Research Article Evaluation of a Trapezoidal Predictive ...

Research ArticleEvaluation of a Trapezoidal Predictive Controller fora Four-Wire Active Power Filter for Utility Equipment ofMetro Railway Power-Land Substations

Sergio Salas-Duarte1 Ismael Araujo-Vargas1

Jazmin Ramirez-Hernandez1 and Marco Rivera2

1Escuela Superior de IngenierıaMecanica y Electrica Unidad Culhuacan Instituto Politecnico Nacional Avenida Santa AnaNo 1000Col San Francisco Culhuacan 04430 Mexico DF Mexico2Universidad de Talca 2 Norte 685 Talca Chile

Correspondence should be addressed to Ismael Araujo-Vargas iaraujoipnmx

Received 14 August 2015 Accepted 20 December 2015

Academic Editor Shengbo Eben Li

Copyright copy 2016 Sergio Salas-Duarte et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The realization of an improved predictive current controller based on a trapezoidal model is described and the impact of thistechnique is assessed on the performance of a 2 kW 216 kHz four-wire Active Power Filter for utility equipment of Metro RailwayPower-Land SubstationsThe operation of the trapezoidal predictive current controller is contrasted with that of a typical predictivecontrol technique based on a single Euler approximation which has demonstrated generation of high-quality line currents eachusing a 400V DC link to improve the power quality of an unbalanced nonlinear load of Metro Railway The results show that thesupply current waveforms become virtually sinusoidal waves reducing the current ripple by 50 and improving its power factorfrom 08 to 0989 when the active filter is operated with a 16 kW load The principle of operation of the trapezoidal predictivecontroller is analysed together with a description of its practical development showing experimental results obtained with a 2 kWprototype

1 Introduction

The use of Active Power Filters (APFs) in the electrical gridis critical for on-land transportation applications such asMetropolitan Railway Substations which reduce the flowingof current harmonics caused by the increased utilizationof nonlinear loads whilst improving the power quality ofthe supply APFs are an attractive solution to comply withthe national and international power quality standards atevery level of the network infrastructure [1ndash3] since high-performance switching devices appear available in themarketto develop power converters [4] In addition the develop-ment of fast and versatile microprocessors has facilitated theimplementation of nonlinear control techniques and therebyAPFs are becoming accurate power processors that reshapeclean sinusoidal supply currents [5ndash9]

Four-wire shunt APFs are a commonplace strategythat exhibit attractive characteristics to inject currents andreshape the line currents drawn by unbalanced nonlinearloads whilst providing a path to cancel the neutral currentby using either an additional switching limb or a split DC link[10 11]These circuits typically incur in the use of a power the-ory to calculate the reference currents [12] such that the filtermay operate as a current amplifier that injects compensatingcurrents to the grid causing a complex transistor switchingscheme since the generated filter currents must track thereferences Predictive control is an attractive method for con-trolling current waveforms in three-phase converters [6 713ndash20] since a piecewise linearmodel of the converter is usedtogether with a cost function to determine an appropriateconverter switching

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2712976 11 pageshttpdxdoiorg10115520162712976

2 Mathematical Problems in Engineering

997888997888rarri1205721205730

997888997888rarri1205721205730

N

997888rarris

997888rarriL

997888rarriL

997888997888997888rarriLoad

997888997888997888rarriL1205721205730

997888997888997888rarriL1205721205730

997888997888997888rarrs1205721205730997888997888997888rarrs1205721205730

997888997888997888rarrs1205721205730

997888997888997888rarrs1205721205730 997888rarrs

q120572

q120573

q0

pT

plossminus

minus

minus

minus

HE(s)Eref

400V

i120572lowast

i120573lowast

i0lowast

+

+

+

+

+

+

Hbal(s)

P-Q theory InverseP-Q theory

Predictivecurrent

controller

Clarktransform

PLL and clarktransform

Clarktransform

IGBTdrivers

Unbalancednonlinear load

Active powerfilter

NDCreject

pT

i0bal E

vge1lowast

vge2lowast

vge6lowast

E

vge1

vge2

vge6

E1 E2

C1 C2

Q1 Q4

Q5 Q2

Q3 Q6

D1

D5 D2

D3 D6

D4

vTC

vSC

vRC

Lf

vdiff

sum

sumsum

sum

sum

Figure 1 Four-wire shunt active filter and its corresponding control block diagram

This paper presents the realization and experimentalverification of a trapezoidal predictive current controller fora four-wire shunt APF that improves the power quality ofunbalanced AC loads in contrast to the typical predictiveEuler control strategy The trapezoidal strategy relies itsoperation on a discrete trapezoidal linear approximation thatmore accurately determines the switching of the active filterfor the one-step ahead current sample such that three signifi-cant advantages are potentially exhibited first the trapezoidalpredictive controller slightly increments the processing timewithout affecting the switching of the power convertersecond in contrast to the typical Euler approximation usedin other works [6 7 13ndash20] the trapezoidalmethod generateslower AC current ripple and third the convergence time andload operating performance are wider than those obtainedusing the typical predictive control strategy which improvesthe reference current tracking and therefore the powerquality Experimental results obtainedwith a 2 kVAprototypeare presented demonstrating that the trapezoidal predictivecontrol may accurately compensate the currents drawn byan unbalanced nonlinear load under static and dynamicconditions

2 Four-Wire Shunt Active Filter

21 Circuit Description The four-wire shunt APF is con-nected in parallel to the unbalanced nonlinear load as shownat the right-hand side of Figure 1 which consists of a splitDC link formed by 119862

1and 119862

2which refer to the AC supply

neutral node119873 to provide a path tomitigate a commonmodecurrent a typical three-phase current-feed active converterformed by transistors 119876

1to 1198766and diodes 119863

1ndash1198636 and three

line filter inductors 119871119891used to generate the filter current

vector 997888rarr119894119871

= [119894119871119877 119894119871119878

119894119871119879]119879 by the difference between the

supply and converter voltage vectors 997888rarrV119904= [V119877119873 V

119878119873V119879119873]119879

and 997888rarrV119862

= [V119877119862119873 V119878119862119873

V119879119862119873]119879 thereby obtaining virtual

sinusoidal supply currents 997888rarr119894119904= [119894119878119877 119894

119878119878119894119878119879]119879

22 Principle of Operation of the Active Filter The principleof operation of the APF of Figure 1 may be described usingthe control block diagram presented at the left-hand side ofFigure 1 An instantaneous active and reactive power theoryP-Q theory block in Figure 1 [12] is used to obtain an effectivecalculation of the reference currents that the APF may injectto the supply to instantaneously mitigate the reactive anddistorted power components drawn by the nonlinear loadand balance the active power per phase The P-Q theory usesthe Clarke transformation of the supply voltage and loadcurrent as shown in

V120572120573120579

=[[[

[

V0

V120572

V120573

]]]

]

= radic2

3

[[[[[[[[

[

1

radic2

1

radic2

1

radic2

1 minus1

2minus1

2

0radic3

2minusradic3

2

]]]]]]]]

]

[[

[

V119877119873

V119878119873

V119879119873

]]

]

Mathematical Problems in Engineering 3

119894120572120573120579

=[[[

[

1198940

119894120572

119894120573

]]]

]

= radic2

3

[[[[[[[[

[

1

radic2

1

radic2

1

radic2

1 minus1

2minus1

2

0radic3

2minusradic3

2

]]]]]]]]

]

[[

[

119894Load119877

119894Load119878

119894Load119879

]]

]

(1)

such that the calculation of the active and reactive instanta-neous powers in the 1205721205730 coordinate system is obtained asrespectively shown in

119901119879

=997888997888rarrV1205721205730

sdot997888997888rarr1198941205721205730

= V120572119894120572+ V120573119894120573+ V01198940 (2)

119902 =997888997888rarrV1205721205730

times997888997888rarr1198941205721205730

=[[

[

119902120572

119902120573

119902120579

]]

]

=

[[[[[[[[[[[

[

[

V120573

V0

119894120573

1198940

]

[V120579

V120572

1198940

119894120572

]

[

V120572

V120573

119894120572

119894120573

]

]]]]]]]]]]]

]

(3)

where 119901119879is the real power or internal product of the voltage

and current vectors and 119902 is the imaginary vector power orexternal product of the voltage and current vectors whichis composed of 119902

120572 119902120573 and 119902

0 Since the load uses a fourth

conductor namely the neutral which is very common in low-voltage distribution system the P-Q calculation may includeboth zero-sequence voltage and current as shown in (2) and(3) Therefore the instantaneous powers defined above maybe combined in a single matrix transformation as shown asfollows

[[[[[

[

119901119879

119902120572

119902120573

1199020

]]]]]

