06486251

8122019 06486251

httpslidepdfcomreaderfull06486251 112

Published in IET Electric Power Applications

Received on 8th February 2012

Revised on 15th June 2012

Accepted on 24th July 2012

doi 101049iet-epa20120038

ISSN 1751-8660

Unbalance and harmonic voltage compensation for astand-alone variable speed constant frequencydouble-output induction generator supplyingnon-linear and unbalanced loadsMonalisa Pattnaik Debaprasad Kastha

Department of Electrical Engineering Indian Institute of Technology Kharagpur Kharagpur 721302 India E-mail monalisapattnaikgmailcom

Abstract This study presents an improved control algorithm for a stand-alone double-output induction machine-based variablespeed constant frequency generator The algorithm ensures sinusoidal load voltage and machine stator current for all types of loads such as balanced unbalanced linear and non-linear in any arbitrary combination This is achieved by incorporatingharmonic and unbalanced load compensation function to the stator side converter Unlike other control strategies reported inthe literature for this purpose the proposed algorithm does not require extraction or compensation of individual harmonic unbalance components of the load voltagestator current Harmonic compensation performance of the algorithm is thoroughlyanalysed theoretically and design guidelines provided The experimental results obtained from a laboratory prototypedemonstrate excellent load voltage regulation property during severe transient operating conditions Total harmonic distortionand unbalance of the load voltage and stator current even with large unbalanced non-linear loads are also found to be muchsmaller compared with similar systems reported earlier in the literature

Nomenclature

vabcs vabcr 3-ph stator and rotor voltages V iabcs iabcr iabcf and iabcl

3-ph stator rotor 1047297lter and load currents A

v sqds v s

qdl stator and load voltage in thestationary ref frame V

veqds v

eqdl stator and load voltage in the

synchronously rotating ref frame V l s

qds l sqdr Stator and rotor 1047298ux in the stationary

ref frame Wbl

eqds le

qdr stator and rotor 1047298ux in thesynchronously rotating ref frame Wb

i sqds i sqdf i sqdl stator 1047297lter and load currents in thestationary ref frame A

ieqds i

eqdf i

eqdl stator 1047297lter and load currents in the

synchronously rotating ref frame Ai sqdr ir

qdr ieqdr rotor current in the stationary rotor

and synchronously rotating refframe A

V d DC-link voltage V r s r primer per phase stator and rotor resistances

(stator referred) Ωl s l primer per phase stator rotor self inductances

(stator referred) H l m per phase magnetising inductance H

l f r f 1047297lter inductance and internalresistance H and Ω

cf 1047297lter capacitance microFωr rotor speed of the induction machine

elect radsωe rotational speed of the stator 1047298ux

elect radsωsl slip frequency elect radsωh harmonic frequency elect radsωcc closed-loop bandwidth of 1047297lter current

control loop elect radsωvc closed-loop bandwidth of stator

voltage control loop elect rads

Superscripts

Reference value

1 Introduction

Double-output induction generators (DOIG) are most suitablefor variable speed wind power generation as they offer

advantages like higher output power decoupled control of the machine active and reactive power and also reduced converter rating DOIG-based variable speed constant frequency (VSCF) generators can be operated in grid

wwwietdlorg

IET Electr Power Appl 2013 Vol 7 Iss 1 pp 27ndash38 27

doi 101049iet-epa20120038 amp The Institution of Engineering and Technology 2013

8122019 06486251


connected and stand-alone modes Various aspects of the grid connected mode of operation of such generators includingspeed sensorless control and harmonic distortionunbalancecompensation have been reported extensively [1ndash6] Inremote locations far away from the grid stand-aloneoperation (or hybrid operation with a diesel generator) of these generators may become necessary Stand-aloneoperation may be needed even in a wind ndashdiesel hybrid

generation system when the diesel generator is turned off [7] to save diesel fuel or because of lsquoislandingrsquo of anotherwise grid-connected generator [8]

The design and performance of a stand-alone VSCFgenerating system using DOIG has been presented for wind

power application in [9] As pointed out by the authorsthemselves 1047297eld orientation in this case could not bemaintained during sudden load change at the generator stator terminal Moreover only a linear balanced load wasconsidered in this paper An inherently speed sensorlesscontrol strategy for the stand-alone generator proposed in[8 10 11] uses a direct voltage control method Thiscontrol method cannot decouple voltage magnitude and frequency control loops The authors do not present anydesign methodology for these controllers which makes

performance prediction dif 1047297cult The load (only resistive)transient results presented in these papers show a signi1047297cant transient disturbance in the stator terminal voltage duringload change Non-linear and unbalanced loads arecompensated by injecting harmonics from the rotor sideconverter which increases the machine current rating and copper loss In addition the resulting pulsating torque putsadditional stress on the wind turbine gear box even whenone discounts the possibility of mechanical resonance in thedrive train The rotor current-based MRAS observer for rotor speed and position of a DOIG-based stand-alonegeneration system reported in [12 13] has not been tested

with non-linear and unbalanced loads The sensorlesscontrol scheme for a stand-alone generator presented in [14]is fundamentally similar to [9] and suffers from the same

problem of undesirable transients in the load terminalvoltage during load change The proposed scheme avoidsthe limitations associated with the harmonic compensationmethod reported in [8 10 11] by incorporating active1047297ltering function in the stator side converter However thereported harmonic compensation performance is not verygood Also unbalanced loads are not considered in this

paper A control system for stand-alone and grid-connected DFIG supplying unbalanced loads discussed in [15] alsouses the stator side converter for harmonic compensationThe proposed strategy requires extraction and control of thenegative sequence variables in the negative sequencesynchronously rotating reference frame This approachcomplicates the implementation of the control algorithm

particularly if the same philosophy is extended for harmonic compensation since each harmonic component will require a separate control reference frame In any case

