Research Article Active Power Filter DC Bus Voltage ...downloads.hindawi.com/journals/jam/2014/835720.pdf · balance principle and a DC bus voltage piecewise reaching law variable

Research ArticleActive Power Filter DC Bus Voltage Piecewise ReachingLaw Variable Structure Control

Baolian Liu Zujun Ding Huanyu Zhao and Defei Jin

Department of Electronic and Electrical Engineering Huaiyin Institute of Technology Huaian Jiangsu 223003 China

Correspondence should be addressed to Zujun Ding dzj king263net

Received 21 July 2014 Accepted 14 August 2014 Published 2 September 2014

Academic Editor Zhiguang Feng

Copyright copy 2014 Baolian Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The DC bus voltage stability control is one key technology to ensure that Active Power Filter (APF) operates stably The externaldisturbances such as power grid and load fluctuation and the system parameters changing may affect the stability of APF DCbus voltage and the normal operation of APF The mathematical model of DC bus voltage is established according to powerbalance principle and a DC bus voltage piecewise reaching law variable structure control algorithm is proposed to solve the aboveproblem and the designmethod is givenThe simulation and experiment results proved that the proposed variable structure controlalgorithm can eliminate the chattering problem existing in traditional variable structure control effectively is insensitive to systemdisturbance and has good robustness and fast dynamic response speed and stable DC bus voltage with small fluctuationThe aboveadvantages ensure the compensation effect of APF

1 Introduction

Recently power electronic technology has been widely devel-oped and applied A lot of harmonic and reactive currentsinject into the power grid as the power electronic devices andnonlinear loads used in industry This can cause partial par-allel resonance and series resonance in electric power systempower grid and electrical equipment problems [1 2] APFis a new power electronic device designed for suppressingharmonic current It can compensate each harmonic currentand reactive power [3]

The APF power circuit usually consists of voltage sourceinverter and the drive circuit It is very important to maintainthe stability of DC side capacitor voltage for the wholecontrol system Low DC side capacitor voltage will reducethe compensation precision of APF Conversely highDC sidecapacitor voltagewillmake the interference harmonic currentof APF increaseTherefore controlling the DC side voltage ofinverter andmaintaining it stable have important significancefor the APF harmonic current compensation effect [4ndash8]

However when the device actually operate the powerconsumption of switching devices loading and unloading

nonlinear load process energy fluctuation in the DC sidecaused by AC side voltage fluctuation and other factorswill make the power grid and DC side capacitor exchangeactive power This will lead to bus voltage fluctuation [9]Nonideal power source voltage will lead to the rising of DCside voltage influence the compensation performance andeven damage the security of the system [10] The DC sidevoltage fluctuation is even larger when the harmonic currentand reactive power are compensated meanwhile This willseriously affect the compensation accuracy [11]

Generally the normal APF DC bus voltage controlmethods are PI control and fuzzy control Because of havingstrong robustness and less depending on the accurate modelthe fuzzy control has attracted extensive application [10 11]In [12] the authors proposed a fuzzy controller to controlthe DC side voltage and illustrated the fuzzy control rules indetail according to the error and change rate of error But thecontrol rules can not be adjusted and the robustness is lim-ited In [13] the fuzzy and PI controller were combined andthe fuzzy controller was used to automatically adjust the PIcontroller parameters The stable and dynamic performancewere improved to a certain extent But the control effect is

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 835720 8 pageshttpdxdoiorg1011552014835720

2 Journal of Applied Mathematics

Three-phase

nonlinear load

usa

usb

usc

1

udc

isa

isb

isc

L

L

L

2 3ica

icb

icc

iiLa

iLb

iLc

RL

RL

RL

Figure 1 Structure diagram of APF converter main circuit

normally seen from the experiment results In [14] the fuzzyinternal model controller was used to control the DC busvoltage but the controller structure is complex

Variable structure control has the characteristics of lowdependence on mathematical model of the controlled objectinsensitive to parameter variations and external disturbances[15 16] Yufeng et al [17] combined adaptive control andvariable structure control to suppress the harmonics andimprove power factor [18] And a DC bus voltage variablestructure controller is designed in this paper based onmodeling the APF DC bus voltage Aimed at the chatteringproblemof traditional variable structure control reaching lawa piecewise reaching law is presented for variable structurecontroller in this paper to optimize the control strategyFinally the simulation and experiment results are given

