second-order sliding-mode controller design and tuning for grid synchronisation and power control of...

5/28/2018 Second-Order Sliding-mode Controller Design and Tuning for Grid Synchronisation and Power Control of a Wind Turbine-driven Doubly Fed In

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Published in IET Renewable Power Generation

Received on 26th January 2012

Revised on 10th November 2012

Accepted on 22nd November 2012

doi: 10.1049/iet-rpg.2012.0026

ISSN 1752-1416

Second-order sliding-mode controller design andtuning for grid synchronisation and power control of awind turbine-driven doubly fed induction generatorAna Susperregui1, Miren Itsaso Martinez1, Gerardo Tapia1, Ionel Vechiu2

1Department of Systems Engineering and Control, Polytechnical University College, University of the Basque Country

UPV/EHU, Plaza de Europa, 1, 20018 Donostia-San Sebastin, Spain2

Ecole Suprievre des Technologies Industrielles Avances (ESTIA) Research, Technopole Izarbel, 64210 Bidart, FranceE-mail: [email protected]

Abstract: This study presents a second-order sliding-mode control (2-SMC) scheme for a wind turbine-driven doubly fedinduction generator (DFIG). The tasks of grid synchronisation and power control are undertaken by two different algorithms,designed to command the rotor-side converter at a xed switching frequency. Effective tuning equations for the parameters of

both controllers are derived. A procedure is also provided that guarantees bumpless transfer between the two controllers at theinstant of connecting the DFIG to the grid. The resulting 2-SMC scheme is experimentally validated on a laboratory-scale7 kW DFIG test bench. Experimental results evidence both the high dynamic performance and the superior robustnessachieved with the proposed control scheme.

Nomenclature

cosj power factorir, is rotor and stator current space-vectorsir,is components of rotor and stator current

space-vectorsLlr, Lls rotor and stator leakage inductancesLm,Lr, Ls magnetizing, rotor and stator inductancesLr rotor transient inductancen general turns ratio

P number of pole pairsPr,Ps, Pt rotor-side, stator-side and total active

powersQs stator-side reactive power

Rr,Rs rotor and stator resistancess slipvg, vs grid and stator voltage space-vectorsvg, vr components of grid and rotor voltage

space-vectorss angle between the stator-ux-oriented and the

stationary reference framess angle between the grid-voltage-oriented and

the stationary reference framesr, r rotor electrical position and speedrm rotor mechanical speed

s,sl synchronous and slip angular frequenciess statorux space-vectors component of statorux space-vector

Subscripts

D,Q

direct- and quadrature-axis components, expressedin the stationary reference frame

x, y direct- and quadrature-axis components, expressedin the stator-ux-oriented reference frame

x,y direct- and quadrature-axis components, expressedin the grid-voltage-oriented reference frame

, direct- and quadrature-axis components, expressedin the rotor natural reference frame

Superscripts

R rated value* reference value

1 Introduction

Standard vector control (VC) schemes conceived to commandwind turbine-driven doubly fed induction generators (DFIGs)typically comprise proportional-integral (PI)-based cascadedcurrent and power control loops [13]. However, the systemtransient performance degrades as the actual values of theDFIG resistances and inductances deviate from those basedon which the control system tuning was carried out duringcommissioning [4]. In addition, the tracking of the

maximum power point achieved by adopting classical VCcongurations exhibits the characteristic lags associatedwith PI-based control schemes [5,6].

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Overcoming those two drawbacks entails improvingrobustness to DFIG parameter variations, as well asachieving high dynamic performance power control. In thisframework, alternative control schemes for DFIGs are being

proposed, among which a strong research line focuses ondirect power control (DPC) [4, 7]. Several others explorethe option of applying sliding-mode control (SMC) [5,810].

In particular, a rst-order SMC (1-SMC) conguration for

decoupled control of DFIG active and reactive powers isdescribed in [5]. In spite of conferring outstandingrobustness to parameter variations and superior tracking ofa rotor speed-dependent optimum power curve, the 1-SMCstructure put forward leads to a variable switchingfrequency of the rotor-side converter (RSC) transistors. Thismay cause broadband harmonics to be injected into thegrid, hence complicating the design of both the

back-to-back converter and the grid-side AC lter [5,7].This drawback is overcome in [10] through application of

the so-called boundary layer method. By adopting thisapproach, the discontinuous rotor voltage control signals towhich 1-SMC leads are rst smoothed out. The resulting

voltage waveforms are then applied to the DFIG rotor bymeans of a modulation technique, such as pulse-widthmodulation (PWM) or space-vector modulation, whichdrives the RSC transistors at a constant switchingfrequency. Even if the described approach solves the

problem of variable switching frequency, it is only at theexpense of severely compromising the characteristics oftracking accuracy, robustness and disturbance rejectionconferred by SMC. Indeed, within the boundary layer, thefeedback system no longer behaves as dictated by slidingmode, and it is simply reduced to a system with no slidingmode [11].