]

=

[[[[[

[

V120572

V120573

V0

0 minusV0

V120573

V0

0 minusV120572

minusV120573

V120572

0

]]]]]

]

[[

[

119894120572

119894120573

1198940

]]

]

(4)

which is defined on the 1205721205730 reference frame 119901119879and 119902 are

instantaneous power signals that have averaged and oscilla-tory components that may be used to calculate a referencecurrent vector for the APF control systemThe average of 119901

119879

119901 corresponds to the energy per time unity that is transferredfrom the supply to the load and becomes the power that thesystem truly uses [8] In this way the ideal condition wouldbe to remove the oscillatory portion of the real power 119901 andthe imaginary power 119902 of power drawn by the load such thatthe calculation of the reference currents for compensatingthe currents drawn by the load may be given with

[[

[

119894120572reflowast


1198940reflowast

]]

]

=1

V2120572120573120579

[[[

[

V120572

0 V120579

minusV120573

V120573

minusV120579

0 V120572

V120579

V120573

minusV120572

0

]]]

]

[[[[[

[

119902120572

lowast

119902120573

lowast

1199020

lowast

]]]]]

]

(5)

which is represented in Figure 1 as the inverse P-Q theoryblock which subtracts 119901 from 119901 to obtain the oscillatorycomponent of 119901 In this fashion the reference currents of(5) are used to operate the three-phase converter of Figure 1as a current amplifier driven by the trapezoidal predictivecurrent controller block shown at the centre of Figure 1

23 DC-Link Voltage Controller The APF requires a fixedDC-link capacitor voltage 119864 greater than the peak value ofthe line-to-line supply voltage for instance 119864 = 400V whena 220V 60Hz supply is being used Since the shunt APFtopology is identical to that of an active three-phase rectifier[14] the circuit boosts the DC-link voltage using an externalvoltage control loop that generates a loss power control signal119901loss which is added to to supply energy for the DC-linkcapacitor and compensate the power losses of theAPF circuitThis is shown in the left bottom side of Figure 1 where a linearcontrol loop calculates 119901loss using the error between the 119864

reference 119864ref and the DC-link voltage 119864 which is obtainedadding the measured DC-link capacitor voltages and com-pensating this error with 119867

119864(119904)

24 DC-Link Capacitor Voltage Balancing Controller Sincethe split DC-link node 119873 is used to draw a compensatingcurrent for the neutral wire of the supply the DC-linkcapacitor voltages may become unbalanced due to the flow ofa small DC current An additional zero-sequence balancingcurrent 119894

0bal is used after the zero-sequence reference currentcalculation to overcome a voltage unbalance between thecapacitors of the split DC link [21] This is shown at the bot-tom of Figure 1 where again a linear control loop calculates1198940bal by compensating the error between the DC-link capaci-tor voltages Vdif with 119867bal(119904)

3 Trapezoidal Predictive Current Controller

31 Discrete Linear Model of the APF Converter A spacevector AC-side model of the APF three-phase converter isderived calculating the filter inductor voltage vector as shownin

997888997888997888rarrV1198711205721205730

= 119871119889997888997888997888rarr1198941198711205721205730

119889119905

=997888997888997888rarrV1199041205721205730

minus997888997888997888rarrV1198881205721205730

(6)

which may be solved to calculate the line current vector 997888rarr119894119871as

shown as follows

997888997888997888rarr1198941198711205721205730

(1199051) =

997888997888997888rarr1198941198711205721205730

(1199051+ 119896119879) + int

1199051+(119896+1)119879

1199051+119896119879

997888997888997888rarrV1198711205721205730

(119905) 119889119905 (7)

A discrete time model of (7) may be obtained by usingthe trapezoidal approximation shown in Figure 2 such that(7) becomes

997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

2119871[997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896

+997888997888997888rarrV1199041205721205730119896+1

minus997888997888997888rarrV1198881205721205730119896+1

]

(8)


tt

1 + kT t1 + (k + 1)T

997888997888997888997888rarrL1205721205730(t1 + kT)

997888997888997888997888rarrL1205721205730(t1 + (k + 1)T)

997888997888997888997888rarrL1205721205730(t) =997888997888997888rarrs1205721205730(t) minus

997888997888997888rarrc1205721205730(t)

Figure 2 Trapezoidal approximation of the volt-seconds integral of(9)

where119879 is the sampling period thatmust be small to obtain anaccurate model approximation of the system Since 997888997888997888rarrV

1199041205721205730119896asymp

997888997888997888rarrV1199041205721205730119896+1

(8) is rewritten as follows

997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

2119871[2

997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (9)

which may produce eight one-step ahead current vectors997888997888997888rarr1198941198711205721205730

0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 since 997888997888997888rarrV

1198881205721205730119896+1has six active 997888997888997888rarrV

1198881205721205730

1 to997888997888997888rarrV1198881205721205730

6 and two neutral vectors 997888997888997888rarrV1198881205721205730

0 and 997888997888997888rarrV1198881205721205730

7 that arelisted in Table 1 with respect to their transistor switchingstates assuming the common mode voltage due to the ACneutral node connection to the DC link [15] 997888997888997888rarr

1198941198711205721205730

0

119896+1to

997888997888997888rarr1198941198711205721205730

7

119896+1are dispersed around the 119896th current sample 997888997888997888rarr119894

1198711205721205730119896

as shown in 1205721205730 frame of Figure 3 such that one of thesemaybecome near to the reference current sample 997888997888997888rarr

1198941198711205721205730

lowast

119896

32 Cost Function of the Current Controller An error currentvector 997888997888997888997888rarr

1198941198711198901205721205730119896

may be used as a cost function to evaluatewhich of the transistor switching states causes the nearestone-step ahead current sample to997888997888997888rarr

1198941198711205721205730

lowast

119896 such that997888997888997888997888rarr119894

1198711198901205721205730119896may

be expressed as shown as follows [16]

997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896=

997888997888997888rarr1198941198711205721205730

lowast

119896minus

997888997888997888rarr1198941198711205721205730

0minus7

119896+1 (10)

The size of (10) may be evaluated using the Euclidean normof 997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896 997888997888997888997888rarr1198941198711198901205721205730

0minus7

1198962 which is equal to

1003817100381710038171003817100381710038171003817

997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896

1003817100381710038171003817100381710038171003817

2

=10038161003816100381610038161003816119894lowast

119871120572119896minus 1198940minus7

119871120572 119896+1

10038161003816100381610038161003816

2

+100381610038161003816100381610038161003816119894lowast

119871120573119896minus 1198940minus7

119871120573 119896+1

100381610038161003816100381610038161003816

2

+10038161003816100381610038161003816119894lowast

1198710119896minus 1198940minus7

1198710 119896+1

10038161003816100381610038161003816

2

(11)

such that theminimum 997888997888997888997888rarr1198941198711198901205721205730

0minus7

1198962 determines the transistor

switching state that may be used at the 119896th instant toproduce an appropriate three-phase filter current trackingwith respect to the current reference vector [17]

33 Control Algorithm of the Four-Wire APF Following thedescription given above a flow diagram of the APF controlalgorithmof Figure 1 is shown in Figure 4This diagram startswith the parameters initialization of the microcontroller andthen enters to an iterative loop control cycle In this cycle allthe voltage and currents variables are sensed such as the sup-ply voltage997888rarrV

119904 the filter current997888rarr119894

119871 the load current 119894Load and

the DC-link voltage 119864 where the AC inputs are converted to1205721205730 plane using (1) Since the APF may operate with a dis-torted voltage or high source impedance [22]997888rarrV

119904is processed

with a Phase Locked Loop (PLL) to obtain a clean three-phasesupply and phase referenceThe next process in the algorithmis the calculation of the two external voltage controllers usedto maintain charged and balanced DC-link capacitors at afixed voltage level which contribute to calculate the referencecurrents through the inverse P-Q theory (4) and (5)Once thereference currents are calculated an ldquoelse-ifrdquo tree is startedto process the trapezoidal predictive current controller withthe eight possible transistor state combinations of the APFconverter which uses the discrete current model of (7) andthe cost function of (10) such that eight one-step aheadcurrent values are evaluated and then weighted against thecurrent reference vector using (10) Finally the converter statevector that minimizes the cost function is determined andthereby the algorithm applies the selected state vector to theAPF converter

4 Experimental Verification

41 Prototype Description A 2 kVA four-wire shunt APFprototype rig was built to evaluate the operation of theAPF of Figure 1 Table 2 lists the operating parameters andcomponents of the rig

