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ContentslistsavailableatScienceDirect

Electric

Power

Systems

Research

jou rn a l h om ep a g e :w w w . e l s e v i e r . c o m / l oc a t e / e p s r

A

sliding-mode

voltage

regulator

for

salient

pole

synchronous

generator

R.L.A.

Ribeiro

,

C.M.S.

Neto,

F.B.

Costa,

T.O.A.

Rocha,

R.L.

Barreto

FederalUniversityofRioGrandedoNorte,CampusUnivesitárioLagoaNova,NatalRN,CEP:59.078-970,Brazil

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received7May2015

Receivedinrevisedform25July2015 Accepted29July2015

Availableonline28August2015 Keywords:

Automaticvoltageregulator Powersystemstabilizer Synchronousgenerator Sliding-modecontrol

a

b

s

t

r

a

c

t

Thispaperpresentsasliding-modecontrolstrategyforvoltageregulationofsynchronousgenerators

connectedtothepowergridthatcombinesslidingmodeandlinearcontrollaws.Thishybridcontrol

schemecontributespreventingsysteminstability,bysuppressingthelow-frequencyoscillationsarising

frompowergridfaultdisturbances.Inaddition,thisstrategydoesnotrequiretheestimationofsystem

parameters.Thiscontrolstrategywasdesignedtobeintroducedeasilyinaconventionalautomatic

volt-ageregulator(AVR)algorithm.Experimentalresultsobtainedwithalaboratorytestset-up,consistingof

thesynchronousgeneratorconnectedtothegridandsubjectedtoseverepowertransients,wasemployed

tovalidatetheproposedcontrolapproach.Comparisonswithaconventionalaswellaswithtwohybrid

controlschemesdemonstratetheadvantagesoftheproposedcontrolstrategy.

©2015ElsevierB.V.Allrightsreserved.

1. Introduction

Despitethenumerousworksproposedinthepastdecade,AVR appliedtosynchronousgeneratorsisstillconsideredan interest-ingproblem.ThemainobjectiveofanAVRsystemistoregulate thegeneratorterminalvoltage,whichimprovesthetransient per-formanceand ensuresthestability of thepowersystematany occurrenceof externaldisturbances orparametricvariation [1]. Typically,proportional-integral-derivative(PID)controllerswith fixedor variableparameters are usedin theAVR [2]. Different approachessuchaspredictivecontrolschemes[3,4]orLyapunov directmethod[5]canbeused.Ingeneral,themodelingadopted inthesecontrolschemesincludesthelinearizedmodelofa syn-chronousgeneratorconnectedtoapowergrid.Predictiveschemes, withfixedparameters,presentagoodtransientstabilityand inher-entrobustnesstotheparametervariations[6].Recently,control schemesthatdonotdependonthesystemmodelsuchasfuzzy logiccontrol[7,8]orneuralnetworks[9–11]havebeenproposed. However,thesecontrollersdonotensurethesystemclosed-loop stability.Itispossibletodesignacontrolschemethatensures sys-temstabilityandgoodtransientperformancebyusingasystem modelingcombinedwithfeedbackcontroltheory,likelinear opti-malcontrol [12],H feedbackcontrol[13]or nonlinearcontrol strategy[14].

∗ Correspondingauthor.Tel.:+558432153731. E-mailaddress:rlucio@ct.ufrn.br(R.L.A.Ribeiro).

AdaptivecontrolstrategieshavealsobeenproposedforAVRto ensuresystemstabilityandtosuppressthelow-frequency oscilla-tions.Intheself-tuningschemes,anadaptiveexternalloopisadded totheconventionalAVR,whichsystemparametersareestimated from the excitation command and generator voltage terminal

[12,15–17].Althoughthesetechniquesdifferontheirdesign pro-cedure,allofthemuseblack-boxdescriptioninthediscrete-time domainandhavecompleximplementations[18].Twosimplified approachesofadaptivestrategieshavebeenintroducedto over-comethis:AVRwithconstrainedleast-squaresalgorithm(CRLS)

[19]andasimpleadaptivecontrol(SAC)basedonthequadratic performanceindex.

