

+

+

M

M

e

s

P

dt

dK

V

P

dt

dH

This is the equation of a 2This is the equation of a 2ndnd order syorder sy

0>MK

Therefore a stable solution is guaranTherefore a stable solution is guaran

,0~

,02 >> eMes PVPH


37/81

An ExampleAn Example

M0.4,pu3.0,pu5.0

,pu01~,pu401~

===

===

HXX

VV

F

RS


38/81

ExampExamp

Let us regulate the midpoint voltaLet us regulate the midpoint voltaper unit using a PI controller of per unit using a PI controller of

+= KVKV IMPF 1~

1~

Note that for purely reactive injecNote that for purely reactive injecshunt compensator, the angle of shunt compensator, the angle of

source is set such that it is in phsource is set such that it is in phmidpoint voltage, i.e.,midpoint voltage, i.e.,

PM

M

F

VV

VV

~

~

~~

=


39/81

ExampExamp

System response to a pertuSystem response to a pertu


40/81

ExampExamp

To improve damping, we now introdTo improve damping, we now introdthat is proportional to the deviatthat is proportional to the deviatmachine speed in the feedback lomachine speed in the feedback lothat the control law is given bythat the control law is given by

( ) ( )VKVKV MIMPF += ~

1~

1~

The last term adds damping to theThe last term adds damping to the


41/81

ExampExamp

System response when the breakeSystem response when the breakeinadvertently.inadvertently.


42/81

Practical ShunPractical Shun

CompensatorCompensator

The above example demonstrateThe above example demonstratefunctioning of afunctioning of a Static CompensStatic Compens((STATCOM) that produces a voSTATCOM) that produces a vol

fundamental component of whichfundamental component of whichwith the midpoint voltage. It conwith the midpoint voltage. It coninverter based Synchronous Voltinverter based Synchronous Volt(SVS) and a connecting transfor(SVS) and a connecting transfor

The first generation shunt compThe first generation shunt compthethe StaticStaticVArVArCompensatorCompensator(SV(SVactually is a variable reactance.actually is a variable reactance.


43/81

Uncompensated SyUncompensated Sy

In the figure below the power systeIn the figure below the power systerepresented by itsrepresented by its TheveninThevenin equivequivfrom the midpoint. It is assumed from the midpoint. It is assumed reactive load is connected at the reactive load is connected at the


44/81

The midpoint voltage increases linThe midpoint voltage increases lincapacitive load current and decrcapacitive load current and decrwith inductive load current.with inductive load current.


45/81

Effects of increasingEffects of increasing VVthth (left) an(left) an


46/81

Components of SComponents of S

The building blocks of an SVC areThe building blocks of an SVC are

Saturated ReactorSaturated Reactor

ThyristorThyristor Switched Capacitor (TSwitched Capacitor (T

ThyristorThyristor Controlled Reactor (TControlled Reactor (T

ThyristorThyristor Switched ReactorSwitched Reactor ThyristorThyristor Controlled TransformControlled Transform

An SVC is made of the combinatioAn SVC is made of the combinatio

more than one of the above commore than one of the above comfixed capacitor banks.fixed capacitor banks.


47/81

Thyristor SwitchThyristor Switch

Capacitor (TSCCapacitor (TSCIn a TSC a capacitor is connected In a TSC a capacitor is connected

with two opposite pole thyristorwith two opposite pole thyristorcurrent flows through the capaccurrent flows through the capacthe opposite poled thyristors arthe opposite poled thyristors ar


48/81

TSCTSC -- EquivalEquival

TSCsTSCs always come in a pack. The efalways come in a pack. The efreactance of the TSC pack can breactance of the TSC pack can bby switching a TSC on or off. Foby switching a TSC on or off. Foanan nn--pack TSC, the effective reapack TSC, the effective rea

kCk

jXeq 1,0,1

==

wherewhere kk is the number ofis the number of TSCsTSCs cc


49/81

TSCTSC -- TransientTransient--

TSC suddenly blocks current or allTSC suddenly blocks current or all

current through it.current through it. Hence severe switching transientsHence severe switching transients

TSC is switched off while the currTSC is switched off while the currthorough it is not zero.thorough it is not zero.

Similarly, the device must be switcSimilarly, the device must be switcparticular instant of the voltage cyparticular instant of the voltage cy

For example, let us consider that aFor example, let us consider that a

supplied by a voltage sourcesupplied by a voltage source vvSS andandcapacitor voltagecapacitor voltage vvCC and a currentand a currentthe capacitor.the capacitor.


50/81

TSCTSC -- TransientTransient--frefre

Then asThen as iiCC == CC((dvdvCC//dtdt),), the currethe currewhenwhen dvdvCC//dtdt= 0, i.e., when the cap= 0, i.e., when the capvoltage reaches its peak.voltage reaches its peak.