performance of the system with non-linear loads has not been reported in this paper The stand-alone DOIG controlmethod proposed in [16] is fundamentally different fromother methods discussed so far in that it uses the stator sideconverter for load voltage regulation However it requires a

battery storage unit and dump loads (with associated power electronic converters) in order to maintain a stable DC-link

voltage for back-to-back converters which makes thescheme somewhat complicated A feedforward voltagecompensation scheme is also presented to maintain

balanced three-phase voltage in the presence of unbalanced

loads However non-linear loads are not considered in this paper Unbalance and harmonic load compensationtechniques proposed in [17ndash19] for the stand-alone DOIGagain use the rotor-side converter for harmonic current injection as in [8 10 11] Consequently they suffer fromthe similar disadvantages of harmonic heating of themachine and pulsating torques as acknowledged by theauthors themselves Although unbalanced ([17 18]) and

harmonic ([18]) compensation currents are controlled in asingle reference frame the references for the compensatingcurrents have to be extracted from the correspondingcomponents of the load voltage using several lsquonotchrsquo 1047297ltersThis along with the requirement of a resonant regulator for each harmonic component ([19]) makes the overall controlscheme complicated Perhaps for this reason only twodominant harmonics (5th and 7th) were compensated in[19] In a subsequent publication [20] the same authorshave simpli1047297ed their control scheme to some extent bycontrolling the injected harmonic currents in thesynchronously rotating reference frame This approachreduced the number of resonant regulators by half for thesame number of harmonics to be compensated The authorsdo not present the THD value of the load voltage whosewaveform shows notable harmonic distortion in one of thesimulation results presented by the authors Moreover noattempt has been made to compensate unbalanced and non-linear loads such as single-phase diode recti1047297ers whichwill require many more resonant regulators to deliver acceptable load voltage waveform The stand-alone DOIGcontroller proposed in [21] uses the stator side converter toregulate load voltage in a closed loop However unlike[16] it proposes a closed-loop controller for the DC-link voltage and thus dispenses with the requirement of the

battery storage unit and the dump load along with their associated power electronic circuitry More importantly for

the 1047297rst time a stator converter control architecture is proposed which can handle linear non-linear balanced and unbalanced loads (in any arbitrary combination) in anuniform manner In fact no special control strategy isrequired for non-linear andor unbalanced loads Excellent load voltage regulation property of the proposed controlalgorithm under severe transient and harmonicunbalanced loading conditions is demonstrated by the simulationresults in this paper The present paper further develops thecontrol strategy proposed in [21] by providing design-oriented analysis of the stator side converter controller It also providesexperimental validation of the controller performance under similar operating conditions as in [21]

The paper is organised as follows In Section 2 adescription of the power circuit and the control system for the DOIG-based stand-alone generator are presentedSection 3 presents a design-oriented analysis of thestator-side converter controller In Section 4 the laboratory

prototype is explained and the experimental results areshown Finally conclusions are drawn in Section 5

2 Description of the DOIG-based stand-aloneVSCF generator

The schematic diagram of the DOIG-based VSCF generator isshown in Fig 1 This system is similar to the one described in

[22] where the stator-side converter was directly connected tothe machine stator terminals and controlled in an open-loopmanner The 1047297lter along with the load was connected to themachine stator terminals through a switch In contrast in

wwwietdlorg

28 IET Electr Power Appl 2013 Vol 7 Iss 1 pp 27ndash38

amp The Institution of Engineering and Technology 2013 doi 101049iet-epa20120038

8122019 06486251


the present scheme the DOIG stator and the load terminalsare connected together at the 1047297lter capacitor junction TheDOIG stator voltage is controlled in a closed-loop manner through the stator-side converter to improve performance

with non-linear loadsThe complete block diagram of the control system is shown

in Fig 2 This controller has been discussed in detail in [21]Hence only a brief overview for the sake of completenesswill be presented here

The controller operates in the stator 1047297eld-oriented referenceframe Magnitude of the stator 1047298ux linkage and the unit

vectors for 1047297eld orientation are computed as

lsd s =

v

sd s minus r si

sd s

dt l

sqs =

v

sqs minus r si

sqs

dt (1)

ls =

(ls

d s)2 + (lsqs)2

(2)

cos u e = lsd s

ls

sin u e = l

sqs

ls

(3)

Of the controlled variables the DC-link voltage is regulated through the rotor-side converter As shown in Fig 3 thenet power 1047298owing into the DC-link capacitor isthe difference between the mechanical power input to the

Fig 1 Schematic diagram of a stand-alone DOIG-based VSCF

system

Fig 2 Control block diagram for the proposed generation system

Fig 3 Active power 1047298 ow in the DOIG-based system

wwwietdlorg



8122019 06486251


generator and the load active power demand Hence DC-link voltage can be controlled by adjusting the torque producingcomponent of the rotor current i

primeeqr

The DC-link voltage controller operates in two distinct modes namely the DC-link voltage build up mode and theVSCF generation mode During the buildup process theload in Fig 1 remains disconnected from the machinemaking the term lsquo P l rsquo in Fig 3 zero in this mode The iprimelowast

qr

switch in Fig 2 is kept in position lsquo2rsquo The rotor-sideinverter is current controlled with

iprimelowastd r = I primelowast

d r 1 minus e(minust t d s)

and iprimelowastqr = I primelowast

qr 1 minus e(minust t qs)

(4)

This particular form of iprimelowastd r and i

primelowastqr is chosen to satisfy the limits

of the rotor and the stator voltages For a given DC-link voltagewith SPWM control of both the inverters these limits are

v s =

ve2

d s + ve2qs

le

vd

2 and vr =

ve2

d r + ve2qr

le

vd

2 (5)