2 The Analysis to Model APF DC BusVoltage Control

Figure 1 gives the topology structure of three-phase shuntAPF The main part is a two-level converter which connectwith power grid through filter inductor where 119877

119871is the

resistor of the filter inductorIn the converter the 1st 2nd and 3rd bridge are used to

produce compensation current 119894119888119886 119894119888119887 119894119888119888of phase A B and

C respectively The corresponding instruction current 119894lowast119888119886

119894lowast

119888119887 119894lowast119888119888produced by arithmetic circuit In ideal condition the

compensation current 119894119888119886 119894119888119887 119894119888119888which is produced according

to instruction current will offset the harmonic current andfundamental reactive currentThus the power grid current issine wave and three-phase symmetrical

In [19] the power consumption of a resistor whichparalleled with DC bus was used to replace the whole APFpower consumption rather than the real power consumptionof the filter inductor and power electronic devices The paperdid not consider the change of APF compensation currentcaused by power grid voltage and load disturbance which

will lead to the change of power consumption The powerconsumption of power electronic devices was considered in[20] but except the on state loss and the filter inductor copperloss

Assume that the fundamental active current increment ofphase A caused by DC bus voltage fluctuation is Δ119894

1198861 Let

119906119904119886= radic2119880 sin (120596119905)

Δ1198941199041198861

= radic2Δ1198681sin (120596119905)

(1)

then the instantaneous power change of phase A caused bythe fluctuation of DC bus voltage is

119901119886= 2Δ119868

1119880sin2 (120596119905) (2)

The total instantaneous power change of the whole systemcaused by the fluctuation of DC bus voltage is

119901out = 119901119886+ 119901119887+ 119901119888 (3)

Assume that the three-phase voltage and current are symmet-ric thus the total instantaneous power change of the wholesystem caused by the fluctuation of DC bus voltage can beexpressed

119901out = 3119880Δ1198681 (4)

The instantaneous power of APF absorbed frompower grid ismainly stored in the bus capacitor tomaintain the bus voltageand the power consumption of equal resistor of filter inductorand power electronic devices

Xie et al [21] denote that (1) the on state loss is pro-portional to forward voltage drop and conduction currentwhile the forward voltage drop will increase along withthe conduction current increasing (2) the switching lossincreases along with the DC bus voltage raise

For the conclusion (1) the on state loss of switchingdevices can be expressed by

119901on = 119877on (1198942

119888119886+ 1198942

119888119887+ 1198942

119888119888) (5)

where 119894119888119896(119896 = 119886 119887 119888) are the APF compensation currents As

the three-phase load is symmetric the 119894119888119896are symmetric so

there is

119901on = 3119877on1198682

119888 (6)

where 119877on is the equal resistor when switching devices ison and 119868

119888is the RMS value of APF compensation current

The real time sampling value is still used for calculation inpractical operation

For the conclusion (2) ignore the APF DC bus voltagefluctuation the switching loss can be expressed as [20]

119901119904= 120574119868119888 (7)

where 120574 is a constant coefficient The total instantaneouspower loss of the equal resistors of the filter inductors is

119901119877= 31198771198711198682

119888 (8)

Journal of Applied Mathematics 3

The instantaneous power absorbed by DC bus capacitor canbe expressed as

119901119889119888= 119906119889119888119894119888= 119906119889119888(119862

119889119906119889119888

119889119905) (9)

The amplitude change of 119906119889119888

is small and the power fluctu-ation of capacitor is mainly decided by the change rate ofcapacitor voltage at steady state But the amplitude change of119906119889119888is maybe larger when the system is disturbed so the 119906

119889119888in

(9) still use real time sample value The instantaneous powerof APF absorbed from power grid is expressed by

119901in = 119901119877+ 119876on + 119876119904 + 119901119889119888 (10)

According to the instantaneous power balance principle119901out = 119901in that is