A 150MHz TMS320F28335 Digital Signal Processor(DSP) was used to implement the control strategy of Figures 1and 4 using a 32-bit data word length for floating point oper-ations ensuring numerical stability Additional hardware wasutilized to interface the DSP with the power converter suchas voltage and current sensors signal conditioners IGBTdrivers and fiber optic links The APF was operated withthe aid of a PLL [22] and driven with either the trapezoidalpredictive controller (9) or a typical predictive controller thatuses the Euler approximation of

997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

119871[997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (12)

to experimentally compare the performance

42 Experimental Results The 2 kVA APF prototype wasverified with the Euler and trapezoidal predictive currentcontrollers and a 127V 60Hz line-to-neutral supply voltageand under three nonlinear load conditions a 16 kW naturallycontrolled three-phase rectifier with a 119871119862 filter Figure 5(a)a 09 kW four-wire unbalanced load Figure 5(b) that con-sisted of two naturally controlled single-phase rectifiers bothwith a 119871119862 output filter and each supplied with different singlephases and a resistive load supplied with a single phase and a1 kW unbalanced load condition Figure 5(c) similar to


Table 1 Normalized converter voltage space vectors with respect to the transistor switching states

Transistor state combination 997888997888997888rarrV1198881205721205730

Normalized converter voltages1198761

1198762

1198763

1198764

1198765

1198766

119881120572E 119881

120573E 119881

0E

0 1 0 1 0 1 997888997888997888rarrV1198881205721205730

0

0 0 minus1

2

1 1 0 0 0 1 997888997888997888rarrV1198881205721205730

1

radic2

30 minus

1

6

1 1 1 0 0 0 997888997888997888rarrV1198881205721205730

2 radic6

6

1

radic2

1

6

0 1 1 1 0 0 997888997888997888rarrV1198881205721205730

3

minusradic6

6

1

radic2minus1

6

0 0 1 1 1 0 997888997888997888rarrV1198881205721205730

4

minusradic2

30

1

6

0 0 0 1 1 1 997888997888997888rarrV1198881205721205730

5

minusradic6

6

minus1

radic2minus1

6

1 0 0 0 1 1 997888997888997888rarrV1198881205721205730

6 radic6

6

minus1

radic2

1

6

1 0 1 0 1 0 997888997888997888rarrV1198881205721205730

7

0 01

2

Table 2 Operating parameters and components of the four-wire APF prototype

Electrical parametersVariable Value

Prototype power rating (laboratory design) 2 kWThree-phase supply 119881

119878119873127V 60Hz

DC-link voltage reference 119864 400VSampling frequency 119891

119878216 kHz

ADC resolution 12 bitsPrototype components

Line filters inductors 119871119891

10mHDC-link capacitors 119862 10mFPower IGBTs BSM100GD60DLC 1200V 30AIGBT drivers Infineon 2ED300CL7-sACDC voltage sensors LEM LV 25-PACDC current sensors Honeywell CSNA111DC-link voltage controller 119867

119864(119904) 3000119904 + 45e4

1199042 + 1000 minus 671e minus 8

DC-link voltage balance controller 119867bal(119904)400119904 + 10

119904

the one used at a power substation of Line B MetropolitanRailway ofMexico City for powering electronic utility equip-ment

Figure 6(a) shows the experimental supply currents 119894119878119877

119894119878119878 and 119894

119878119879and one supply phase voltage V

119877119873 obtained

with the load condition of Figure 6(a) The experimental linecurrent waveforms were typical of a three-phase 6-pulserectifier without the operation of the APF but when theAPF was turned on using the Euler predictive controller thesupply currents became virtually sinusoidal waveforms withthe supply currents Total Harmonic Distortion (THD) andthe total power factor being improved from 29 to 15 andfrom 095 to 098 respectively which confirmed that the APFwas properly operating Twomain characteristics were foundin the experimental supply current waveforms of Figure 6(a)a 42 kHz high-frequency ripple and a small glitch occurringat every rising and falling slope of the load current waveform

The first was attributed to the Euler approximation used withthe predictive control switching that continuously tracks thereference currents [23] which was confirmed with a dynamiccondition of stepping the output filter inductance from50mH to 100mH Figure 6(b) shows that the operation ofthe APF with the Euler predictive current control and the P-Q theory is maintained throughout the filter inductance stepsince the sinusoidal waveformquality of the supply currents isstable as shown in Figure 6(b) ensuring reliable operation ofthe APF and therefore the predictive controller is likely to becompliant under dynamic conditions a typical requirementfor control techniques however the amplitude of the filtercurrent waveforms slightly fell from 2A to 15 A duringthe transient response with the supply current THD beingbarely degraded around 24 The second characteristic wasconfirmed by contrasting the measured filter current 119894

119871119877

with its digital reference 119894119871119877

lowast as shown in the left-hand side


iL120573

997888997888997888rarriL1205721205730

3

k+1997888997888997888rarriL1205721205730

2

k+1

997888997888997888rarriL1205721205730

4

k+1

997888997888997888rarriL1205721205730

1

k+1

997888997888997888rarriL1205721205730

5

k+1

997888997888997888rarriL1205721205730

6

k+1

997888997888997888rarriL1205721205730kminus1997888997888997888rarr

iL1205721205730k

Path of 997888997888rarriLref (t)

997888997888997888rarriL1205721205730

07

k+1

997888997888997888rarriL1205721205730

lowast

k

iL120572

iL0

Figure 3 One-step ahead current samples 997888997888997888rarr1198941198711205721205730

0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 around 997888997888997888rarr

1198941198711205721205730119896

and 997888997888997888rarr1198941198711205721205730

lowast

119896in the 1205721205730 current frame

of Figure 6(c) revealing that the predictive current controlslightly follows the reference during the high 119889119894119889119905 periodsof the load current waveforms due to the simple Eulerapproximation used in the algorithm reducing the trackingaccuracy of the references and therefore introducing smallglitches to the supply current waveforms This undesiredphenomenon was improved by changing the Euler approx-imation of the predictive controller to the trapezoidal tech-nique as shown in the right-hand side of Figure 6(c) whichreveals in its amplification shown in Figure 6(d) that thetrapezoidal predictive controller reduces the current rippleamplitude by approximately 50 increasing its frequencyrate from 42 kHz to 95 kHz and producing a lower supplycurrent THD 102 and a power factor of 0989 in contrastto the Euler technique with the supply current waveformsbecoming virtually free of high-frequency glitches and ripple


119894119878119878 and 119894

119878119879and the neutral current 119894

119873obtained with the

unbalanced load condition of Figure 5(b) at 09 kW This fig-ure shows that the line currents are typical of an unbalancednonlinear load before the APF is activated and after the APFis on the supply currents become virtually balanced sinu-soidal waveforms of 24 A with the supply current THD andthe power factor being improved from an unbalanced 43 toa balanced 25 and from 093 to 0972 respectively revealingagain that the active filter prototype is correctly operatingwith a four-wire load In addition the same figure showsthat the power quality of the supply currents becomes muchmore improved when the predictive controller is changedfrom the Euler to the trapezoidal technique with the supplycurrent THD and the power factor becoming 15 and 098

respectively These experimental current waveforms haveagain a high-frequency ripple being 42 kHz when the APFis operated with the typical Euler technique and 95 kHz withthe proposed trapezoidal strategy In Figure 7(a) the neutralcurrent 119894

119873was virtuallymitigated after theAPFwas activated

becoming more reduced when the APF was driven with thetrapezoidal controllerThis experiment revealed the effective-ness of the 4-wire P-Q theory used in this work togetherwith the predictive current control switching

Figure 7(b) shows the experimental supply currents 119894119878119877

119894119878119878 and 119894



front-end controlled rectifier drive andmonophasic resistiveload of Figure 5(c) at 09 kW Before the APF is activated asshown in Figure 7(b) the experimental supply currents arecompletely unbalanced distorted and phase-displaced dueto the biphasic connection of the front-end rectifier of themotor drive and the resistive load connection however whenthe APF is on using firstly the Euler predictive techniqueand then the trapezoidal version the supply currents becomeagain balanced with virtual sinusoidal waveforms such thattheir THD was improved from an unbalanced 40 to abalanced 23 and 18 for the Euler and trapezoidalmethodsrespectively and the power factor from 08 to 097 and 098again for the Euler and trapezoidal methods respectively InFigure 7(b) the neutral current 119894

119873was again virtually miti-

gated after the APFwas activated becoming almost cancelledwhen the APF used the trapezoidal controller

A power analyser was used to measure the supply activepower 119875 the supply apparent power 119878 the per-phase supplycurrent THD 119868