A more detailed discussion on the DC-link voltage buildup process can be found in [21 22] With the choice of i

primelowastd r and i

primelowastqr as given by (4) the DC-link voltage gradually starts

increasing from a low initial value When it reaches itssteady-state reference value (V d ) the iprimelowast

qr switch in Fig 2moves to position lsquo1rsquo and the load switch is closed The rotor q-axis current iprimee

qr is used to control the power balance of thesystem A PI controller for DC-link voltage has been used togenerate the reference value of iprimee

qr The reference value of iprimeed r

is a free variable which can be used to control the stator-side power factor of the machine

For stator voltage and frequency control it is noted that aslong as the stator 1047298ux magnitude and its rotational frequencyare controlled to be constant the stator voltage magnitude

is affected only by the stator resistance drop which iscomparatively small Therefore by controlling both thesevariables a fairly constant stator terminal voltage ismaintained In the stator 1047297eld-oriented reference frame

pls +ls

t s

= ved s +

l mt s

iprimeed r (6)

v els = veqs +

l mt s

iprimeeqr (7)

where τs = (l s r s) If ved s and v

eqs are controlled to follow

velowastd s =

l

t s

minusl mt s

iprimeed r (8)

velowastqs = v

lowastels minus

l mt s

iprimeeqr (9)

Then from (6) and (7) pls + (lst s) = l amp v lowaste = v e The

right ndashhand-side expressions of (8) and (9) are used asreferences in the stator-side voltage controller The block diagram of the stator-side controller and the 1047297lter is shownin Fig 4 The stator-side voltage controller is described indetail in [21]

3 Analysis of the stator side voltage

controller

Figs 5a and b show the q eminusd e axes equivalent circuit of the

machine incorporating the effect of the rotor current controllers at any harmonic frequency ωh In these 1047297gureslsquo K pr rsquo and lsquo K ir rsquo are the proportional and integral gains of therotor-side current controllers respectively It is alsoassumed that the rotor current references

iprimelowast

qr and iprimelowastd r

do not

contain any lsquoωhrsquo frequency component From Fig 5a

iprimeqrh = 0 Hence iqsh = minus

l ml s

iprimeqrh = 0

Also for dominant load current harmonic components thefollowing relation will usually hold

v hl m ≫ K pr + r primer + r s

≫ v h l primelr + l ls

minus

K ir

v h

Fig 4 Block diagram of the stator voltage controller

wwwietdlorg



8122019 06486251


Therefore from Fig 5b

id sh ≃vd sh

R where R = K pr + r s + r primer

These relations are used to simplify the stator-side voltagecontroller block diagram (Fig 4) to yield Fig 5c In this1047297gure the block ((ωcc) s + ωcc) represents closed-looptransfer function of the 1047297lter current controllers It is also

noted that at any harmonic frequency ωh the stator voltagereferences vlowast

qsh and vlowastd sh

are zero Furthermore assumed

v cc Rcf ≫ 1 Fig 5c further simpli1047297es to Fig 5d after somesimple block diagram manipulation From Fig 5d (see (10))

where

T v( s) =G v

scf

v cc

s + v cc

is the loop gain of the stator voltage control loop Theoff-diagonal terms in the matrix on the left-hand side of

(10) are much smaller compared with the diagonal terms for any s = j ωh since usually v cc ≫ v e Neglecting these off diagonal terms

vqsh ≃ minus1

1 + T v( s) v cc

s + v cc

iqlh

v cccf

(11)

vd sh ≃ minus 11 + T v( s) v cc

s + v cc

id lh

v cccf

(12)

Denoting the gain crossover frequency of |T v ( j ω)| by ωvc thefollowing steady-state relationships are obtained

V qsh = minus 1

1 minus j (v vcv h)

1 + j (v hv cc)

lowast1

1 + j (v hv cc)lowastv e

v cc

lowast X cf I qlh

(13)

Fig 5 Harmonic equivalent circuit and block diagram of the machine and the stator voltage controller

1 + T v( s)

1 + ( sv cc)

(v ev cc)

minus(v ev cc) 1 + T v( s)

1 + ( sv cc)

vqsh

vd sh

= minus

1

v cc cf

iqlh

id lh

(10)

wwwietdlorg



8122019 06486251


or

V qsh = minus1

1 minus j ((v vc)v h) minus ((v h)v cc) lowastv e

v cc

lowast X cf I qlh

(14)

V d sh = minus1

1 minus j (v vcv h) minus (v hv cc) lowast

v e

v cc

lowast X cf I d lh (15)

I qsh = 0 I d sh =V d sh

R = minus

1

1 minus j (v vcv h) minus (v hv cc)

lowastv e

v cc

lowast X cf

R I d lh

(16)

In these equations above all variables in capital denote the phasor representation of the corresponding variable in asteady state X cf = (1 ωecf ) is the fundamental frequencyreactance of the 1047297lter capacitor All the phasors in (13)ndash(16)can in general be decomposed into positive and negativesequence components That is

F q = F q+ + F qminus F d = F d + + F d minus (17)

where lsquo F rsquo may be any of the voltagecurrent phasorsappearing in (13)ndash(16) The following relations hold among

the sequence components in a steady state

F q+ = minus j F d + = minus j F + F qminus = j F d minus = j F minus (18)

Then from (14) and (15)

minus j V sh+ minus V shminus

=

j

1 minus j ((v vc)v h) minus ((v h)v cc) lowast v e

v cc

lowast X cf I lh+ minus I lhminus

(19)

V sh+ + V shminus

= minus

1

1 minus j ((v vc)v h) minus ((v h)v cc)

lowastv e

v cc

lowast X cf I lh+ + I lhminus

(20)

[ V sh+

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

lowastv e

v cc

lowast X cf

I lh+

(21)

Similarly from (16)

[ I sh+

= I shminus

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X cf

R lowast

1

2 I lh+ + I lhminus

(22)

In any given application I lh+ can be found from the giventhree-phase load current waveforms Equations (21) and

(22) can then be used to choose the values of X cf ωcc and ωvc so that the THD of the stator voltage and current aremaintained within prescribed limits Once the value of cf ischosen in this manner the value of l f can be selected such