3119880Δ1198681= 3119877eq119868

2

119888+ 120574119906119889119888+ 119906119889119888(119862

119889119906119889119888

119889119905) (11)

where 119877eq = 119877119871+ 119877on

3 Variable Structure Controller Design

31 Improved Reaching Law Design An exponential ratereaching law presented in [22] is as follows

119904119896+1

= (1 minus 120572119879) 119904119896 minus 120576119879 sgn (119904119896) (12)

where 0 lt 120572 lt 1 120576 gt 0119879 is the control period and 119896 and 119896+1denote the present and the next control period respectivelyLet

sgn (119904119896) =

119904119896

10038161003816100381610038161199041198961003816100381610038161003816

(13)

It follows from (12) and (13) that

119904119896+1

= [1 minus 120572119879 minus120576119879

10038161003816100381610038161199041198961003816100381610038161003816

] 119904119896 (14)

In order tomake the switching function decrease in exponen-tial that is the value of switching function 119904 at present controlperiod 119879 less than the last 119879 the following condition shouldbe satisfied

100381610038161003816100381610038161003816100381610038161003816

1 minus 120572119879 minus120576119879

10038161003816100381610038161199041198961003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

lt 1 (15)

Soloing (15) gives

10038161003816100381610038161199041198961003816100381610038161003816 gt

120576119879

(2 minus 120572119879) (16)

According to (16) we know that the system will decreasein exponential only when the switching function value isrelatively big enough The system becomes the switchingcontrol when the condition in (16) is not satisfied Thuswe can conclude that the sliding mode band of normalexponential reaching law is ribbon and its bandwidth is Δ

1=

120576119879(2 minus 120572119879) The system will reach a chattering near the

slidingmode surface in the process ofmoving to slidingmodesurface in switching band and this will not ensure the systemapproaches sliding mode surface finally while the high fre-quency chattering can excite the high frequency componentwhich is unconsidered when establishing the system modelThereby it may increase the burden of controller [23 24]

A power reaching law presented in [25] is as follows

119904119896+1

= (1 minus 120572119879) 119904119896 minus 12057311987910038161003816100381610038161199041198961003816100381610038161003816

120582minus1119904119896 (17)

It follows from (13) and (17) that

119904119896+1

= (1 minus 120572119879) 119904119896 minus 12057311987910038161003816100381610038161199041198961003816100381610038161003816

120582 sgn (119904119896) (18)

where 0 lt 120582 lt 1 120573 gt 0This is a variable exponential reaching law actually The

switching control part is big at the beginning and will makethe system approach sliding mode surface quickly But theamplitude of switching control is small when it is near thesliding mode surface Thus the system could converge tosliding mode surface finally And the system is stable atthe origin The switching band is sector This reaching lawnot only could keep the basic requirements of crossing overswitching surface step by step at quasi-sliding mode but alsocould effectively inhibit or weaken the chattering

A piecewise reaching law is proposed to design thevariable structure controller for DC bus voltage based on theabove two kinds of reaching law

119904119896+1

= (1 minus 120572119879) 119904119896 minus 120576119879 sgn (119904

119896)

10038161003816100381610038161199041198961003816100381610038161003816 gt Δ

(1 minus 120572119879) 119904119896 minus 12057311987910038161003816100381610038161199041198961003816100381610038161003816

120582 sgn (119904119896)

10038161003816100381610038161199041198961003816100381610038161003816 le Δ

(19)

where Δ is the boundary value of sliding mode band selectedfor the two reaching laws The quasi-sliding mode band ofreaching law in (18) is

119909119896|1003817100381710038171003817119904119896

1003817100381710038171003817 lt Δ2 Δ

2= (

120573119879

2 minus 120572119879)

1(1minus120582)

(20)

For the reaching law no 1 in (19) in order to make the slidingmode attenuate in exponential before entering the switchingband Δ

1 Δ ge Δ

1is needed Similarly for the reaching law no

2 in order to make the sliding mode attenuate in exponentialbefore entering the switching band Δ

2 Δ le Δ

2is needed

Based on the above analysis we can get Δ2ge Δ1

Taking 120582 = 05 and letting Δ2= Δ = Δ

1 then

(120573119879

2 minus 120572119879)