Duetothelowsensitivitytoparametricvariationsand exter-naldisturbances,thesliding-modecontrol(SMC)[20,21]canbe alsoconsidered.Thesecontrollershavebeenproposedforsingle machineconnectedtoapowergrid[22]orformultimachine sys-tems[23].ThedrawbackoftheSMCreferstothechatteringdue totheunmodeledexciter dynamics[24].Anapproach basedon high-orderslidingmode(HOSM)algorithms[25] hasbeen pro-posedforreducingtheseeffects.Although,thisalgorithmstillhas asignificantcomplexityofimplementation.Hybridstructuresthat combineSMCandadaptivecontrolhavegainedconsiderable inter-est[26],becausetheyallytherobustnessofSMCtoparametric uncertaintiesandthegrowingofstabilitymarginduetothe adap-tivecontrol.Thisstructurehasbeenusedwithactivepowerfilter applications,implemented withtheassociation ofthe adaptive poleplacementcontrol(VS-APPC)[26]ormodelreferencescheme (VS-MRAC)[27].However,thesecontrolstructuresstillpresenta

http://dx.doi.org/10.1016/j.epsr.2015.07.016 0378-7796/©2015ElsevierB.V.Allrightsreserved.

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Chopper

1

Chopper 2

r

s

l

s

r

s

l

s

r

l

l

l

v

t

123

i

g

123

i

fd

i

a

e

fd

e

a

e

8

k

2 CP

r

sc

k

1

Fault Emulator

Synchronous

Generator

DC Motor

n

Power Grid

Usb

Optical

DSP

Sensors

Fig.1.Blockdiagramoftheexperimentaltestset-up.

considerablecomplexityofimplementation. In ordertoachieve robustness,hybridstrategiesthatcombineSMCand proportional-integral(PI)controllershavealsobeenintroduced[28].Inthese structures,thediscontinuouscontrolisapproximatedbyan adap-tivePIcontrolstructure,employedforminimizingthechattering oftheSMC.Themaindrawbackofthesestrategiesreferstothe persistenceofthechattering.However,thesestrategieshavethe advantageofnotrequiringthesystemparameterestimation.

Inthesamedirection,thispaperproposesanAVRforasalient polesynchronousgeneratorbasedonahybridcontrolstructure, composedoftheintegrationofSMCapproachand PIcontroller (SM-PI).Differentlyfromthestrategiespresented previously,in theproposedscheme,theSMCdeterminesthePIcontrollergains fromasuitablesliding-surface.Thisslidingsurfaceisobtainedfrom theerrorofgeneratorterminalvoltage.Thechatteringduetothe SMCisminimizedbyPIcontroller bandwidth.Besidesthe sim-plicityofimplementation,theSM-PIexhibitsexcellentrobustness toparametricuncertaintiesandimmunitytounmodeled distur-bances.Experimentalresults,obtainedfromlaboratorytestsetup consistingofthesynchronousgeneratorconnectedtothemains andsubjectedtoseveretransientsinthepowersystem,validate theproposedcontrolapproach.

2. Systemdescriptionandmodeling

Fig. 1 presents the block diagram of the laboratory-based powersystem.It comprises an electromechanicalcomposedby asynchronousgeneratorpulledbyaDCmotor.Thisgeneratoris connectedtoapowergridemulatorthatiscomposedoftwo trans-missionlinesconnectedtothepointofcommoncoupling(PCC), whichisconnectedtoathree-phaseRLload,equippedwith con-trolledby-passswitchesandexternalresistorstoemulatefaults. BothDCmotorarmatureandsynchronousgeneratorfieldare feed-ingthroughH-bridgebuckconverter.Theproposedcontrolstrategy aswellasPWMschemeswereimplementedina floating-point digitalsignalprocessor(DSP).

3. Voltagegeneratorcontrolsystem

Thestandardsalientpolesynchronousgeneratormodelis rep-resentedby a stator composedof three stator windings, and a rotorconsistedofonefieldandtwodamperwindings.According to[29],thesmallsignalmodelformachineterminalvoltage devia-tions

v

t(s)canberelatedtothechangesofgeneratorfieldvoltage

efd(s)andtheloadangleı(s)asfollows:



v

t(s)=Gvt(s)efd(s)+Gı(s)ı(s) (1)

K

3

K

6

1+ K

s

3 d0'

(

)

1+

s

d0'

(

a

'

)

1+

s

d0'

(

'

)

a

SM-PI

e

fd

G (s)

v

t

v

t*

+

+

+

-v

t’

v

t

Synchronous

Generator

with the Grid

ε

Fig.2. Blockdiagramoftheautomaticvoltageregulatorsystem.