Fig 6 Block diagram of the prototype VSCF generator

wwwietdlorg



8122019 06486251


that the stator-side inverter switching frequency ripple issuf 1047297ciently (at least minus40 dB) attenuated at the stator terminal

4 Description of the laboratory prototype

A block diagram of the prototype VSCF generator is shownin Fig 6 Speci1047297cations of the power circuit elements aregiven in Table 1 The control algorithm discussed so far isimplemented using RT-LAB real-time simulation platformfrom Opal-RT The controller gains are given in Table 2The prime mover for the laboratory tests is avoltage-controlled DC shunt motor

41 Experimental procedure and results

Both DC-link voltage and machine 1047298ux have to build up before the system can start supplying load Fig 7 showssome important system variables during the DC-link voltage

buildup phase For this experiment the prime mover speed is set at 1240 RPM and the DC-link capacitor is initiallycharged to a voltage of 30 of its 1047297nal value The

rotor-side current controllers are initially supplied with 1047297xed current commands as given in (4) Figs 7a and c showthat DC-link voltage and machine 1047298ux build up in acontrolled under damped manner without any overunder shoot After the DC-link voltage builds up to the referencevalue iprimee

qrref is set from the closed-loop DC-link voltagecontroller

After the DC-link voltage build up a 438 kW (782 of mc rating) balanced 3-ph resistive load is applied at 202 s(Fig 8d ) in a single step The waveforms (Figs 8a and e)show that the system copes with this large step increase inthe load demand without signi1047297cant transience in load voltage magnitude and frequency The q-axis rotor current (Fig 8b) increases substantially to increase active power

1047298ow through the rotor However the d -axis rotor current does not show any transient disturbance indicating proper 1047297eld orientation Both the converter currents are observed toremain within their respective rated values (Table 1) eventhough their current ratings do not exceed 50 of themachine stator current rating The dynamic performances of the system observed from these experimental results aresuperior compared with similar results presented in theearlier literature ([8 9 14]) where a smaller load step (50of machine rating) produced a larger (more than 10)1047298uctuation in load voltage

Fig 9 shows load voltage during increase and decrease of prime mover speed The prime mover speed is changed from

535 rads (1277 RPM) to 310 rads (740 RPM) and back withthe generator lightly loaded The rotor current shows smoothtransition through synchronous speed (Fig 9c) The primemover speed variation has no effect on load voltage(Figs 9b and d )

Table 1 Specifications of the power circuit components

Induction machine (statorreferred)

stator (8 pole nabla connected) 220 V 50 Hz 22 A (RMS)rotor (8 pole Y connected) 300 V 9 A (RMS)rated power 56 kWrated speed 720 rpm

stator resistancephase (r s) 087 Ωrotor resistancephase (r primer) 112 Ωstator reactancephase (x s) 124 Ωrotor reactancephase (x primer) 124 Ωmagnetising reactancephase(x m)

113 Ω

converter ratingstator side 230 V (RMS) 11 A(RMS)

rotor side 230 V (RMS) 9 A(RMS)

filter parametersinductor(l f ) 135 mH 30 A capacitor(c f ) 35 μF450 V Δ

connectedESR (r f ) 05 Ω

Table 2 Controller parameters

filter current controllers (i eq f i ed f ) K p = 15 K i = 300

stator voltage controllers (v eq s v ed s) K p = 02 K i = 2

rotor current controllers (i eq r i ed r) K p = 30 K i = 6000

Fig 7 DC-link voltage build up at super-synchronous speed (1240 rpm)

a DC-link voltageb q-axis actual and reference rotor current c stator 1047298ux linkaged d -axis actual and reference rotor current

wwwietdlorg



8122019 06486251


To verify the harmonic compensation performance of thestator voltage controller a 42 kW (75 of mc rating)

balanced 3-phase non-linear (diode recti1047297er feeding aresistive load on the DC side) load is applied in a singlestep The waveforms and the Fast Fourier Transform (FFT)

of the non-linear load current load voltage and stator current are shown in Figs 10andash f The load voltage is found to remain constant with negligible distortion The totalharmonic distortion (THD) of load voltage (205) and

stator current (097) are much lower compared withsimilar results presented in [14] where the load voltageTHD was 257 and the stator current THD was 663 for a much lower level (28 of mc rating) of non-linear loading

THD of the stator voltage and current for the loading

condition in Fig 10 can also be predicted from (21) and (22) With a three-phase diode recti1047297er load I lhplusmn V shplusmn and

I shplusmn in these equations correspond to (6n plusmn 1)(n = 1 23 hellip) order harmonics in load current stator voltage and

Fig 8 Experimental waveforms during 3-ph resistive load increase

a DC-link and RMS load voltageb qe

-d e-axis actual and reference rotor currentsc Stator and rotor current d Load current e Load voltage f Filter current

Fig 9 Speed transient performance

a DC-link voltage and actual speed during speed transient b Load voltage during speed decreasec Rotor currentsd Load voltage during speed increase

wwwietdlorg



8122019 06486251


stator current respectively Equations (21) and (22) are 1047297rst converted to per unit form by dividing both sides of theseequations by the fundamental component of stator phasevoltage (V s1) and phase current ( I s1) respectively to yield

V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc

lowast X l lowast X cf

PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)

where ωh = 6nωe is the harmonic frequency in the stator 1047297eld-oriented reference frame

| X cf |PU = ( X cf | I l|rated (V s1)) is the per unit 1047297lter capacitor reactance with load voltage and power as the base

X l = ( I l1| I l |rated ) is the pu loading I l 1 is the fundamentalcomponent of the load current