2

=120576119879

2 minus 120572119879 (21)

It can be deduced from (21) that

120573 = radic120576 (2 minus 120572119879)

119879 (22)

32 Design of APF DC Bus Voltage Variable Structure Con-troller Select the error of DC bus voltage and its integrationas state variables Let

1199091= int (119880

lowast

119862minus 119906119889119888) 119889119905 119909

2= 1= 119880lowast

119862minus 119906119889119888 (23)

4 Journal of Applied Mathematics

According to (11) the system state equation can be expressedas

[1

2

] = [0 1

0 0] [

1199091

1199092

] + [

[

0

minus3119880

119862119906119889119888

]

]

Δ1198681+ [

[

0

3119877eq

119862119906119889119888

1198682

119888+ 120574

]

]

(24)

It can be written in matrix form as

[1

2

] = 1198601015840[1199091

1199092

] + 1198611015840Δ1198681+ 1198631015840 (25)

where 1198601015840 1198611015840 are coefficient matrix and 1198631015840 is disturbance

quantity Discretize (25) one can get that

119909119896+1

= 119860119909119896+ 119861Δ119868

1119896+ 119863 (26)

where 119860 = 119868 + 1198791198601015840 119861 = 119879119861

1015840 119863 = 1198791198631015840 in which 119868 is unit

diagonal matrix Let

119904119896+1

= [1198881 1] 119909119896+1 = 119862119909119896+1

= 119862 [119860119909119896+ 119861Δ119868

1119896+ 119863] (27)

It follows from (17) and (27) that

Δ1198681119896=

(119862119861)minus1[119865 minus 120576119879 sgn (119904

119896)]

10038161003816100381610038161199041198961003816100381610038161003816 gt

120576119879

(2 minus 120572119879)

(119862119861)minus1[119865 minus 120573119879radic

10038161003816100381610038161199041198961003816100381610038161003816 sgn (119904119896)]

10038161003816100381610038161199041198961003816100381610038161003816 le

120576119879

(2 minus 120572119879)

(28)

where 119865 = minus119862119860119909119896minus 119862119863 + (1 minus 120572119879)119904

119896

4 Simulation and Experiment

41 Simulation In order to verify the effectiveness of the pro-posed piecewise reaching law variable structure controllerwe establish the simulation model in MATLABSIMULINKThe system parameters are as follows phase voltage is 220V(RMS) switching frequency is 10 KHz DC bus voltage is700V and filter inductor and capacitor are 5mHand 4700120583Frespectively Three-phase bridge diode rectifier connectedwith resistor is as the load

Figures 2 and 3 give the DC bus voltage response waveand the voltage and current wave of power grid phase AwhenAPF is with nonload operation From Figure 2 it can be seenthat the DC bus voltage rises to the setting value within twosine wave cycles and the response speed is better than thatof fuzzy controller But the overshoot is smaller than that ofthe PI controller The voltage fluctuation is very small afterstable operation All these proved that the performance of theproposed variable structure controller is good

It can be seen from Figure 3 that the charging currentof capacitor is big the reason is that the capacitor voltageis zero at the beginning and the big charging current canmake the capacitor voltage rises quickly then the dynamicresponse time is shortened The charging current becomesstable within two sine wave cycles which is consistent withthat of the DC bus voltage and is small after stable operationIts waveform is similar to the sine wave and its phase is almost

300200100

500600400(V

)

020 1006040 80

(ms)120

700800

140 160 180

Figure 2 Dynamic response curve of DC bus voltage (nonload)

50 100 150 200 250 300Time (ms)

0

50

100

minus50

usa10 isa

Figure 3 Waves of power grid voltage and current

the same as that of power grid voltage The active powergeneration frompower grid is used formaintaining the powerconsumption of APF itself

In order to verify the dynamic regulation performance ofAPF DC bus voltage variable structure controller in the eventof external disturbance the simulation research is carried outon the condition that the power grid voltage and load appearfluctuation