Thus for transientThus for transient--free switchingfree switching,

ensured that the capacitor voltageensured that the capacitor voltageits positive peak or negative peak fits positive peak or negative peak fturn on or turn off.turn on or turn off.

The transientThe transient--free switching is shfree switching is sh

next slide, in which the instant of next slide, in which the instant of and off are also indicated.and off are also indicated.


51/81

TSCTSC -- TransientTransient--ff

The capacitor voltageThe capacitor voltage vvCC is kept atis kept atthe supply voltage when the switchthe supply voltage when the switchindicating an open circuit.indicating an open circuit.


52/81

Thyristor SwitchThyristor Switch

Reactor (TCR)Reactor (TCR)In a TCR a reactor is connected in seIn a TCR a reactor is connected in se

opposite poled thyristors. One of topposite poled thyristors. One of tthyristors conducts in each half cythyristors conducts in each half cyfrequency.frequency.


53/81

TCRTCR--VV

The gating signal to each thyristorThe gating signal to each thyristor

by an angleby an angle (often called the firi(often called the firiconduction angle) from the zero crconduction angle) from the zero crsource voltage.source voltage.


54/81

The conduction angle must be in thThe conduction angle must be in th

180180. For. For == 9090, the current, the currentconduction and will lag the voltage conduction and will lag the voltage

TCRTCR--VV


55/81

TCRTCR--VI VI

ForFor = 18= 1800, the current will b, the current will b


56/81

TCRTCR-- FundameFundame

The TCR fundamental reactance iThe TCR fundamental reactance i

from the following equationfrom the following equation

( ) ( )

=

=

for0

forcoscossin1

tL

VdV

Lim

t

m

L

The fundamental current is given The fundamental current is given above equation asabove equation as

( ) ( = sinsin tX

Vi

L

mLf


57/81

TCRTCR-- FundameFundame

Since the TCR fundamental curreSince the TCR fundamental curre

the voltage by 90the voltage by 90, we have the fu, we have the fufrequencyfrequency susceptancesusceptance of the TCof the TC

( ) Ls

Lf

XV

I

BL

sin

~

~

==

TheThe susceptancesusceptance is zero foris zero for = 0= 0andand reciprocal of the chosen valuereciprocal of the chosen value

= 180= 180 (( = 90= 90).).


58/81

TCRTCR-- FundameFundamen


59/81

TCRTCR

The TCR current will not contain aThe TCR current will not contain a

even harmonics, but only the odd heven harmonics, but only the odd hThe peak of the harmonic currentThe peak of the harmonic currentbyby

( ){ }

( )

( ){ }

( )

scos

12

1sin

12

1sin4_

+

+

+=

n

n

n

n

X

VI

L

mnL

The normalized harmonic spectrumThe normalized harmonic spectrumvalues ofvalues of are shown in the nextare shown in the next

As the firing angle increases, the As the firing angle increases, the the harmonic current increases.the harmonic current increases.


60/81

TCTC

BotBot

concon((//withwithresresu

redredpeapeafunfuncurcurincrincrharharconcon


61/81

Typical SVC ScheTypical SVC Sche


62/81

SVC VI CharacteriSVC VI Characteri


63/81

SVC VI SVC VI


64/81

SVC VI SVC VI

Assume that the system is oAssume that the system is o

with a voltagewith a voltage VV00.. If the system voltage increaIf the system voltage increa

increase toincrease to VV11 without SVC. without SVC.

the SVC moves the operatinthe SVC moves the operatinBBby absorbing inductive curby absorbing inductive curholds the voltage atholds the voltage at VV33..

Similarly the SVC holds the Similarly the SVC holds the

VV44 for a decrease in the sysfor a decrease in the sysvoltage.voltage.


65/81


66/81

SVSSVS --VoltagVoltag


67/81

The transformer primaries provideshift of 30.


68/81

1212--StSt

WaveformWaveform

SpSp


69/81

MultiMulti--Step SVSStep SVS

In a similar way 6In a similar way 6

nn--step output step output v

be obtained by connectingbe obtained by connecting nnbasbasinverters and by providing phaseinverters and by providing phasethrough transformer connectionthrough transformer connection

For example, a 24For example, a 24--step inverterstep inverterconstructed by phase shifting econstructed by phase shifting eafour 6four 6--step inverters by 15step inverters by 15..

Similarly a phase shift of 7.5Similarly a phase shift of 7.5 bbe

basic inverter output will producbasic inverter output will producstep output waveform.step output waveform.


70/81

MM

Note that the firing pulses of thNote that the firing pulses of thbasic inverters must also be phabasic inverters must also be phaby 15by 15 or 7.5or 7.5 to obtain 24 or 48to obtain 24 or 48output waveforms respectively.output waveforms respectively.