with Gvt(s)= K3K6a(1+sd0 ) (1+sad0 )(1+sK3d0) (2) and G(s)=K5− K4d(1+scd0 ) (1+sad0 )(1+sK3d0 )+ K4qK6d (1+sbq0 ) (3)

ThetermsGvt(s)andGı(s)representedbyEq.(1)presentsa

sec-ondorderdynamicbehaviordeterminedbypoless1=−1/K3d0

ands2=−1/ad0 .Italsohasathirdorderdynamicbehavior

deter-mined by poles s1=−1/K3d0, s2=−1/ad0 and s3=−1/bq0 ,

respectively.Theblockdiagramoftheproposedcontrolsystemis presentedinFig.2.TheproposedSM-PIcontrolsthegenerator ter-minalvoltage,whichdeterminesthefieldvoltageefd(s)thatis synthesizedbyaPWMbuckconverter,omittedintheblock dia-gramofFig.2.Theeffectontheterminalvoltageduetotheload angledeviationGı(s)ı(s)isconsideredasadisturbancetobe

compensatedbytheSM-PIcontroller. 3.1. SM-PIcontroller

TheproposedcontrolschemeisarobustPIcontrollerinwhich itscontrollergainsarecalculatedbySMCstrategy.Theyare deter-minedbyswitchinglawsaccordingtoaslidingsurfaceobtained fromthecontrollooperroranditsderivative.Thecontroller sta-bility is verifiedby using theLyapunov stability theory. In the generatordynamicmodelofEq.(2),thezeroats=−1/d0

approx-imately cancels the pole at s=−1/ad0 , which adresses to a first-ordersystemgivenby[29]:

Gvt(s) ∼=



v

t(s) efd(s)=

bv

(3)

where 

v

t(s)=

v

t(s)+

v

tı(s), av=1/(K3d0 ), bv=aK6/(d0 )

and

v

tı(s)=Gı(s)ı.Thetransferfunctionoftheproposed

SM-PIcontrollerisgivenby: Gc(s)=

˜kps+ ˜ki

s (5)

where ˜kpand ˜kiarethecontrollergains.Theclosedloopdynamics

ofthegeneratorterminalvoltageisgivenby: 

v

t(s)= Gc(s)Gvt(s) 1+Gc(s)Gvt(s) 

v

t(s)+ Gı(s) 1+Gc(s)Gvt(s) ı(s) (6)

andtheerrorofgeneratorterminalvoltagecontrolloopis:

ε(s)=1 1

+Gc(s)Gvt(s)



v

t(s)− Gı(s)

1+Gc(s)Gvt(s)

ı(s) (7)

AsGc(s)istype-oneandGvt(s)andGı(s)aretype-zerothetwo

parcelsofsystemdescribebyEq.(7)aretype-one,whichensures zerosteady-stateerrorforsmallstepvoltagereferenceandsmall steploadangledisturbance,whichisreasonableforthis applica-tion,andthestabilityofthegeneratorterminalvoltageisassured. Duringthetransientstate,thegain ˜kpswitchesbetweenkavp −

2k−p andkavp +2kp+.Uponreachingthesteadystate, ˜kpiskept

con-stantatkav

p .Asimilarstatementappliesto ˜ki.

Definingaslidingsurfacedescribedby

=cvε+ ˙ε (8)

whereε=

v

t−

v

t, ˙ε isitsderivative,and cisapositive

con-stant.Thederivative ˙εiscomputedbyusingtheEulerapproach.To provethestabilityoftheproposedSM-PIattheorigin(=0),let theLyapunovcandidatebe:

V (ε)=12ε2 (9)

anditstimederivativethatcanbeexpressedas:

˙V(ε)=ε ˙ε=ε(−cvε)=−cvε2<0 (10)

Sincetheconstantcvispositive,theproposedcontrolis

asymp-toticallystable.Basedonthesestabilityrestrictions,thecontroller gainscanbedeterminedbyusingthefollowingswitchinglaws ˜kp=[(1+sgn())k+p −(1−sgn())k−p]+kavp (11)

˜ki=[(1+sgn())k+i −(1−sgn())k−i]+k av

i (12)

wherek+p,k−p,ki+,ki−,kpavandkavi arepositiveconstantsdetermined

asafunctionofthedesiredsystemperformance(thesegainscan beobtainedbyusingastandardPIdesignmethodologye.g.root locus).Themathematicalfunctionsgn()returnsthevalues1for >0or−1for<0.