119877119878THD 119868

119878119878THD and 119868

119879119878THD and the total

power factor during the experiments described above The


Start

Controller parameter initialization

Clark transform

PLL algorithm at kth instant

PQ theory calculation at kth instant

DC-link controller at kth instant

DC-link balance controller at kth instant

Combination = j

No

No

Yes

Yes

For j = 0 middot middot middot 7

Apply transistor state combination to APF

Measurement of 997888rarrsk997888rarriLk

997888997888997888997888rarriLoadk E1k E2k

Prediction of 997888997888997888rarriL1205721205730

0ndash7

k+1

997888997888997888997888rarriLe1205721205730

j

klt997888997888997888997888rarriLe1205721205730

jminus1

k

997888997888997888997888997888997888997888rarriLe1205721205730min

= 0 and combination = 0997888997888997888997888997888997888997888rarri Le1205721205730min

=997888997888997888997888rarriLe1205721205730

j

k

Saving 997888997888997888997888rarrc1205721205730Combinationk

for next sampling

j = 7

997888997888997888997888rarriLe1205721205730

j

kEvaluation of

Figure 4 Algorithm flow diagram of the predictive current controller

results are contrasted in the comparative bar plot of Figure 8calculated with a 2 kVA APF rating a 127V 60Hz line-to-neutral supply voltage and a 400V APF DC link Figure 8shows that 119875 slightly rise by approximately 5 100W whenthe APF prototype was used to improve and balance thesupply currents among the experiments the additional powerloss occurring in the transistors filter inductors and DC-linkcapacitors of the APF converter due to the high-frequencyoperation In comparison with the unbalanced load the cur-rent THD reduction is representative when the APF is usedtogether with the trapezoidal predictive controller to com-pensate the drawn currents of the balanced load as shown inFigure 8 whereas the current THD is slightly improved whenthe APF is used together with the Euler strategy and the loadcases of Figure 5 nevertheless the drawn current through theneutral wire is noticeably cancelled when the APF is usedto correct the power quality of the four-wire AC loads ofFigures 5(b) and 5(c) In contrast with the balanced load thedistribution between apparent and active power for the APF

with the unbalanced load becomes equilibrated due to cor-rection of current phase displacement

Closer inspection of the microprocessor operationrevealed that the total period to perform the algorithmof Figure 4 was around 29 120583s with the Euler predictivecontroller which is well below the sampling period to ensureminimal delay effects of the control system Unlike theexperimental verification of the Euler predictive controllerthe experimental verification of the APF with the trapezoidalpredictive controller resulted in a slight increment ofalgorithm processing time of Figure 4 from 29120583s to 30 120583swhich was imperceptible during the experimental verifica-tion of the APF

The presented trapezoidal predictive controller is slightlymore complex than the typical predictive strategy to per-form the current reference tracking and generates a slightincrease of power losses which could make it inadequatefor implementation in low-rated rigs nevertheless the ripplecurrent reduction closer current tracking and power quality


N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)

Figure 5 Nonlinear loads used to experimentally verify the APF and predictive current controller of Figure 1 (a) Balanced three-wire load(b) unbalanced four-wire load and (c) unbalanced load used at a Metro Power Substation


C1 100Vdiv 2800V offsetC2 DC1M 100Adiv minus1150V offsetC3

C3

DC 100Adiv minus2820A offsetC4 BwL DC 100Adiv 730A offset

Tbase minus595ms

160msdiv160MS 100MSs

Trigger C1 DC

Stop 0VEdge Positive

BwL DC

APF on with the Euler controller

1M

APF on with the Euler controllerlA

C1

C2

C4

(a)

C3

C1 F B D1 100Vdiv minus29500VC2 DC 200Adiv 0mA offsetC3 INV DC1M 50Adiv minus14A offsetC4 BwL DC 200Adiv 4640A offset

Tbase minus9104ms160msdiv

160MS 100MSs

Trigger C1 DCStop minus68VEdge Positive

DC filter inductor step

C1

C2

C4

(b)

C1 F B D 500Adiv 480mA offsetC2 F B D1 500Vdiv 5080V offsetC3 F B D 500Adiv 5080A offsetC4 F B D1 500Adiv minus4820V offset

Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive

APF on with the trapezoidal controllerAPF on with the Euler controller APF ith th t id l t llAPF on ithhhhhhhhh thehhhhhe Euler controllerl

C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive

APF on with the trapezoidal controllerAPF on with the Euler controller APF on wwwwwwith the trapezoidal controllerlde nAPF on with the Euler controllerh

(d)

Figure 6 Experimental verification of the APF prototype with the 16 kW balanced load of (a) (a) Measured waveforms V119877119873

(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894

119878119879(blue) before and after the activation of the APF (b) Measured response of V

119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877

(yellow) to a filter inductance step from 50mH to 100mH (c) Measured response of 119894119871119877

(blue) and 119894119878119877

(green) to a predictive currentcontroller step from Euler to the trapezoidal approximation which is contrasted against the reference 119894

119871119877

lowast (red) and load current 119894Load119877 (d)Time amplification of (c) at the instant of the predictive controller step 127V 60Hz supply 400VAPFDC link and a 216 kHz APF samplingfrequency

improvement are important advantages to consider over thetraditional predictive controller technique Furthermore thedigital implementation is acceptable for fastmicrocontrollerssuch as a DSPs and hybrid digital controllers and will beused once the trapezoidal control strategy is implementedto control other power converter systems which would besuitable to obtain high power quality results

5 Conclusions

The utilization of a trapezoidal predictive technique to gen-erate the filter currents of a four-wire shunt APF allows the

power quality improvement of using unbalanced nonlinearloads for on-land utility applications such that the supplycurrents become virtual sinusoidal waves The latter makesthe current control strategy attractive for easy and straightimplementation on future power converters that requirehigh-performance power quality nevertheless the controltechnique is suitable for a wide range of power converterapplications In this work the trapezoidal predictive con-troller was experimentally verified and evaluated with thefour-wire APF under three load conditions in the first theload was set up with a three-wire balanced nonlinear circuitto preliminary check the basic operation of the control


C1 BwL DC1M 500Vdiv minus520V offsetC2 DC1M 500Vdiv 470V offsetC3 DC 500Adiv 1335A offsetC4 F B D 100Adiv minus29900A

minus15885 s Trigger C2 DC

100V

Tbase


StopEdge Positive

APF on with thetrapezoidal controller

APF on with theEuler controller

C3

C1

C2

C4 APF PFAAAAAA F on with thhhhh thehhhhhhhhhhhhhhFFFtrapezoiooo dal controllerno

APF APFPFPFAPAPAAPFAAPAPFPFFAPAPFAPFPFPAPFFFFFPFFFAAPPFFFFFAA FFFFAPFFFFFFAAPFFA FA oon withhhhhhhhhhithithithththhhhhhhhhhhith iii hhhhhhhthhitiiithhhhhhithiitthhhhhhiiiiththhhhthhth thehhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhA wwEuler controlleroon

(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive

C1 BwL DC1M 500Vdiv minus570V offsetC2 DC1M 500Vdiv 1410V offsetC3 BwL DC1M 100Vdiv minus32400VC4 BwL DC 500Adiv 445A offset



C3

C1

C2

C4APF on withhhhh theAPF on with theF

trapezoie dal controllo ereeeapezoidal cont o eeeetzAPF on with theAPF on with thewwEuleulul r coontrorr llerrleEulll coontrolleooon

(b)

Figure 7 (a) Measured supply current waveforms 119894119878119877

(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894

119873(green) before and after the activation of the APF

with the Euler and Trapezoidal predictive controllers and the load condition of Figure 5(a) (b) Measured supply current waveforms 119894119878119877(red)

119894119878119878(yellow) 119894

119878119879(green) and 119894

119873(blue) before and after the activation of the APF with the Euler and trapezoidal predictive controllers and the

load condition of Figure 5(b) 127V 60Hz supply 400V APF DC link and 216 kHz APF sampling frequency

00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)

Active power P (pu)Apparent power S (pu)

Power factoriSR THD

iSS THDiST THD

Figure 8 Bar plot of the power quality results for the AC loadcircuits of Figure 5 with the APF prototype operating with theEuler and trapezoidal predictive techniques 127V 60Hz supply and2 kVA APF rating

technique such that sinusoidal supply current waves weregenerated and the power quality was improved A currentTHD of 10 and a power factor of 0989 were measured inthe first experiment showing a noticeable improvement incontrast to the traditional predictive technique

In the second and third load conditions the load wasfour-wire unbalanced nonlinear load with the load currentsbeing much distorted and producing a neutral current pathin both load conditions The supply current waveforms wereall improved and balanced when the APF and the trapezoidal

predictive controller were activated with the neutral currentbeing mitigated the current THD was 15 and 18 respec-tively and the power factorwas 098 in both experiments witha 127V 60Hz three-phase supply voltage The shape of thesupply current waveforms and the power quality were signif-icantly improved in comparison with the original load cur-rents and power quality with the active power being slightlyincreased due to the high-frequency switching losses of theAPF power transistors