The lowest order harmonics in aThe lowest order harmonics in ainverter is 6inverter is 6nn1 in the1 in the ac side aac side athe dc side.the dc side.


71/81

2424--Step OutStep Out

and Harmoand Harmo


72/81

33--Level InverteLevel Inverte


73/81

55--Level InverteLevel Inverte


74/81

STATCOMSTATCOM

A STATCOA STATCOof a SVS tof a SVS tsupplied bysupplied bystorage castorage ca

The SVS isThe SVS isin shunt wiin shunt wisystem bussystem bus

coupling trcoupling trwith a leakwith a leakreactance reactance


75/81

STATCOMSTATCOM --VV

IfIf ==, then the direction of t, then the direction of t

purely reactive currentpurely reactive current IIqq will dewill dethe voltage magnitudesthe voltage magnitudes VV11 andand V

IfIf VV11 >> VV22 then the current flowsthen the current flowsac system to the SVS and the coac system to the SVS and the coabsorbs reactive (inductive) powabsorbs reactive (inductive) pow

IfIf VV22 >> VV11 then the current flowsthen the current flowsSVS to the ac system and the coSVS to the ac system and the co

generates reactive (capacitive) generates reactive (capacitive) pthe ac system.the ac system.


76/81

STATCOMSTATCOM -- DC CaDC Ca

However pure reactive injection oHowever pure reactive injection o

is neither possible nor desirable.is neither possible nor desirable. Since the converter is supplied bySince the converter is supplied by

capacitor, the voltage across the capacitor, the voltage across the cwill fall if the STATCOM is notwill fall if the STATCOM is not lolo

The dc capacitor voltage can be reThe dc capacitor voltage can be rereplenishing the losses due to swreplenishing the losses due to swiin the coupling transformer circuiin the coupling transformer circuipower from the ac system.power from the ac system.

ThereforeTherefore must lagmust lag by a by a ssuch that the dc capacitor voltagesuch that the dc capacitor voltageconstant.constant.


77/81

STATSTAT

In a multiIn a multi--step converter, the fustep converter, the fucomponent of the output voltage component of the output voltage determined by the magnitude of determined by the magnitude of capacitor voltage.capacitor voltage.

Therefore the voltage magnitudeTherefore the voltage magnitudeincreased or decreasedincreased or decreased visvis----vis vis magnitude ofmagnitude of VV11 by charging or dby charging or dthe dc capacitor through the conthe dc capacitor through the con

This makes the control loop slow.This makes the control loop slow.


78/81

STATSTAT

Pulse width modulation (PWM) caPulse width modulation (PWM) ca

effectively be used in a multileveeffectively be used in a multileveThis has a better control responThis has a better control respon

In a sinusoidal PWM, the fundamIn a sinusoidal PWM, the fundamcomponent of the output voltagecomponent of the output voltage

can be changed by changing eithecan be changed by changing eithecapacitor voltage or the modulatcapacitor voltage or the modulat The capacitor voltage imbalance The capacitor voltage imbalance

diodediode--clamped topology makes itclamped topology makes it

restrictive. Alternatively flying crestrictive. Alternatively flying ctopology can be used.topology can be used.


79/81

STST

ChCh


80/81

Comparison BetweeComparison Betwee

and STATCOMand STATCOM


81/81

For Further Reading

Flexible AC Transmission Systems

The following two books cover most of the aspects of FACTS

[1] N. G. Hingorani and L. Gyugyi, Understanding FACTS: Concepts and Technology of

Flexible AC Transmission Systems, IEEE Press, New York, 2000.[2] Y. H. Song and A. T. Johns (eds.),Flexible AC Transmission Systems (FACTS), Institute of

Electrical Engineers, London, 2001.

The following book mainly deals with thyristor based FACTS devices like SVC and TCSC.

[3] R. M. Mathur and R. K. Varma, Thyristor Based FACTS Controllers for Electric PowerTransmission Systems, IEEE Press and Wiley Interscience, New York, 2002.

The following book is a classic. It discusses most of the early thyristor based technology.

[4] T. J. E. Miller (ed), Reactive Power Control in Electric Systems, John Wiley, New York,

1982.

The following book covers some of the aspects of shunt and series compensation and in general

is a good reference book on Power Systems

[5] P. S. Kundur,Power System Stability and Control, McGraw-Hill, New York, 1994.

Power Quality and Custom Power

The following books cover many aspects of power quality problems.

[6] R. C. Dugan, M. F. McGranaghan and H. W. Beaty, Electric Power Systems Quality, 2nd

ed., McGraw-Hill, New York, 2003.

[7] J. Arrillaga, N. R. Watson and S. Chen,Power Quality Assessment, John Wiley, New York,

2000.

Voltage sag and interruptions problems have been discussed in great detail in the following

book

ghosh monday july 23 2003

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