For the values given in Tables 1 and 2 f vc = 318Hz f cc = 15kHz X cf

PU

= 35 X l = 075( X cf R) = 0995 From the FFT of the load current and stator current waveforms [shown in Figs 10a) and 10e)]( I l 1 I s1) = 10 Harmonic distortion of the stator voltage at different harmonic frequencies computed from (23) isshown in Table 3 along with simulated and experimentalvalues The simulated values were obtained by simulatingthe entire VSCF generation system in aMATLAB-SIMULINK environment under an identicalloading condition as in Fig 10 More detailed simulation

results of this system can be found in [21]Harmonic distortion of the stator current can also be

computed from (24) The stator current THD came out to be 054 while the simulated and experimental values were

Fig 10 Experimental waveforms with 3-ph non-linear load

a Load Current b FFT of the load current c Load Voltaged FFT of the load voltagee Stator current f FFT of the stator current

wwwietdlorg



8122019 06486251


061 and 097 respectively It should be noted that themachine stator current in addition to the residual load harmonic currents also carries harmonic components of themagnetisation current produced because of iron saturationin a practical machine In the present experimental setupTHD of the no load stator current was experimentally found to be 07 of the fundamental stator current in Fig 10e

Therefore the experimental values of the stator current THDcame out to be higher than the corresponding theoreticaland simulated values

Next the system is tested by connecting a single-phasenon-linear load (1-ph diode recti1047297er feeding an RL-load) of 20 kW (36 of mc rating) between two lines in a singlestep The waveforms and the FFT of the unbalanced non-linear load current load voltage and stator current are

shown in Figs 11andash f ) respectively The load voltage isfound to have remained unchanged and undistorted whilesupplying this single-phase load The THD of the load voltage is 192 and THD of the stator current is 204while that of the load current is 1964 The negativesequence component of the stator voltage is 133 of thefundamental positive sequence voltage and that of the stator current is 148 To the best of our knowledge this

particular type of load has not been considered in thereported literature so far Hence no comparative THD1047297gures from the literature could be presented for this case

Table 3 Stator voltage harmonic compensation performance

Harmonic order Theory Simulation Experiment

5th 129 108 1507th 092 091 07011th 076 054 06513th 066 034 04617th 049 035 05919th 043 022 047THD 199 175 206


a Load current

b FFT of the load current c Load voltaged FFT of the load voltagee Stator current f FFT of the stator current

wwwietdlorg



8122019 06486251


In both the cases of three-phase and single-phasenon-linear loads the harmonic as well as unbalancecomponent of the load current are supplied completely bythe stator-side converter This is evident from the 1047297lter current waveforms shown in Figs 12a and b for three-phase and single-phase loads respectively As a result the stator voltage and current remain almost sinusoidal withvery low distortion It is also observed that even with thislarge non-linear and unbalanced loading the stator sideconverter currents do not exceed their rating as given inTable 1

The harmonic and unbalanced current supplied by thestator-side converter does have an effect on the DC-link voltage waveform as is evident from Figs 12c and d However even in the worst case (Fig 12d ) the DC-link voltage ripple is negligibly small (lt2)

5 Conclusion

This paper has presented a stator 1047298ux-oriented control schemefor a DOIG-based stand-alone VSCF generator The proposed controller maintains correct 1047297eld orientation during load (non-linear and unbalanced) transients and exhibitsimproved load voltage regulation property compared withthe available literature Even with a large non-linear (up to75 of the machine rating) and unbalanced load (up to36 of the machine rating) the load voltage unbalance and harmonic distortions are negligible Compared with

previously reported methods of harmonic and unbalanced load compensation the control strategy proposed in this

paper is simpler and does not adversely affect machineoperation by producing harmonic heating or pulsatingtorque Neither the machine nor the converter current ratings are exceeded in any case A design-oriented analysis

of the stator voltage controller is presented Theexperimental results from a laboratory prototype matchreasonably well with analytical and simulation resultswithin the limits of experimental error

6 References

1 Pena R Clare JC Asher GM lsquoDoubly fed induction generator using back-to-back PWM converters and its application to variable

speed wind-energy generationrsquo IEE Proc Electr Power Appl 1996143 (3) pp 231ndash241

2 Longya X Wei C lsquoTorque and reactive power control of a doubly fed induction machine by position sensorless schemersquo IEEE Trans Ind Appl 1995 31 (3) pp 636ndash642

3 Datta R Ranganathan VT lsquoA simple position-sensorless algorithmfor rotor-side 1047297eld-oriented control of wound-rotor inductionmachinersquo IEEE Trans Ind Electron 2001 48 (4) pp 786ndash793

4 Mohammed OA Liu Z Liu S lsquoA novel sensorless control strategyof doubly-fed induction machinesrsquo Proc IEEE Int Conf on ElectricMachines Drives San Antonio TX USA May 2005 pp 315ndash319

5 Hopfensperger B Atkinson DJ Lakin RA lsquoStator-1047298ux-oriented control of a doubly-fed induction machine with and without positionencoder rsquo IEE Proc Electr Power Appl 2000 147 (4) pp 241ndash250

6 Abolhassani M Enjeti P Toliyat H lsquoIntegrated doubly-fed electricalternatoractive 1047297lter (IDEA) a viable power quality solution for

wind energy conversion systemsrsquo IEEE Trans Energy Convers2008 23 (2) pp 642ndash650

7 Saha TK Kastha D lsquoDesign optimization and dynamic performanceanalysis of a stand-alone hybrid wind-diesel electrical power generationsystemrsquo IEEE Trans Energy Convers 2010 25 (4) pp 1209ndash1217

8 Iwanski G Koczara W lsquoDFIG based power generation system withUPS function for variable speed applicationrsquo IEEE Trans Ind Electron 2008 55 (8) pp 3047ndash3054

9 Pena R Clare JC Asher GM lsquoA doubly fed induction generator using back-to back PWM converters supplying an isolated load from avariable speed wind turbinersquo IEE Proc Electr Power Appl 1996143 (5) pp 380ndash387

10 Iwanski G Koczara W lsquoSensorless stand alone variable speed system

for distributed generationrsquo IEEE Power Electronics Specialists Conf2004 vol 3 pp 1915ndash1921