The practical realization of the presented trapezoidalpredictive controller could consider the use of an extendedsampled-data horizon either forward or backward to achievea faster convergence and reduce the current ripple amplitudeThis would be convenient for developing power converterswith new generation of switching power devices for otherapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Council of Sci-ence and Technology (CONACyT) the Instituto PolitecnicoNacional (IPN) of Mexico the Institute of Science andTechnology of Mexico City (ICyT) and the Universidad deTalca Chile for their encouragement and the realizationof the prototype Additionally the authors acknowledge theMetropolitan Railway Transportation System of Mexico City(SCT Metro) for the support offered to obtain power qualitymeasurements at Line B installations


References

[1] IEEEApplicationGuide for IEEE Standard 1547 ldquoIEEE standardfor interconnecting distributed resources with electric powersystemsrdquo IEEE Standard 15472-2008 2008

[2] S W Mohod andM V Aware ldquoA STATCOMmdashcontrol schemefor grid connected wind energy system for power qualityimprovementrdquo IEEE Systems Journal vol 4 no 3 pp 346ndash3522010

[3] IEEE ldquoIEEE recommended practices and requirements forharmonic control in electrical power systemsrdquo IEEE Standard519-1992 1992

[4] J D vanWyk and F C Lee ldquoOn a Future for Power ElectronicsrdquoIEEE Journal of Emerging and Selected Topics in Power Electron-ics vol 1 no 2 pp 59ndash72 2013

[5] P Kanjiya V Khadkikar and H H Zeineldin ldquoOptimal controlof shunt active power filter to meet IEEE Std 519 currentharmonic constraints under nonideal supply conditionrdquo IEEETransactions on Industrial Electronics vol 62 no 2 pp 724ndash734 2015

[6] P Acuna L Moran M Rivera R Aguilera R Burgos and VG Agelidis ldquoA single-objective predictive control method for amultivariable single-phase three-level NPC converter-basedactive power filterrdquo IEEE Transactions on Industrial Electronicsvol 62 no 7 pp 4598ndash4607 2015

[7] P Acuna L Moran M Rivera R Aguilera R Burgos and VG Agelidis ldquoA single-objective predictive control method fora multivariable single-phase three-level NPC converter-basedactive power filterrdquo IEEE Transactions on Industrial Electronicsvol 62 no 7 pp 4598ndash4607 2015

[8] P Jintakosonwit H Fujita and H Akagi ldquoControl and per-formance of a fully-digital-controlled shunt active filter forinstallation on a power distribution systemrdquo IEEE Transactionson Power Electronics vol 17 no 1 pp 132ndash140 2002

[9] Z Xiao X Deng R Yuan P Guo and Q Chen ldquoShunt activepower filter with enhanced dynamic performance using novelcontrol strategyrdquo IET Power Electronics vol 7 no 12 pp 3169ndash3181 2014

[10] P Acuna L Moran M Rivera J Rodriguez and J DixonldquoImproved active power filter performance for distributionsystems with renewable generationrdquo in Proceedings of the 38thAnnual Conference on IEEE Industrial Electronics Society(IECON rsquo12) pp 1344ndash1349 Montreal Canada October 2012

[11] P Acuna L Moran M Rivera J Dixon and J RodriguezldquoImproved active power filter performance for renewable powergeneration systemsrdquo IEEE Transactions on Power Electronicsvol 29 no 2 pp 687ndash694 2014

[12] H Akagi E H Watanabe and M Aredes Instantaneous PowerTheory and Applications to Power Conditioning IEEE PressSeries on Power Engineering Wiley-IEEE Press 2007

[13] R P Aguilera and D E Quevedo ldquoPredictive control of powerconverters designs with guaranteed performancerdquo IEEE Trans-actions on Industrial Informatics vol 11 no 1 pp 53ndash63 2015

[14] P Cortes J Rodrıguez P Antoniewicz andM KazmierkowskildquoDirect power control of an AFE using predictive controlrdquo IEEETransactions on Power Electronics vol 23 no 5 pp 2516ndash25232008

[15] J Scoltock T Geyer and U K Madawala ldquoModel predictivedirect power control for grid-connected NPC convertersrdquo IEEETransactions on Industrial Electronics vol 9 no 62 pp 5319ndash5328 2015

[16] M Lopez J Rodriguez C Silva and M Rivera ldquoPredictivetorque control of a multidrive system fed by a dual indirectmatrix converterrdquo IEEE Transactions on Industrial Electronicsvol 62 no 5 pp 2731ndash2741 2015

[17] A Bhattacharya and C Chakraborty ldquoA shunt active powerfilter with enhanced performance using ANN-based predictiveand adaptive controllersrdquo IEEE Transactions on Industrial Elec-tronics vol 58 no 2 pp 421ndash428 2011

[18] R Vargas U Ammann B Hudoffsky J Rodriguez and PWheeler ldquoPredictive torque control of an induction machinefed by a matrix converter with reactive input power controlrdquoIEEE Transactions on Power Electronics vol 25 no 6 pp 1426ndash1438 2010

[19] B S Riar T Geyer and U K Madawala ldquoModel predictivedirect current control of modular multilevel converters model-ing analysis and experimental evaluationrdquo IEEE Transactionson Power Electronics vol 30 no 1 pp 431ndash439 2015

[20] L Tarisciotti P Zanchetta A Watson J C Clare M Deganoand S Bifaretti ldquoModulated model predictive control for athree-phase active rectifierrdquo IEEE Transactions on IndustryApplications vol 51 no 2 pp 1610ndash1620 2015

[21] A Luo X Xu L Fang H Fang J Wu and C Wu ldquoFeedback-feedforward PI-type iterative learning control strategy forhybrid active power filter with injection circuitrdquo IEEE Trans-actions on Industrial Electronics vol 57 no 11 pp 3767ndash37792010

[22] Y F Wang and Y W Li ldquoThree-phase cascaded delayed signalcancellation PLL for fast selective harmonic detectionrdquo IEEETransactions on Industrial Electronics vol 60 no 4 pp 1452ndash1463 2013

[23] M Rivera C Rojas J Rodrıguez P Wheeler B Wu and JEspinoza ldquoPredictive current control with input filter resonancemitigation for a direct matrix converterrdquo IEEE Transactions onPower Electronics vol 26 no 10 pp 2794ndash2803 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


997888997888rarri1205721205730

997888997888rarri1205721205730

N

997888rarris

997888rarriL

997888rarriL

997888997888997888rarriLoad

997888997888997888rarriL1205721205730

997888997888997888rarriL1205721205730

997888997888997888rarrs1205721205730997888997888997888rarrs1205721205730

997888997888997888rarrs1205721205730

997888997888997888rarrs1205721205730 997888rarrs

q120572

q120573

q0

pT

plossminus

minus

minus

minus

HE(s)Eref

400V

i120572lowast

i120573lowast

i0lowast

+

+

+

+

+

+

Hbal(s)

P-Q theory InverseP-Q theory

Predictivecurrent

controller

Clarktransform

PLL and clarktransform

Clarktransform

IGBTdrivers

Unbalancednonlinear load

Active powerfilter

NDCreject

pT

i0bal E

vge1lowast

vge2lowast

vge6lowast

E

vge1

vge2

vge6

E1 E2

C1 C2

Q1 Q4

Q5 Q2

Q3 Q6

D1

D5 D2

D3 D6

D4

vTC

vSC

vRC

Lf

vdiff

sum

sumsum

sum

sum

Figure 1 Four-wire shunt active filter and its corresponding control block diagram

This paper presents the realization and experimentalverification of a trapezoidal predictive current controller fora four-wire shunt APF that improves the power quality ofunbalanced AC loads in contrast to the typical predictiveEuler control strategy The trapezoidal strategy relies itsoperation on a discrete trapezoidal linear approximation thatmore accurately determines the switching of the active filterfor the one-step ahead current sample such that three signifi-cant advantages are potentially exhibited first the trapezoidalpredictive controller slightly increments the processing timewithout affecting the switching of the power convertersecond in contrast to the typical Euler approximation usedin other works [6 7 13ndash20] the trapezoidalmethod generateslower AC current ripple and third the convergence time andload operating performance are wider than those obtainedusing the typical predictive control strategy which improvesthe reference current tracking and therefore the powerquality Experimental results obtainedwith a 2 kVAprototypeare presented demonstrating that the trapezoidal predictivecontrol may accurately compensate the currents drawn byan unbalanced nonlinear load under static and dynamicconditions