3.2. DesigncriteriaoftheSM-PIcontroller

Thedesigncriteriaemployedinthisworkisbasedinthepole assignment,based onthesolutionoftheDiophantineequation. Thus,thetransferfunctionsofthesynchronousgenerator(Gvt(s))

andthevoltageregulatorSM-PI(Gc(s))canbewrittenintermsof

thefollowingpolynomials Gvt(s)= Z(s) R(s), Gc(s)= P(s) L(s), (13) where Z(s)=aK6/(d0 ), R(s)=s+1/(K3d0 ), P(s)= ˜kps+ ˜ki and

L(s)=s.Theclosed-looptransferfunctionofgeneratorterminal volt-agecanthenbegivenby

Tet(s)=

Z(s)P(s)

Z(s)P(s)+R(s)L(s) (14)

whosecharacteristicequationis

Z(s)P(s)+R(s)L(s)=0 (15)

ThedesignobjectiveistochoosesuitablepolynomialsP(s)andL(s) yielding

Z(s)P(s)+R(s)L(s)=A∗(s) (16)

where A*(s) is a desired monic Hurwitz polynomial and the

superscript ={fs(fast),av(average), sl(slow)} refersto the per-formancecriteriaemployedfordeterminingdesiredpolynomials. OncedefinedthesuitablepolynomialsA*(s),thevoltageregulator

parameterscanbeobtainedfromthesolutionoftheDiophantine equation.The polynomialsAsl*(s) and Afs*(s) shouldbe sized so

thatthecontrolsystemcompensatestheloadanglevariation dis-turbancequickly.Thedesigncriterionfirstlydeterminestheslow polynomial(Asl*(s))fromthenominalparametersofthecontrol

plant(Eq.(4)).Thus,consideringtheperformanceindexesof maxi-mumovershootanddampingcoefficient,thefollowingpolynomial canbeobtained Asl∗(s)=s2+2asl m+2(aslm) 2 (17) whereasl m=4/tslss(2%)andt sl ss(2%)=t nom

ss(2%)isthenominalsettlingtime

ofthesteady-state system.To determinetheamplitudes ofthe switching laws given by (11)and (12), two other polynomials (Aav*(s)andAfs*(s))arealsodefined,correspondingtothereduction

of42.85%and60%ofthenominalsteady-statetime(tss(2%)sl ).Taking intoaccountthethreepolynomialsandsolvingtheDiophantine equationsforeachcase,thefollowinggainscanbecalculated: kp= 2am−av bv , k  i = 2(am)2 bv , (18) wherekp andk 

i,with={fs,av,sl}refertothecontrollergains

obtainedfromthepolynomialsA*(s).Inaddition,thecoefficients

k+p,k−p,ki+,andk−i aredeterminedasfollows k+p =k fs p −kavp 2 , k + i = kfsi −kav i 2 (19) k−p =k av p −kslp 2 , k−i = kav i −ksli 2 (20)

ThevalueofthisconstantdetermineshowtheSM-PIactsduring thetransient.

4. ExperimentalevaluationoftheSM-PIvoltageregulator

The proposed SM-PI voltage regulator has been evaluated experimentallybyusingalaboratoryset-upwhosestructurewas presentedinFig.1,whichconsistsof5kVAthree-phasesalientpole synchronousgeneratorpulledbya7kWDCmotorandanelectrical energysubstationof45kVA–220/127V(rms),aspowergrid.The synchronousgeneratorisconnectedtoathree-phasepowergrid emulatorcomposedoftwotransmissionlinesegments(rs=0.1

andls=2mH).InthemiddleofthislinePCC,thereisabalanced

three-phaseRLload,star-connected(rl=20andll=60mH).Their

loadphasescanbeconnectedtoexternalresistorsbycontrolled switchesforemulatingsystemshort-circuitfaultsorsteady-state workingpointchanges.ThetorqueoftheDCmotorisregulated toensuretheoperationof thegeneration insynchronism with thepowergrid.Thefieldwindingofthesynchronousmachineis fedviaaH-BridgebuckconvertercommandedbyaPWM strat-egywhose referencevoltage isgenerated by theSM-PIvoltage regulator.Bothcontrolstrategiesforregulatingthetorqueofthe DCmotorandgeneratorterminalvoltagewereimplementedona DSP(TMS320F28335)platform.TheA/DconvertersoftheDSPcard wereconnectedtoameasurementunit,composedofhall-effect

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Table1

Synchronousgeneratorparameters.