The DC bus voltage and power grid current responsecurves under the condition that the power grid voltage dumps10 and APF operates in stable state with load are given inFigure 4 It can be seen from Figure 4 that the bus voltagedroop is little and recovers to the setting value quicklythrough the regulation of control algorithm The bus voltagedroop caused by the power grid voltage droop increases thecharging current of capacitor At the same time the loadpower is decreased and the reason is the same But the valueof load current drop is larger than that of the charging currentincrement So the peak current of power grid is decreasedfrom 3875A to 3625A The power grid current transitionprocess is smooth

Figures 5 and 6 give the DC bus voltage the powergrid voltage and current response curves when the APFconnectswith load after power on inwhich the load suddenlyincreases to 2 times of the original at time 150ms

It can be seen from Figure 5 that the time of DC busvoltage getting to the setting value is less than 70ms Theresponse time is bigger than that time in Figure 2 because ofthe current generation fromAPF including the compensationharmonic and reactive currents The bus voltage is droppedwhen the load is suddenly increased at time 150ms but it isregulated to setting value by the variable structure controllerat about time 220ms The droop of bus voltage and theregulation time are larger than that of Figure 4 The reasonis that the value of load increment (100) is larger than thedroop (10) of power grid voltage

Journal of Applied Mathematics 5

40 20012080 160Time (ms)

0100200300400500600700800

(V)

(A)

(V)

3625A3875A50

minus50

400

200

minus400

minus200

278V311V

udc

Figure 4 Wave of DC bus voltage and power grid current (powergrid voltage dumps 10)

300200100

500600

400(V)

050 250150100 200

(ms)300

700800

Figure 5 Dynamic responsewave ofDCbus voltage (load changes)

It can be seen from Figure 6 that the phase current tracksthe phase voltage well at stable and dynamic operation Thepower grid current is sine wave and its phase is almost thesame as that of voltage The transition process is stable andthe measured power factor is 099

From the above simulation results and analysis we canconclude that the dynamic and static performance of theproposed variable structure controller are excellent The DCbus voltage is stable and thus ensures the current compensa-tion performance of APFThe problem discussed in the thirdparagraph of Section 1 is solved well

42 Experiment In order to verify the effect of the proposedcontrol algorithm the experiment operates on the existingAPF platformThe system uses TMS320F28335 as the controlcore to execute the control algorithmThe power grid voltageits frequency and the filter inductor are 220V (RMS) 50Hzand 5mH respectively As the actual power grid voltage isbigger than 220Vnormally theDC bus voltage is set as 750V

(V)

40 20012080Time (ms)

160 240 300

usa isa300

200

100

minus300

minus200

minus100

0

Figure 6 Waves of power grid voltage and current (load changes)

100

Vd

iv

5 sdiv

50msdiv

Figure 7 DC bus voltage dynamic response curve

not as 700V in simulation The load is the same as that ofsimulation (Figures 7ndash12 are based on this load)

The DC bus voltage response curve after power on isgiven in Figure 7 At the beginning the power grid chargesthe capacitor through a 51Ω current limiting resistor theAPF operated under uncontrolled rectification and the busvoltage rises rapidly to the stable state After a certain delaythe 51Ω resistor is shorted then the bus voltage rises slightlyThen the PWM boost voltage process is started and thecontrol period is 2ms It can be seen from Figure 7 that thetime of the DC bus voltage increasing from uncontrolledrectification to the last setting value 750V is about 50ms themeasured actual bus voltage is 745V and the fluctuation is5 V

Figure 8 gives the wave of power grid voltage and currentin which the APF is with no load and in the stable state Itcan be seen that the current is about 2A which is consistentwith the simulation result As the bus voltage is stable thecurrent fluctuation is small The current rippled around theinstruction current in a certain range the reason is that thecurrent is small and the switching frequency of the powerelectronic devices is limited But the current wave outlineis similar to sine wave The phase of current lags that ofthe voltage The reason is that there exists reactive powerexchanging between the filter inductor and the power grid

The waves of DC side voltage load current power gridcurrent and the output harmonic current of APF under

6 Journal of Applied Mathematics

udc (200Vdiv) usa (200Vdiv)isa (5Adiv)

udc (200Vdiv)usa (200Vdiv)

isa (5Adiv)