2 Four-Wire Shunt Active Filter

21 Circuit Description The four-wire shunt APF is con-nected in parallel to the unbalanced nonlinear load as shownat the right-hand side of Figure 1 which consists of a splitDC link formed by 119862

1and 119862

2which refer to the AC supply

neutral node119873 to provide a path tomitigate a commonmodecurrent a typical three-phase current-feed active converterformed by transistors 119876

1to 1198766and diodes 119863

1ndash1198636 and three

line filter inductors 119871119891used to generate the filter current

vector 997888rarr119894119871

= [119894119871119877 119894119871119878

119894119871119879]119879 by the difference between the

supply and converter voltage vectors 997888rarrV119904= [V119877119873 V

119878119873V119879119873]119879

and 997888rarrV119862

= [V119877119862119873 V119878119862119873

V119879119862119873]119879 thereby obtaining virtual

sinusoidal supply currents 997888rarr119894119904= [119894119878119877 119894

119878119878119894119878119879]119879

22 Principle of Operation of the Active Filter The principleof operation of the APF of Figure 1 may be described usingthe control block diagram presented at the left-hand side ofFigure 1 An instantaneous active and reactive power theoryP-Q theory block in Figure 1 [12] is used to obtain an effectivecalculation of the reference currents that the APF may injectto the supply to instantaneously mitigate the reactive anddistorted power components drawn by the nonlinear loadand balance the active power per phase The P-Q theory usesthe Clarke transformation of the supply voltage and loadcurrent as shown in

V120572120573120579

=[[[

[

V0

V120572

V120573

]]]

]

= radic2

3

[[[[[[[[

[

1

radic2

1

radic2

1

radic2

1 minus1

2minus1

2

0radic3

2minusradic3

2

]]]]]]]]

]

[[

[

V119877119873

V119878119873

V119879119873

]]

]


119894120572120573120579

=[[[

[

1198940

119894120572

119894120573

]]]

]

= radic2

3

[[[[[[[[

[

1

radic2

1

radic2

1

radic2

1 minus1

2minus1

2

0radic3

2minusradic3

2

]]]]]]]]

]

[[

[

119894Load119877

119894Load119878

119894Load119879

]]

]

(1)


119901119879

=997888997888rarrV1205721205730

sdot997888997888rarr1198941205721205730

= V120572119894120572+ V120573119894120573+ V01198940 (2)

119902 =997888997888rarrV1205721205730

times997888997888rarr1198941205721205730

=[[

[

119902120572

119902120573

119902120579

]]

]

=

[[[[[[[[[[[

[

[

V120573

V0

119894120573

1198940

]

[V120579

V120572

1198940

119894120572

]

[

V120572

V120573

119894120572

119894120573

]

]]]]]]]]]]]

]

(3)



120572 119902120573 and 119902



[[[[[

[

119901119879

119902120572

119902120573

1199020

]]]]]

]

=

[[[[[

[

V120572

V120573

V0

0 minusV0

V120573

V0

0 minusV120572

minusV120573

V120572

0

]]]]]

]

[[

[

119894120572

119894120573

1198940

]]

]

(4)



119879


[[

[



1198940reflowast

]]

]

=1

V2120572120573120579

[[[

[

V120572

0 V120579

minusV120573

V120573

minusV120579

0 V120572

V120579

V120573

minusV120572

0

]]]

]

[[[[[

[

119902120572

lowast

119902120573

lowast

1199020

lowast

]]]]]

]

(5)




119864(119904)





997888997888997888rarrV1198711205721205730

= 119871119889997888997888997888rarr1198941198711205721205730

119889119905

=997888997888997888rarrV1199041205721205730

minus997888997888997888rarrV1198881205721205730

(6)


shown as follows

997888997888997888rarr1198941198711205721205730

(1199051) =

997888997888997888rarr1198941198711205721205730

(1199051+ 119896119879) + int

1199051+(119896+1)119879

1199051+119896119879

997888997888997888rarrV1198711205721205730

(119905) 119889119905 (7)


997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

2119871[997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896

+997888997888997888rarrV1199041205721205730119896+1

minus997888997888997888rarrV1198881205721205730119896+1

]

(8)


tt

1 + kT t1 + (k + 1)T

997888997888997888997888rarrL1205721205730(t1 + kT)

997888997888997888997888rarrL1205721205730(t1 + (k + 1)T)


997888997888997888rarrc1205721205730(t)



1199041205721205730119896asymp

997888997888997888rarrV1199041205721205730119896+1


997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

2119871[2

997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (9)


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 since 997888997888997888rarrV


1198881205721205730

1 to997888997888997888rarrV1198881205721205730


0 and 997888997888997888rarrV1198881205721205730


1198941198711205721205730

0

119896+1to

997888997888997888rarr1198941198711205721205730

7


1198711205721205730119896


1198941198711205721205730

lowast

119896


1198941198711198901205721205730119896


1198941198711205721205730

lowast

119896 such that997888997888997888997888rarr119894

1198711198901205721205730119896may


997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896=

997888997888997888rarr1198941198711205721205730

lowast

119896minus

997888997888997888rarr1198941198711205721205730

0minus7

119896+1 (10)


0minus7

119896 997888997888997888997888rarr1198941198711198901205721205730

0minus7


1003817100381710038171003817100381710038171003817

997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896

1003817100381710038171003817100381710038171003817

2

=10038161003816100381610038161003816119894lowast

119871120572119896minus 1198940minus7

119871120572 119896+1

10038161003816100381610038161003816

2

+100381610038161003816100381610038161003816119894lowast

119871120573119896minus 1198940minus7

119871120573 119896+1

100381610038161003816100381610038161003816

2

+10038161003816100381610038161003816119894lowast

1198710119896minus 1198940minus7

1198710 119896+1

10038161003816100381610038161003816

2

(11)


0minus7







119904is processed





997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

119871[997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (12)







1198762

1198763

1198764

1198765

1198766

119881120572E 119881

120573E 119881

0E

0 1 0 1 0 1 997888997888997888rarrV1198881205721205730

0

0 0 minus1

2

1 1 0 0 0 1 997888997888997888rarrV1198881205721205730

1

radic2

30 minus

1

6

1 1 1 0 0 0 997888997888997888rarrV1198881205721205730

2 radic6

6

1

radic2

1

6

0 1 1 1 0 0 997888997888997888rarrV1198881205721205730

3

minusradic6

6

1

radic2minus1

6

0 0 1 1 1 0 997888997888997888rarrV1198881205721205730

4

minusradic2

30

1

6

0 0 0 1 1 1 997888997888997888rarrV1198881205721205730

5

minusradic6

6

minus1

radic2minus1

6

1 0 0 0 1 1 997888997888997888rarrV1198881205721205730

6 radic6

6

minus1

radic2

1

6

1 0 1 0 1 0 997888997888997888rarrV1198881205721205730

7

0 01

2




119878119873127V 60Hz


119878216 kHz




119864(119904) 3000119904 + 45e4

1199042 + 1000 minus 671e minus 8


119904



119894119878119878 and 119894


119877119873 obtained



119871119877




iL120573

997888997888997888rarriL1205721205730

3

k+1997888997888997888rarriL1205721205730

2

k+1

997888997888997888rarriL1205721205730

4

k+1

997888997888997888rarriL1205721205730

1

k+1

997888997888997888rarriL1205721205730

5

k+1

997888997888997888rarriL1205721205730

6

k+1


iL1205721205730k


997888997888997888rarriL1205721205730

07

k+1

997888997888997888rarriL1205721205730

lowast

k

iL120572

iL0


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 around 997888997888997888rarr

1198941198711205721205730119896

and 997888997888997888rarr1198941198711205721205730

lowast




119894119878119878 and 119894








119894119878119878 and 119894







119877119878THD 119868

119878119878THD and 119868




Start


Clark transform





Combination = j

No

No

Yes

Yes




997888997888997888997888rarriLoadk E1k E2k


0ndash7

k+1

997888997888997888997888rarriLe1205721205730

j

klt997888997888997888997888rarriLe1205721205730

jminus1

k

997888997888997888997888997888997888997888rarriLe1205721205730min


=997888997888997888997888rarriLe1205721205730

j

k


for next sampling

j = 7

997888997888997888997888rarriLe1205721205730

j

kEvaluation of







N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)




C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119894120572120573120579

=[[[

[

1198940

119894120572

119894120573

]]]

]

= radic2

3

[[[[[[[[

[

1

radic2

1

radic2

1

radic2

1 minus1

2minus1

2

0radic3

2minusradic3

2

]]]]]]]]

]

[[

[

119894Load119877

119894Load119878

119894Load119879

]]

]

(1)