S=5kVA e∞=220/127V rf=1.2

lf=79mH lmd=117.62mH lmq=69.4mH

xd=50 xq=31.6 xd=16.5

fs=60Hz P=6poles H=0.09kgm2

rsc=5 a=1 xs=1.5

voltageandcurrentsensors.Thesignalstakenfromthesesensors

passthroughlow-passfilterswithacutofffrequencyoffc=2.5kHz.

Thecontrolalgorithmwasexecutedwithasamplingtimeof100

␮s.Theparametersofthesynchronousgeneratoremployedinthe

laboratoryprototypeareprovidedinTable1.

Theperformanceoftheproposedcontrolsystemwasassessed through experimental tests, according to the following steps: performance undervoltagereference variation, systemstability undershort-circuitfault,andvariationofthesteady-stateworking point.

4.1. Performanceundervoltagereferencevariation

Theperformanceofthevoltageregulator,wasaccomplishedby anexperimentaltestinwhichthereferencevoltageisincreased andreducedbystepsprofiles.Fig.3showstheexperimentalresults ofthegeneratorterminalvoltagewhenitsreferenceischanged. Inthisgraph,thereferencevoltageofthegeneratorisinitiallyin steady-state(

v

t=380V)andint=28.3sitisincreasedbythestep

of

v

t=15V.andreducedint=31.25s,bynegativevoltagestep

of

v

t=15V.Inbothstepchanges,thegeneratorterminalvoltage

takesapproximatelyt=0.5s,withoutovershoot,forreachingits reference,whichdemonstratesgoodperformanceunderreference voltagevariation.

4.2. Systemstabilityundershort-circuitfault

TheperformanceoftheSM-PIcontrollertoensurethesystem stabilityandtomaintainthesynchronizationofthegenerator dur-ingtheoccurrenceofseveredisturbancesinthepowersystemwas experimentallyevaluated.Inthisexperiment,asevereelectricload variationwasimposedonthesynchronousgeneratorbymeansof theshort-circuittransientcarriedonthepowergridemulator.At thePCC,resistorsof5wereinsertedinparallelwiththe three-phaseloadthroughcontrolledswitchesforincreasingthegenerator phasecurrent byaboutfivetimes.Theperformance ofthe pro-posedSM-PIwasalsoevaluatedincomparisonwiththreedifferent controlschemes:standardPID[2]andhybridstrategiesVS-APPC

[26]andVS-MRAC[27].Figs.4-8showtheexperimentalresults

28 28.5 29 29.5 30 30.5 31 31.5 32 32.5 33

Time (s)

375

V

o

ltage (

V)

380

385

390

395

400

v t

t

( )

SM-PI

Reference

Fig.3. Experimentalresultoftheterminalvoltageofthegeneratorregulatedfrom aSM-PIcontrollerduringastepinthevoltagereference.

28 Rms current (A) 6 8 10 12 14 16 2 4 0 ig1( )t t1 t2

Fault clearing time Fault inception time

t1 t2

28.5 29 29.5 30 30.5

Time (s)

Fig.4.ExperimentalresultofthegeneratorphaseRMScurrent1.

28 28.5 29 29.5 30 30.5 31 31.5

V

o

ltage (V

)

0

10

20

30

40

50

60

70

80

Δ

e

t

( )

fd

Time (s)

Fig.5. ExperimentalresultoftheSM-PIoutputduringtheshort-circuittest.

regardingtheshort-circuittest.Fig.4presentsthermsvalueofthe generatorphasecurrentig1duringtheshort-circuitonthepower

grid.Inthistest,att=28.3s,themachinephasecurrentisincreased from2Ato14A,withoutovershoot,duringt=0.5s.

Fig.5presentstheoutputcontrolsignaloftheSM-PIrequiredto obtainthedynamicperformanceofthesystemtransientcondition. Thedv/dtimposedbytheshort-circuittestdoesnotchangethe magnitudeofthechatteringinthecontrolsignalgeneratedbythe proposedcontrolscheme.Moreover,thischatteringeffectdoesnot increasetheoscillationofthegeneratorterminalvoltagebecause itissmoothbythePWMdc–dcbuckconverter.