11 Iwanski G Koczara W lsquoSensorless direct voltage control of thestand-Alone slip-ring induction generator rsquo IEEE Trans Ind Electron

2007 54 (2) pp 1237ndash1239

12 Cardenas R Pena R Proboste J Asher G Clare J lsquoMRASobserver for sensorless control of standalone doubly fed inductiongeneratorsrsquo IEEE Trans Energy Convers 2005 20 (4) pp 710ndash718

13 Cardenas R Pena R Clare J Asher G Proboste J lsquo

MRASobservers for sensorless control of doubly-fed induction generatorsrsquo IEEE Trans Power Electron 2008 23 (3) pp 1075ndash1084

14 Jain AK Ranganathan VT lsquoWound rotor induction generator withsensorless control and integrated active 1047297lter for feeding nonlinear

Fig 12 Experimental waveforms for a 3-ph and 1-ph non-linear load

a Filter current with 3-ph non-linear load b Filter current with1-ph non-linear load c DC-link voltage with 3-ph non-linear load d DC-link voltage with 1-ph non-linear load

wwwietdlorg



8122019 06486251


loads in a stand-alone grid rsquo IEEE Trans Ind Electron 2008 55 (1) pp 218ndash228

15 Pena R Cardenas R Escobar E Clare J Wheeler P lsquoControlstrategy for a doubly-fed induction generator feeding an unbalanced grid or stand-alone load rsquo Electr Power Syst Res 2009 79 (2) pp 355ndash364

16 Bhattacharya T Umanand L lsquo Negative sequence compensationwithin fundamental positive sequence reference frame for a stiff micro-grid generation in a wind power system using slipring induction machinersquo IET Electr Power Appl 2009 3 (6)

pp 520ndash53017 Phan VT Lee HH Chun TW lsquoAn improved control strategy using

PI-resonant controller for unbalanced stand-alone doubly-fed inductiongenerator rsquo J Power Electron 2010 10 (2) pp 194ndash202

18 Phan VT Lee HH lsquoImproved predictive current controlfor unbalanced stand-alone doubly-fed induction generator-based

wind power systemsrsquo IET Electr Power Appl 2011 5 (3) pp 275ndash287

19 Phan VT Lee HH lsquoStationary frame control scheme for astand-alone doubly fed induction generator system with effectiveharmonic voltages rejectionrsquo IET Electr Power Appl 2011 5 (9) pp 697ndash707

20 Phan VT Lee HH lsquoControl strategy for harmonic elimination instand-alone DFIG applications with nonlinear loadsrsquo IEEE Trans Power Electron 2011 26 (9) pp 2662ndash2675

21 Pattnaik M Kastha D lsquoControl of double output induction machine

based stand alone variable speed constant frequency generator withnonlinear and unbalanced loadsrsquo IEEE PES General MeetingMinneapolis Minnesota USA July 2010

22 Isha TB Kastha D lsquoTransient performance of a stand-alone variablespeed constant frequency generation systemrsquo Power Conversion Conf Nagoya Japan April 2007 pp 622ndash628

wwwietdlorg



8122019 06486251


connected and stand-alone modes Various aspects of the grid connected mode of operation of such generators includingspeed sensorless control and harmonic distortionunbalancecompensation have been reported extensively [1ndash6] Inremote locations far away from the grid stand-aloneoperation (or hybrid operation with a diesel generator) of these generators may become necessary Stand-aloneoperation may be needed even in a wind ndashdiesel hybrid

generation system when the diesel generator is turned off [7] to save diesel fuel or because of lsquoislandingrsquo of anotherwise grid-connected generator [8]

The design and performance of a stand-alone VSCFgenerating system using DOIG has been presented for wind

power application in [9] As pointed out by the authorsthemselves 1047297eld orientation in this case could not bemaintained during sudden load change at the generator stator terminal Moreover only a linear balanced load wasconsidered in this paper An inherently speed sensorlesscontrol strategy for the stand-alone generator proposed in[8 10 11] uses a direct voltage control method Thiscontrol method cannot decouple voltage magnitude and frequency control loops The authors do not present anydesign methodology for these controllers which makes

performance prediction dif 1047297cult The load (only resistive)transient results presented in these papers show a signi1047297cant transient disturbance in the stator terminal voltage duringload change Non-linear and unbalanced loads arecompensated by injecting harmonics from the rotor sideconverter which increases the machine current rating and copper loss In addition the resulting pulsating torque putsadditional stress on the wind turbine gear box even whenone discounts the possibility of mechanical resonance in thedrive train The rotor current-based MRAS observer for rotor speed and position of a DOIG-based stand-alonegeneration system reported in [12 13] has not been tested

with non-linear and unbalanced loads The sensorlesscontrol scheme for a stand-alone generator presented in [14]is fundamentally similar to [9] and suffers from the same

problem of undesirable transients in the load terminalvoltage during load change The proposed scheme avoidsthe limitations associated with the harmonic compensationmethod reported in [8 10 11] by incorporating active1047297ltering function in the stator side converter However thereported harmonic compensation performance is not verygood Also unbalanced loads are not considered in this

paper A control system for stand-alone and grid-connected DFIG supplying unbalanced loads discussed in [15] alsouses the stator side converter for harmonic compensationThe proposed strategy requires extraction and control of thenegative sequence variables in the negative sequencesynchronously rotating reference frame This approachcomplicates the implementation of the control algorithm

particularly if the same philosophy is extended for harmonic compensation since each harmonic component will require a separate control reference frame In any case

performance of the system with non-linear loads has not been reported in this paper The stand-alone DOIG controlmethod proposed in [16] is fundamentally different fromother methods discussed so far in that it uses the stator sideconverter for load voltage regulation However it requires a

battery storage unit and dump loads (with associated power electronic converters) in order to maintain a stable DC-link