119901119879

=997888997888rarrV1205721205730

sdot997888997888rarr1198941205721205730

= V120572119894120572+ V120573119894120573+ V01198940 (2)

119902 =997888997888rarrV1205721205730

times997888997888rarr1198941205721205730

=[[

[

119902120572

119902120573

119902120579

]]

]

=

[[[[[[[[[[[

[

[

V120573

V0

119894120573

1198940

]

[V120579

V120572

1198940

119894120572

]

[

V120572

V120573

119894120572

119894120573

]

]]]]]]]]]]]

]

(3)



120572 119902120573 and 119902



[[[[[

[

119901119879

119902120572

119902120573

1199020

]]]]]

]

=

[[[[[

[

V120572

V120573

V0

0 minusV0

V120573

V0

0 minusV120572

minusV120573

V120572

0

]]]]]

]

[[

[

119894120572

119894120573

1198940

]]

]

(4)



119879


[[

[



1198940reflowast

]]

]

=1

V2120572120573120579

[[[

[

V120572

0 V120579

minusV120573

V120573

minusV120579

0 V120572

V120579

V120573

minusV120572

0

]]]

]

[[[[[

[

119902120572

lowast

119902120573

lowast

1199020

lowast

]]]]]

]

(5)




119864(119904)





997888997888997888rarrV1198711205721205730

= 119871119889997888997888997888rarr1198941198711205721205730

119889119905

=997888997888997888rarrV1199041205721205730

minus997888997888997888rarrV1198881205721205730

(6)


shown as follows

997888997888997888rarr1198941198711205721205730

(1199051) =

997888997888997888rarr1198941198711205721205730

(1199051+ 119896119879) + int

1199051+(119896+1)119879

1199051+119896119879

997888997888997888rarrV1198711205721205730

(119905) 119889119905 (7)


997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

2119871[997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896

+997888997888997888rarrV1199041205721205730119896+1

minus997888997888997888rarrV1198881205721205730119896+1

]

(8)


tt

1 + kT t1 + (k + 1)T

997888997888997888997888rarrL1205721205730(t1 + kT)

997888997888997888997888rarrL1205721205730(t1 + (k + 1)T)


997888997888997888rarrc1205721205730(t)



1199041205721205730119896asymp

997888997888997888rarrV1199041205721205730119896+1


997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

2119871[2

997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (9)


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 since 997888997888997888rarrV


1198881205721205730

1 to997888997888997888rarrV1198881205721205730


0 and 997888997888997888rarrV1198881205721205730


1198941198711205721205730

0

119896+1to

997888997888997888rarr1198941198711205721205730

7


1198711205721205730119896


1198941198711205721205730

lowast

119896


1198941198711198901205721205730119896


1198941198711205721205730

lowast

119896 such that997888997888997888997888rarr119894

1198711198901205721205730119896may


997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896=

997888997888997888rarr1198941198711205721205730

lowast

119896minus

997888997888997888rarr1198941198711205721205730

0minus7

119896+1 (10)


0minus7

119896 997888997888997888997888rarr1198941198711198901205721205730

0minus7


1003817100381710038171003817100381710038171003817

997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896

1003817100381710038171003817100381710038171003817

2

=10038161003816100381610038161003816119894lowast

119871120572119896minus 1198940minus7

119871120572 119896+1

10038161003816100381610038161003816

2

+100381610038161003816100381610038161003816119894lowast

119871120573119896minus 1198940minus7

119871120573 119896+1

100381610038161003816100381610038161003816

2

+10038161003816100381610038161003816119894lowast

1198710119896minus 1198940minus7

1198710 119896+1

10038161003816100381610038161003816

2

(11)


0minus7







119904is processed





997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

119871[997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (12)







1198762

1198763

1198764

1198765

1198766

119881120572E 119881

120573E 119881

0E

0 1 0 1 0 1 997888997888997888rarrV1198881205721205730

0

0 0 minus1

2

1 1 0 0 0 1 997888997888997888rarrV1198881205721205730

1

radic2

30 minus

1

6

1 1 1 0 0 0 997888997888997888rarrV1198881205721205730

2 radic6

6

1

radic2

1

6

0 1 1 1 0 0 997888997888997888rarrV1198881205721205730

3

minusradic6

6

1

radic2minus1

6

0 0 1 1 1 0 997888997888997888rarrV1198881205721205730

4

minusradic2

30

1

6

0 0 0 1 1 1 997888997888997888rarrV1198881205721205730

5

minusradic6

6

minus1

radic2minus1

6

1 0 0 0 1 1 997888997888997888rarrV1198881205721205730

6 radic6

6

minus1

radic2

1

6

1 0 1 0 1 0 997888997888997888rarrV1198881205721205730

7

0 01

2




119878119873127V 60Hz


119878216 kHz




119864(119904) 3000119904 + 45e4

1199042 + 1000 minus 671e minus 8


119904



119894119878119878 and 119894


119877119873 obtained



119871119877




iL120573

997888997888997888rarriL1205721205730

3

k+1997888997888997888rarriL1205721205730

2

k+1

997888997888997888rarriL1205721205730

4

k+1

997888997888997888rarriL1205721205730

1

k+1

997888997888997888rarriL1205721205730

5

k+1

997888997888997888rarriL1205721205730

6

k+1


iL1205721205730k


997888997888997888rarriL1205721205730

07

k+1

997888997888997888rarriL1205721205730

lowast

k

iL120572

iL0


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 around 997888997888997888rarr

1198941198711205721205730119896

and 997888997888997888rarr1198941198711205721205730

lowast




119894119878119878 and 119894








119894119878119878 and 119894







119877119878THD 119868

119878119878THD and 119868




Start


Clark transform





Combination = j

No

No

Yes

Yes




997888997888997888997888rarriLoadk E1k E2k


0ndash7

k+1

997888997888997888997888rarriLe1205721205730

j

klt997888997888997888997888rarriLe1205721205730

jminus1

k

997888997888997888997888997888997888997888rarriLe1205721205730min


=997888997888997888997888rarriLe1205721205730

j

k


for next sampling

j = 7

997888997888997888997888rarriLe1205721205730

j

kEvaluation of







N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)




C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










tt

1 + kT t1 + (k + 1)T

997888997888997888997888rarrL1205721205730(t1 + kT)

997888997888997888997888rarrL1205721205730(t1 + (k + 1)T)


997888997888997888rarrc1205721205730(t)



1199041205721205730119896asymp

997888997888997888rarrV1199041205721205730119896+1


997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

2119871[2

997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (9)


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 since 997888997888997888rarrV


1198881205721205730

1 to997888997888997888rarrV1198881205721205730


0 and 997888997888997888rarrV1198881205721205730


1198941198711205721205730

0

119896+1to

997888997888997888rarr1198941198711205721205730

7


1198711205721205730119896


1198941198711205721205730

lowast

119896


1198941198711198901205721205730119896


1198941198711205721205730

lowast

119896 such that997888997888997888997888rarr119894

1198711198901205721205730119896may


997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896=

997888997888997888rarr1198941198711205721205730

lowast

119896minus

997888997888997888rarr1198941198711205721205730

0minus7

119896+1 (10)


0minus7

119896 997888997888997888997888rarr1198941198711198901205721205730

0minus7


1003817100381710038171003817100381710038171003817

997888997888997888997888rarr1198941198711198901205721205730

0minus7

119896

1003817100381710038171003817100381710038171003817

2

=10038161003816100381610038161003816119894lowast

119871120572119896minus 1198940minus7

119871120572 119896+1

10038161003816100381610038161003816

2

+100381610038161003816100381610038161003816119894lowast

119871120573119896minus 1198940minus7

119871120573 119896+1

100381610038161003816100381610038161003816

2

+10038161003816100381610038161003816119894lowast

1198710119896minus 1198940minus7

1198710 119896+1

10038161003816100381610038161003816

2

(11)


0minus7







119904is processed





997888997888997888rarr1198941198711205721205730119896+1

cong997888997888997888rarr1198941198711205721205730119896

+119879119904

119871[997888997888997888rarrV1199041205721205730119896

minus997888997888997888rarrV1198881205721205730119896+1

] (12)