Fig.6presentsthreedifferenttestsforthesynchronous gener-atorspeedωrundershort-circuit.Ineachexperiment,theSM-PI

iscomparedwiththreedifferentcontrolstrategiesforevaluating itsperformance. Inthesteady-stateoperation, thespeedof the synchronousgeneratorisωr=600rpm.Thefaultinceptiontimeis

t=28.3s.Fig.6ashowstheexperimentalresultsofthegenerator speedωrwhentheterminalvoltageisregulatedbytheSM-PIor

astandardPID.Inthistest,thestabilizationtimewhentheSM-PI isemployedistss=0.86s,boththeovershootandtheundershoot

areapproximately4.33%.WhenthePIDisused,thesettlingtimeis tss=0.96s,theovershootis3%andtheundershootis2%.Inthe

sec-ondtest,theSM-PIiscomparedwiththecontrolschemeVS-APPC aspresentedinFig.6b.Sincethetestsareperformedunderthe sameconditions,thestabilizationtimeofSM-PIremainsthesame. However,whentheVS-APPCisused,thesettlingtimebecomes tss=1.88s,theovershootis3.33%andtheundershootis3%.Finally, Fig.6cpresentsthesametestwiththeVS-MRAC.Inthiscase,the

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r

t

r

t

28 28.5 29 29.5 30 30.5 31 31.5

Time (s)

560

Speed

(R

P

M

)

570

580

590

600

610

620

630

640

28 28.5 29 29.5 30 30.5 31 31.5

Time (s)

560

Speed

(R

P

M

)

570

580

590

600

610

620

630

640

r

SM-PI

PID

(a)

(b)

SM-PI

VS-APPC

SM-PI

VS-MRAC

28 28.5 29 29.5 30 30.5 31 31.5

Time (s)

560

Speed

(R

P

M

)

570

580

590

600

610

620

630

640

(c)

t

Fig.6.Experimentalresultofthegeneratorspeedduringtheshort-circuittest:(a) PIDandSM-PI;(b)VS-APPCandSM-PI;(c)VS-MRACandSM-PI.

stabilizationtimeistss=1.58s,theovershootis3.5%andthe

under-shootis3.33%.ThesetestsdemonstratethattheSM-PIprovidedthe bestperformance.

Fig.7showstheexperimentalresultsoftheAVRimplemented by the SM-PI and also by the control strategies PID, VS-APPC andVS-MRACforthesameoperationalcondition.Thecontrollers presentedthefollowingstabilizationtimes,overshootsand under-shoots:PID(tss=1.99s,theovershootis2.89%andtheundershoot

is6.97%,SM-PI(tss=0.75s,theovershootis3.55%andthe

under-shootis4.6%)VS-APPC(tss=1.39s,theovershootis3.68%andthe

undershootis6.84%)andVS-MRAC(tss=1.64s,theovershootis

3.55%andtheundershootis4.47%).

Basedontheseresults,theSM-PIpresentedsuperior perfor-mance.Theseexperimentalresultsdemonstrate thatthevoltage oscillationspresented in thegenerator terminal voltage,in the steady-state,arealmostthesameforallcontrollers.

v

t

( )

t

v

t

( )

t

28 28.5 29 29.5 30 30.5 31 31.5

Time (s)

350

V

o

ltage (V)

355

360

365

370

375

380

385

390

395

400

28 28.5 29 29.5 30 30.5 31 31.5

Time (s)

350

V

oltage (V

)

355

360

365

370

375

380

385

390

395

400

v

tt

( )

t

SM-PI

PID

(a)

(b)

28 28.5 29 29.5 30 30.5 31 31.5

Time (s)

350

V

oltage (V

)

355

360

365

370

375

380

385

390

395

400

(c)

SM-PI

VS-APPC

SM-PI

VS-MRAC

Fig.7. Experimentalresultofterminalvoltagevectorvtofthegeneratorregulated bycontrollers:(a)SM-PIandPID;(b)SM-PIandVS-APPC;(c)SM-PIandVS-MRAC duringtheshort-circuittest.

Fig.8presentstheexperimentalresultsoftheSM-PIaverage gainskp(Fig.7a)andki(Fig.7b)duringtheshort-circuittest.The

controller gainskp and ki obtainedfromEqs.(11)–(12),filtered

byafirstorderlow-passfilterwithacutofffrequencyof100Hz. Thevariationofthecontrollergainkpaddressestotheovershoots

of64.29%andundershootsof46.43%.Aswellas,changesof con-trollerthecontrollergainskiresultsinovershootsof78.95%and

undershootsof47.37%. Thesegraphsshowthatcontrollergains varyregularlytoobtaintherequiredsystemdynamicperformance forthetransientcondition,whichdemonstratestheiradaptivefor systemdynamicchanges.