voltage for back-to-back converters which makes thescheme somewhat complicated A feedforward voltagecompensation scheme is also presented to maintain

balanced three-phase voltage in the presence of unbalanced

loads However non-linear loads are not considered in this paper Unbalance and harmonic load compensationtechniques proposed in [17ndash19] for the stand-alone DOIGagain use the rotor-side converter for harmonic current injection as in [8 10 11] Consequently they suffer fromthe similar disadvantages of harmonic heating of themachine and pulsating torques as acknowledged by theauthors themselves Although unbalanced ([17 18]) and

harmonic ([18]) compensation currents are controlled in asingle reference frame the references for the compensatingcurrents have to be extracted from the correspondingcomponents of the load voltage using several lsquonotchrsquo 1047297ltersThis along with the requirement of a resonant regulator for each harmonic component ([19]) makes the overall controlscheme complicated Perhaps for this reason only twodominant harmonics (5th and 7th) were compensated in[19] In a subsequent publication [20] the same authorshave simpli1047297ed their control scheme to some extent bycontrolling the injected harmonic currents in thesynchronously rotating reference frame This approachreduced the number of resonant regulators by half for thesame number of harmonics to be compensated The authorsdo not present the THD value of the load voltage whosewaveform shows notable harmonic distortion in one of thesimulation results presented by the authors Moreover noattempt has been made to compensate unbalanced and non-linear loads such as single-phase diode recti1047297ers whichwill require many more resonant regulators to deliver acceptable load voltage waveform The stand-alone DOIGcontroller proposed in [21] uses the stator side converter toregulate load voltage in a closed loop However unlike[16] it proposes a closed-loop controller for the DC-link voltage and thus dispenses with the requirement of the

battery storage unit and the dump load along with their associated power electronic circuitry More importantly for

the 1047297rst time a stator converter control architecture is proposed which can handle linear non-linear balanced and unbalanced loads (in any arbitrary combination) in anuniform manner In fact no special control strategy isrequired for non-linear andor unbalanced loads Excellent load voltage regulation property of the proposed controlalgorithm under severe transient and harmonicunbalanced loading conditions is demonstrated by the simulationresults in this paper The present paper further develops thecontrol strategy proposed in [21] by providing design-oriented analysis of the stator side converter controller It also providesexperimental validation of the controller performance under similar operating conditions as in [21]

The paper is organised as follows In Section 2 adescription of the power circuit and the control system for the DOIG-based stand-alone generator are presentedSection 3 presents a design-oriented analysis of thestator-side converter controller In Section 4 the laboratory

prototype is explained and the experimental results areshown Finally conclusions are drawn in Section 5

2 Description of the DOIG-based stand-aloneVSCF generator

The schematic diagram of the DOIG-based VSCF generator isshown in Fig 1 This system is similar to the one described in

[22] where the stator-side converter was directly connected tothe machine stator terminals and controlled in an open-loopmanner The 1047297lter along with the load was connected to themachine stator terminals through a switch In contrast in

wwwietdlorg



8122019 06486251







lsd s =

v

sd s minus r si

sd s

dt l

sqs =

v

sqs minus r si

sqs

dt (1)

ls =

(ls

d s)2 + (lsqs)2

(2)

cos u e = lsd s

ls

sin u e = l

sqs

ls

(3)



system



wwwietdlorg



8122019 06486251



primeeqr


qr






(4)




v s =

ve2

d s + ve2qs

le

vd

2 and vr =

ve2

d r + ve2qr

le

vd

2 (5)











pls +ls

t s

= ved s +

l mt s

iprimeed r (6)

v els = veqs +

l mt s

iprimeeqr (7)



velowastd s =

l

t s

minusl mt s

iprimeed r (8)

velowastqs = v

lowastels minus

l mt s

iprimeeqr (9)




controller



iprimelowast


do not



l ml s

iprimeqrh = 0




minus

K ir

v h


wwwietdlorg



8122019 06486251



id sh ≃vd sh




qsh and vlowastd sh



where

T v( s) =G v

scf

v cc

s + v cc



vqsh ≃ minus1

1 + T v( s) v cc

s + v cc

iqlh

v cccf

(11)


s + v cc

id lh

v cccf

(12)


V qsh = minus 1

1 minus j (v vcv h)

1 + j (v hv cc)

lowast1


v cc

lowast X cf I qlh

(13)


1 + T v( s)

1 + ( sv cc)

(v ev cc)


1 + ( sv cc)

vqsh

vd sh

= minus

1

v cc cf

iqlh

id lh

(10)

wwwietdlorg



8122019 06486251


or

V qsh = minus1


v cc

lowast X cf I qlh

(14)

V d sh = minus1


v e

v cc



R = minus

1


lowastv e

v cc

lowast X cf

R I d lh

(16)








=

j


v cc


(19)

V sh+ + V shminus

= minus

1


lowastv e

v cc


(20)

[ V sh+

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

lowastv e

v cc

lowast X cf

I lh+

(21)

Similarly from (16)

[ I sh+

= I shminus

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X cf

R lowast

1

2 I lh+ + I lhminus

(22)




wwwietdlorg



8122019 06486251


















113 Ω











wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251







lsd s =

v

sd s minus r si

sd s

dt l

sqs =

v

sqs minus r si

sqs

dt (1)

ls =

(ls

d s)2 + (lsqs)2

(2)

cos u e = lsd s

ls

sin u e = l

sqs

ls

(3)



system



wwwietdlorg



8122019 06486251



primeeqr


qr






(4)




v s =

ve2

d s + ve2qs

le

vd

2 and vr =

ve2

d r + ve2qr

le

vd

2 (5)











pls +ls

t s

= ved s +

l mt s

iprimeed r (6)

v els = veqs +

l mt s

iprimeeqr (7)



velowastd s =

l

t s

minusl mt s

iprimeed r (8)

velowastqs = v

lowastels minus

l mt s

iprimeeqr (9)




controller



iprimelowast


do not



l ml s

iprimeqrh = 0




minus

K ir

v h


wwwietdlorg



8122019 06486251



id sh ≃vd sh




qsh and vlowastd sh



where

T v( s) =G v

scf

v cc

s + v cc



vqsh ≃ minus1

1 + T v( s) v cc

s + v cc

iqlh

v cccf

(11)


s + v cc

id lh

v cccf

(12)