1198762

1198763

1198764

1198765

1198766

119881120572E 119881

120573E 119881

0E

0 1 0 1 0 1 997888997888997888rarrV1198881205721205730

0

0 0 minus1

2

1 1 0 0 0 1 997888997888997888rarrV1198881205721205730

1

radic2

30 minus

1

6

1 1 1 0 0 0 997888997888997888rarrV1198881205721205730

2 radic6

6

1

radic2

1

6

0 1 1 1 0 0 997888997888997888rarrV1198881205721205730

3

minusradic6

6

1

radic2minus1

6

0 0 1 1 1 0 997888997888997888rarrV1198881205721205730

4

minusradic2

30

1

6

0 0 0 1 1 1 997888997888997888rarrV1198881205721205730

5

minusradic6

6

minus1

radic2minus1

6

1 0 0 0 1 1 997888997888997888rarrV1198881205721205730

6 radic6

6

minus1

radic2

1

6

1 0 1 0 1 0 997888997888997888rarrV1198881205721205730

7

0 01

2




119878119873127V 60Hz


119878216 kHz




119864(119904) 3000119904 + 45e4

1199042 + 1000 minus 671e minus 8


119904



119894119878119878 and 119894


119877119873 obtained



119871119877




iL120573

997888997888997888rarriL1205721205730

3

k+1997888997888997888rarriL1205721205730

2

k+1

997888997888997888rarriL1205721205730

4

k+1

997888997888997888rarriL1205721205730

1

k+1

997888997888997888rarriL1205721205730

5

k+1

997888997888997888rarriL1205721205730

6

k+1


iL1205721205730k


997888997888997888rarriL1205721205730

07

k+1

997888997888997888rarriL1205721205730

lowast

k

iL120572

iL0


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 around 997888997888997888rarr

1198941198711205721205730119896

and 997888997888997888rarr1198941198711205721205730

lowast




119894119878119878 and 119894








119894119878119878 and 119894







119877119878THD 119868

119878119878THD and 119868




Start


Clark transform





Combination = j

No

No

Yes

Yes




997888997888997888997888rarriLoadk E1k E2k


0ndash7

k+1

997888997888997888997888rarriLe1205721205730

j

klt997888997888997888997888rarriLe1205721205730

jminus1

k

997888997888997888997888997888997888997888rarriLe1205721205730min


=997888997888997888997888rarriLe1205721205730

j

k


for next sampling

j = 7

997888997888997888997888rarriLe1205721205730

j

kEvaluation of







N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)




C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1198762

1198763

1198764

1198765

1198766

119881120572E 119881

120573E 119881

0E

0 1 0 1 0 1 997888997888997888rarrV1198881205721205730

0

0 0 minus1

2

1 1 0 0 0 1 997888997888997888rarrV1198881205721205730

1

radic2

30 minus

1

6

1 1 1 0 0 0 997888997888997888rarrV1198881205721205730

2 radic6

6

1

radic2

1

6

0 1 1 1 0 0 997888997888997888rarrV1198881205721205730

3

minusradic6

6

1

radic2minus1

6

0 0 1 1 1 0 997888997888997888rarrV1198881205721205730

4

minusradic2

30

1

6

0 0 0 1 1 1 997888997888997888rarrV1198881205721205730

5

minusradic6

6

minus1

radic2minus1

6

1 0 0 0 1 1 997888997888997888rarrV1198881205721205730

6 radic6

6

minus1

radic2

1

6

1 0 1 0 1 0 997888997888997888rarrV1198881205721205730

7

0 01

2




119878119873127V 60Hz


119878216 kHz




119864(119904) 3000119904 + 45e4

1199042 + 1000 minus 671e minus 8


119904



119894119878119878 and 119894


119877119873 obtained



119871119877




iL120573

997888997888997888rarriL1205721205730

3

k+1997888997888997888rarriL1205721205730

2

k+1

997888997888997888rarriL1205721205730

4

k+1

997888997888997888rarriL1205721205730

1

k+1

997888997888997888rarriL1205721205730

5

k+1

997888997888997888rarriL1205721205730

6

k+1


iL1205721205730k


997888997888997888rarriL1205721205730

07

k+1

997888997888997888rarriL1205721205730

lowast

k

iL120572

iL0


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 around 997888997888997888rarr

1198941198711205721205730119896

and 997888997888997888rarr1198941198711205721205730

lowast




119894119878119878 and 119894








119894119878119878 and 119894







119877119878THD 119868

119878119878THD and 119868




Start


Clark transform





Combination = j

No

No

Yes

Yes




997888997888997888997888rarriLoadk E1k E2k


0ndash7

k+1

997888997888997888997888rarriLe1205721205730

j

klt997888997888997888997888rarriLe1205721205730

jminus1

k

997888997888997888997888997888997888997888rarriLe1205721205730min


=997888997888997888997888rarriLe1205721205730

j

k


for next sampling

j = 7

997888997888997888997888rarriLe1205721205730

j

kEvaluation of







N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)




C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










iL120573

997888997888997888rarriL1205721205730

3

k+1997888997888997888rarriL1205721205730

2

k+1

997888997888997888rarriL1205721205730

4

k+1

997888997888997888rarriL1205721205730

1

k+1

997888997888997888rarriL1205721205730

5

k+1

997888997888997888rarriL1205721205730

6

k+1


iL1205721205730k


997888997888997888rarriL1205721205730

07

k+1

997888997888997888rarriL1205721205730

lowast

k

iL120572

iL0


0

119896+1to 997888997888997888rarr

1198941198711205721205730

7

119896+1 around 997888997888997888rarr

1198941198711205721205730119896

and 997888997888997888rarr1198941198711205721205730

lowast




119894119878119878 and 119894








119894119878119878 and 119894







119877119878THD 119868

119878119878THD and 119868




Start


Clark transform





Combination = j

No

No

Yes

Yes




997888997888997888997888rarriLoadk E1k E2k


0ndash7

k+1

997888997888997888997888rarriLe1205721205730

j

klt997888997888997888997888rarriLe1205721205730

jminus1

k

997888997888997888997888997888997888997888rarriLe1205721205730min


=997888997888997888997888rarriLe1205721205730

j

k


for next sampling

j = 7

997888997888997888997888rarriLe1205721205730

j

kEvaluation of







N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)




C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start


Clark transform





Combination = j

No

No

Yes

Yes




997888997888997888997888rarriLoadk E1k E2k


0ndash7

k+1

997888997888997888997888rarriLe1205721205730

j

klt997888997888997888997888rarriLe1205721205730

jminus1

k

997888997888997888997888997888997888997888rarriLe1205721205730min


=997888997888997888997888rarriLe1205721205730

j

k


for next sampling

j = 7

997888997888997888997888rarriLe1205721205730

j

kEvaluation of







N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)




C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadS

iLoadT

100mH

10mF

C R

50ohms

(a)

N

Active power filter

vRN vSN vTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

100mH

10mF

50mH

C R

50ohms

100 ohms

75ohms

D1

D2

D3

D1 D3

D4

D2D4

10mF

(b)

L

IM

N

Active power filter

vRN

VRN

vSN

VSN

vTN

VTN

iLR

iSR

iLS iLT

iSSiST

iLoadR

iLoadSiLoadT

CC

50ohms

D1Q1 Q3 Q5

Q2Q6Q4 D2

D3 D5

D4 D6

(c)




C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











C3


Tbase minus595ms


Trigger C1 DC


BwL DC


1M


C1

C2

C4

(a)

C3



160MS 100MSs



C1

C2

C4

(b)


Timebase 78ms


C2 HFR

158V

Trigger

StopEdge Positive


C1

C2

C4

(c)

C1

C2

C4

Timebase 780ms


C2 HFR

158V


Trigger

StopEdge Positive


(d)


(yellow) 119894119878119877

(green) 119894119878119878(red) and 119894


119877119873(yellow) 119894

119871119877(red) 119894Load119877 (green)

and 119894119878119877


(blue) and 119894119878119877


119871119877



5 Conclusions






100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












100V

Tbase


StopEdge Positive



C3

C1

C2



(a)

minus1600ms200msdiv

500MS 25MSs

C3 HFR30V

TriggerTbase

StopEdge Positive




C3

C1

C2



(b)


(blue) 119894119878119878(yellow) 119894

119878119879(red) and 119894



119894119878119878(yellow) 119894

119878119879(green) and 119894



00102030405060708091

0102030405060708090

100

Bala

nced

load

of F

igur

e5(

a) w

ithou

t APF

Bala

nced

load

of F

igur

e5(

a) w

ith A

PF an

d th

eEu

ler m

etho

dBa

lanc

ed lo

ad o

f Fig

ure

5(a)

with

APF

and

the

trap

ezoi

dal m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

b) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Unb

alan

ced

load

of F

igur

e5(

c) w

ithou

t APF

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

eEu

ler m

etho

d

Unb

alan

ced

load

of F

igur

e5(

c) w

ith A

PF an

d th

etr

apez

oida

l met

hod

Supp

ly cu

rren

t TH

D (

)


Power factoriSR THD

iSS THDiST THD








Acknowledgments



References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Evaluation of a Trapezoidal Predictive ...

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Transcript of Research Article Evaluation of a Trapezoidal Predictive ...