20msdiv

2msdivuudc ((200200Vdiv)Vdiv

uusa (2002002 Vdiv)Vdiv)disaa ((55Adiv)Ad vd 22msdiv

Figure 8 APF compensation current (nonload)

100

Vd

iv50

Ad

iv50

Ad

iv50

Ad

iv

Time (10msdiv)

Figure 9 Waves of DC bus voltage load current power gridcurrent and APF compensation current

load operation are given in Figure 9 The measured averageDC bus voltage is 740V and the fluctuation is 15 V Itsstable error is small From the current wave it can be seenthat the harmonic current components of load current arecompensated by APF The power grid current is sine waveand with no peak burr The harmonic current from APF hadno high frequency component The experiment result provesthat the compensation effect is good

Figures 10 and 11 give the phase A current spectrumbefore and after compensation respectively It can be seenfrom Figure 10 that the content of 3rd 5th 7th 9th 11th13th and 17th harmonic are big especially the 5th harmonicThe harmonic component after compensation is very smallThe THD value is 267 before compensation and is 60after compensation The THD is reduced a lot and thecompensation effect is obvious

In order to verify the anti-interference performance ofthe improved reaching law variable structure controller at theevent of load disturbance the load is suddenly increased to 2times of the original Figure 12 gives the responsewaves ofDCbus voltage and power grid current It can be seen from thewave that the variable structure controller can quickly makethe system reach a new stable state and the regulation time isno more than 30ms and the transition process is smooth

Figure 13 gives the power grid voltage and current wavesafter compensation in which the load is three-phase bridge

1 5 9 13 17 21 25 29 33 37 4541 49

25

5000009

Harmonics THD 267f K 45

U

THDDC

A A

Figure 10 Phase A current spectrum before compensation

25

5000007

Harmonics THD 60f K 28

U

1 5 9 13 17 21 25 29 33 37 4541 49THDDC

A A

Figure 11 Phase A current spectrum after compensation

diode rectifier connected with resistor-inductor load It canbe seen that the harmonic and reactive currents are alsocompensated well the power grid current is sine wave andits phase is the same as that of the power grid voltage

5 Conclusion

The APF DC bus voltage is very important for APF stableand reliable operation The DC bus voltage mathematicalmodel has been analyzed and established according to powerbalance A DC bus voltage piecewise reaching law variablestructure controller has been designed aimed at variousdisturbances The simulation and experiment results haveproved that the proposed control strategy has some goodperformances such as maintaining the DC bus voltagestable no chattering insensitiveness to external disturbancequick dynamic response and small fluctuation All of theseensured the steady amplitude sine wave little distortionsmooth transition process of the power grid current and thecompensation effect of APF

Journal of Applied Mathematics 7

u100

Vd

ivi50

Ad

iv

udc

isa

Time (10msdiv)

Figure 12 Waves of DC bus voltage and power grid current (loadchanges)

50

Ad

iv

usa isa

Time (10msdiv)

Figure 13 Waves of power grid voltage and current after compen-sation (resistor-inductor load)

Conflict of Interests

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

Acknowledgments

This research has been supported byNational Natural ScienceFoundation of China (61201410) Natural Science Fund forColleges and Universities in Jiangsu Province (11KJB470002)and Huaian Science and Technology Plan Project (SocialDevelopment HAS2011049) The authors are grateful to thereviewers for their valuable comments

References

[1] W Yong and W Shuyun ldquoReasearch on matlab simulation ofharmonic and reactive current detectionrdquo Electrical Measure-ment amp Instrumentation vol 42 pp 21ndash23 2005

[2] S Yin X Li H Gao and O Kaynak ldquoData-based techniquesfocused on modern industry an overviewrdquo IEEE Transactionson Industrial Electronics 2014

[3] Z-K Hu M-Y Hu W-H Gui C-H Yang and Z-M HeldquoNovel self-tuning predictive control method of shunt activepower filterrdquo Electric Machines and Control vol 14 no 3 pp18ndash30 2010

[4] T Zhong J Yingda Y Yongsheng and W Guo ldquoDirect-sidevoltage control of mid-point capacitor three phase four wireshunt active power filterrdquo Electrical Measurement amp Instrumen-tation vol 48 41 no 545 p 44 2011