V qsh = minus 1

1 minus j (v vcv h)

1 + j (v hv cc)

lowast1


v cc

lowast X cf I qlh

(13)


1 + T v( s)

1 + ( sv cc)

(v ev cc)


1 + ( sv cc)

vqsh

vd sh

= minus

1

v cc cf

iqlh

id lh

(10)

wwwietdlorg



8122019 06486251


or

V qsh = minus1


v cc

lowast X cf I qlh

(14)

V d sh = minus1


v e

v cc



R = minus

1


lowastv e

v cc

lowast X cf

R I d lh

(16)








=

j


v cc


(19)

V sh+ + V shminus

= minus

1


lowastv e

v cc


(20)

[ V sh+

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

lowastv e

v cc

lowast X cf

I lh+

(21)

Similarly from (16)

[ I sh+

= I shminus

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X cf

R lowast

1

2 I lh+ + I lhminus

(22)




wwwietdlorg



8122019 06486251


















113 Ω











wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



primeeqr


qr






(4)




v s =

ve2

d s + ve2qs

le

vd

2 and vr =

ve2

d r + ve2qr

le

vd

2 (5)











pls +ls

t s

= ved s +

l mt s

iprimeed r (6)

v els = veqs +

l mt s

iprimeeqr (7)



velowastd s =

l

t s

minusl mt s

iprimeed r (8)

velowastqs = v

lowastels minus

l mt s

iprimeeqr (9)




controller



iprimelowast


do not



l ml s

iprimeqrh = 0




minus

K ir

v h


wwwietdlorg



8122019 06486251



id sh ≃vd sh




qsh and vlowastd sh



where

T v( s) =G v

scf

v cc

s + v cc



vqsh ≃ minus1

1 + T v( s) v cc

s + v cc

iqlh

v cccf

(11)


s + v cc

id lh

v cccf

(12)


V qsh = minus 1

1 minus j (v vcv h)

1 + j (v hv cc)

lowast1


v cc

lowast X cf I qlh

(13)


1 + T v( s)

1 + ( sv cc)

(v ev cc)


1 + ( sv cc)

vqsh

vd sh

= minus

1

v cc cf

iqlh

id lh

(10)

wwwietdlorg



8122019 06486251


or

V qsh = minus1


v cc

lowast X cf I qlh

(14)

V d sh = minus1


v e

v cc



R = minus

1


lowastv e

v cc

lowast X cf

R I d lh

(16)








=

j


v cc


(19)

V sh+ + V shminus

= minus

1


lowastv e

v cc


(20)

[ V sh+

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

lowastv e

v cc

lowast X cf

I lh+

(21)

Similarly from (16)

[ I sh+

= I shminus

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X cf

R lowast

1

2 I lh+ + I lhminus

(22)




wwwietdlorg



8122019 06486251


















113 Ω











wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



id sh ≃vd sh




qsh and vlowastd sh



where

T v( s) =G v

scf

v cc

s + v cc



vqsh ≃ minus1

1 + T v( s) v cc

s + v cc

iqlh

v cccf

(11)


s + v cc

id lh

v cccf

(12)


V qsh = minus 1

1 minus j (v vcv h)

1 + j (v hv cc)

lowast1


v cc

lowast X cf I qlh

(13)


1 + T v( s)

1 + ( sv cc)

(v ev cc)


1 + ( sv cc)

vqsh

vd sh

= minus

1

v cc cf

iqlh

id lh

(10)

wwwietdlorg



8122019 06486251


or

V qsh = minus1


v cc

lowast X cf I qlh

(14)

V d sh = minus1


v e

v cc



R = minus

1


lowastv e

v cc

lowast X cf

R I d lh

(16)








=

j


v cc


(19)

V sh+ + V shminus

= minus

1


lowastv e

v cc


(20)

[ V sh+

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

lowastv e

v cc

lowast X cf

I lh+

(21)

Similarly from (16)

[ I sh+

= I shminus

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X cf

R lowast

1

2 I lh+ + I lhminus

(22)




wwwietdlorg



8122019 06486251


















113 Ω











wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251


or

V qsh = minus1


v cc

lowast X cf I qlh

(14)

V d sh = minus1


v e

v cc



R = minus

1


lowastv e

v cc

lowast X cf

R I d lh

(16)








=

j


v cc


(19)

V sh+ + V shminus

= minus

1


lowastv e

v cc


(20)

[ V sh+

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

lowastv e

v cc

lowast X cf

I lh+

(21)

Similarly from (16)

[ I sh+

= I shminus

= 1 +v vc

v h

2

+v h

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X cf

R lowast

1

2 I lh+ + I lhminus

(22)




wwwietdlorg



8122019 06486251


















113 Ω











wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251


















113 Ω











wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251



V s 6n+1( )

PU

= 1 + v vc

6nv e

2

+ 6nv ev cc

2

minus2v vc

v cc

minus(12)v e

v cc


PU

6n+ 1 (23)

I s 6nminus1( )

PU

= I s 6n+1( )

PU

= 1 +v vc

6nv e

2

+6nv e

v cc

2

minus2v vc

v cc

minus(12)

v e

v cc

lowast X l6n

lowast X cf

R lowast

I l1

I s1

(24)





PU






wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251











a Load current


wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251




5 Conclusion





6 References















2007 54 (2) pp 1237ndash1239







wwwietdlorg



8122019 06486251














wwwietdlorg



8122019 06486251














wwwietdlorg



Documents

06486251