TheparametersofthecontrollersPID,SM-PI,VS-APPC,and VS-MRACarepresentedinTable2.

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28 28.5 29 29.5 30 30.5 31 31.

5

Time (s)

0.05

0.1

0.15

0.2

0.25

k

p

( )

t

( )

a

28 28.5 29 29.5 30 30.5 31 31.

5

Time (s)

2

4

6

8

10

12

14

16

18

k

i

( )

t

( )

b

Fig.8.ExperimentalresultoftheSM-PIcontrollergainaveragevalue ¯kpand ¯ki duringtheoccurrenceofshort-circuitfault.

Table2

Controllersparameters.

PID VS-APPC VS-MRAC SM-PI

kc=0.054 ¯a=45 ¯ 1=0.02 kslp=0.028 ksli =2.679 i=0.012 ¯b =15 ¯ 2=0.15 kavp =0.128 kavi =8.202 d=0.009 bn=300 am=35 kfsp=0.229 kfsi =16.74

N=50 am=35 cv=30

4.3. PerformanceoftheproposedSM-PIundersteady-state

systemworkingpointvariation

TheperformanceoftheproposedSM-PIvoltageregulatorwas

evaluated when the generator wassubjected toa steady-state

workingpoint change.Inthis test, att=28.7s, thegeneratoris

subjectedtoaloadstepchangeofapproximately1.8kW.Thistest

demonstratesthatSM-PIcontrollerhasagoodperformancewhen

thesystemoperates underloadanglechangestransients.Fig.9

presentstheexperimentalresultsoftheactivepowergenerated bythemachineduringtheloadchangetest.Thegeneratoractive powerincreasedfrom2.3kWtoapproximately4.1kW, increment-ingthetorqueimposedbytheDCmotor.

Fig.10showstheexperimentalresultoftheterminalvoltage

v

t,

regulatedbytheSM-PIcontroller,duringtheintervalinwhichthe steady-stateworkingpointwaschanged.Atthetimeinwhichthe loadvariationisintroduced,thereisadropinthegenerator termi-nalvoltage.However,theSM-PIchangeditsdynamicbehaviorfor convergingthegeneratorterminalvoltagetoitsreference,which takesapproximatelyt=0.2sandtheundershootis0.66%.

28 28.5 29 29.5 30 30.5 31 31.5

2

Active power (kW

)

2.5

3

3.5

4

4.5

P

0

( )

t

Time (s)

Fig.9.Experimentalresultoftheactivepowerduringthesteady-sateoperating pointchange.

28 28.5 29 29.5 30 30.5 31 31.

5

v

t

( )

t

350

V

oltage (V)

355

360

365

370

375

380

385

390

395

400

Time (s)

Fig.10.Experimentalresultoftheterminalvoltageofthegeneratorregulatedfrom aSM-PIcontrollerduringthesteady-sateoperatingpointchange.

5. Conclusion

Thispaperproposedasliding-modecontrolstrategyfor regulat-ingtheterminalvoltageofthesynchronousgeneratorconnected toapowergrid.Thiscontrolschemewasbasedonahybrid struc-turecomposedoftheintegrationoftheSMCandthestandardPI regulator,whichresultsintheSM-PIcontroller.Inthisapproach, thecontrollergainswereissuedfromaswitchinglawdefinedby theslidingsurfaceforthegeneratorterminalvoltage.

RegardingthecontrollerSM-PI,theproofofstability andthe designcriteriafordeterminingthecontrollergainswereprovided. The performance of proposed SM-PI wascompared withthree differentcontrollers(PID,VS-APPC,andVS-MRAC).TheSM-PI pre-sentedsuperiorperformanceforthesameoperationalcondition. Duetoitssimplicity,theSM-PIcanbeintroducedintothecontrol algorithmoftheconventionalAVRwithoutmuchcomputational effort.Theexperimentalresultsdemonstratedthatthecontrolleris efficientindampingthevoltageoscillationsofthegenerator termi-nalvoltageduetotheoccurrenceofseveretransientsinthepower system.

References

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Referências

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