[5] F Z Peng ldquoHarmonic sources and filtering approachesrdquo IEEEIndustry Applications Magazine vol 7 no 4 pp 18ndash25 2001

[6] S Yin S Ding X Xie and H Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[7] A Chandra B Singh B N Singh and K Al-Haddad ldquoAnimproved control algorithm of shunt active filter for voltageregulation harmonic elimination power-factor correction andbalancing of nonlinear loadsrdquo IEEE Transactions on PowerElectronics vol 15 no 3 pp 495ndash507 2000

[8] M Salo and H Tuusa ldquoA vector controlled current-sourcePWM rectifier with a novel current damping methodrdquo IEEETransactions on Power Electronics vol 15 no 3 pp 464ndash4702000

[9] P-F Chang J-L Zeng T Wang and X-M Chen ldquoResearchon DC-side voltage control methods of three-phase four-wireactive power filtersrdquo Automation of Electric Power Systems vol29 no 8 pp 75ndash78 2005

[10] D-G Liu A Luo and Z-K Shuai ldquoNew issues and solvingschemes for controlling the DC-side voltage of new injectiontype hybrid active power filterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 28 no 30 pp 27ndash34 2008

[11] C Yu L Huiyun and L Qionglin ldquoAn APF design with newDC voltage control strategyrdquo Electrical Applications vol 25 no7 pp 79ndash82 2006

[12] F Bourourou K khettab F Senani and S Guettouche ldquoActivepower filter DC voltage regulation with fuzzy logic controllerrdquoin Proceedings of the 4th International Conference on PowerEngineering Energy and Electrical Drives pp 87ndash91 IstanbulTurkey May 2013

[13] L Zhihua L Zhen and L Zhenbin ldquoSelf- adaptive fuzzy PIcontrol of DC voltage in active power filterrdquo Chinese Journal ofPower Sources vol 34 no 6 pp 582ndash585 2010

[14] L Kong X Zhang X Li and C Li ldquoResearch on DC-sidevoltage fuzzy internal model of shunt active power filterrdquoTransactions of China Electrotechnical Society vol 26 no 1 pp224ndash228 2011

[15] M Hamerlain T Youssef and K Bouyoucef ldquoReducing thechattering using the generalized variable structure controlapplied to a manipulator armrdquo in Proceedings of the CanadianConference on Electrical and Computer Egineering (CCECE 00)vol 2 pp 597ndash604 Halifax Canada May 2000

[16] K Huang W Wang and X Wang ldquoModeling and simulationof PWM rectifier based on sliding-mode controlrdquo Power SystemTechnology vol 33 no 8 pp 18ndash23 2009

[17] W Yufeng S Bao and Z Chengbo ldquoApplication research ofadaptive discrete sliding mode control for APFrdquo Electric Drivevol 42 20 no 8 p 23 2012

[18] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[19] Z Ding B Liu and Y Zhang ldquoOptimal control for DCside voltage of active power filter based on auto-disturbancerejection controlrdquo Power System Technology vol 37 no 7 pp2030ndash2034 2013

[20] Z Dong L V Zhengyu and C Guozhu ldquoCapacitor voltagecontrol of shunt active power filterrdquo Power Electronics vol 41no 10 pp 76ndash79 2007

[21] B Xie K Dai S Zhang and Y Kang ldquoOptimization control ofDC link voltage for shunt active power filterrdquo Proceedings of theChinese Society of Electrical Engineering vol 31 no 9 pp 23ndash292011

[22] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

8 Journal of Applied Mathematics

[23] O Xuwen and Y Huajie ldquoSliding mode control researchof PMSM based on variable exponential rate reaching lawrdquoMicromotors vol 44 no 90 pp 31ndash34 2011

[24] Y Mi W L Li and Y W Jing ldquoVariable structure control for adiscrete-time system based on power reaching lawrdquoControl andDecision vol 23 no 6 pp 643ndash646 2008

[25] C L Zhai and Z M Wu ldquoA variable structure control methodfor discrete time systemsrdquo Journal of Shanghai Jiaotong Univer-sity vol 34 no 5 pp 719ndash722 2000

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