In this context, with the objective of keeping a constantswitching frequency without renouncing to the robustness

and tracking ability features gained with 1-SMC, aPWM-based second-order SMC (2-SMC) scheme for DFIGcontrol is presented in this paper. This control congurationis not only aimed at accurately tracking time-varyingreference values for the active and reactive powers to beexchanged between the DFIG stator and the grid, but alsoat ensuring the synchronisation required for smoothconnection of the DFIG stator to the grid [1215].In addition, the problem of bumpless transfer between thecontrollers in charge of synchronisation and power control,

at the instant of connecting the DFIG stator to the grid, issolved straightforwardly.

Not only the design and implementation of the resultingglobal control scheme are addressed in detail, but also itsanalytical tuning. As an alternative to the conventionaltrial-and-error method frequently adopted to adjust 2-SMCalgorithms [16], tuning equations for the global 2-SMCsystem put forward are derived based on DFIG parameters

and intuitive specications for the closed-loop timeresponse.

Results of experimental evaluation of the proposed 2-SMCscheme on a 7 kW DFIG prototype are reported.

2 Overview of DFIG models for powercontrol and grid synchronisation

When the DFIG stator is connected to the grid, its rotor-sidemodel, expressed in the stator-ux-oriented xy referenceframe depicted in Fig.1, is given by [17]

irx=

vrx

Lr Rr

Lrirx

Lm

LsLr|cs

| +vsliry (1)

iry=vry

LrRr

Lriryvsl irx+

LmLsL

r

|cs|

(2)

where Lr= Lr L2m

/ Ls

and sl= s r. Regarding theactive and reactive powers owing between the DFIG statorand the grid, they may be expressed as [17]

Ps= 3

2

LmLs

|vg|iry ; Qs=3

2

|vg|Ls

|cs| Lmirx

(3)

Expressions (1)(3) consider that currents owing into the

DFIG stator and rotor are positive, which implies thatgenerated active and reactive powers are represented asnegative.

On the other hand, given that power generation is notprotable for rotor speeds below a given value, the DFIGstator is not connected to the grid until that threshold isexceeded. The connection is not trivial, since the gridvoltage and that induced at the open stator of the DFIGmust coincide both in magnitude and phase to avoidshort-circuits. Therefore, prior to connection, the DFIGmust be commanded to achieve grid synchronisation [18].

Let a new xy reference frame be adopted when thestator is disconnected from the grid, whose y quadrature

axis is collinear with the grid voltage space-vector, asdisplayed in Fig. 2. As proved in [14], the open-stator

Fig. 2 xyreference frame for synchronisation

Fig. 1 Stator-ux-oriented xy reference frame, together with the

stationary DQ reference frame, and the rotor natural frame,

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model for the DFIG rotor side, expressed in that referenceframe, is given by

irx=vrx

LrRr

Lrirx+vsliry (4)

iry=vry

LrRr

Lriryvslirx (5)

It turns out that, in steady state, if disconnected from thegrid, the DFIG stator ux and voltage space-vectors are,respectively, collinear with x and y axes [14]. As a result,vss, as depicted in Fig. 2. xy and xyreference framesmust hence be aligned to achieve synchronisation. Yet, fora complete match up, vs must be not only collinear with vg,

but also identical in magnitude. As shown in [14], thosetwo requirements are satised if rotor current regulation iscarried out around set-points

irx= |vg

|Lmvs ; iry=0 (6)

3 Second-order SMC scheme

A global 2-SMC scheme for the DFIG is described in thissection. Owing to the different dynamic behaviours andcontrol targets to be dealt with when its stator isdisconnected from or connected to the grid, the relevantDFIG model is considered to design the control law foreach of those two cases.

3.1 Selection of switching variables

Examination of the set-points in (6) evidences that rotorcurrent regulation must be carried out for gridsynchronisation. Therefore when the DFIG stator isdisconnected from the grid, the switching functions givennext are assumed

sx= irxirx+cx

irxirx

dt (7)

sy= iryiry+cy iryiry dt (8)

where the integral terms, weighted by cx and cy positiveconstants, are added for steady-state response improvement,as suggested in [19].

On the other hand, seeking to achieve close trackingof the maximum power point, Ps , as well as highdynamic performance control of the reactive power, thefollowing switching variables are adopted when connectedto the grid

sP=PsPs+cP

PsPs

dt (9)

sQ=QsQs+cQ QsQs dt (10)where cPand cQ are also positive constants to be tuned.

3.2 SynchronisationDFIG disconnected fromthe grid

3.2.1 Design: Bearing in mind that the irx and iry

set-points in (6) are constant, the following time derivativesof switching functions sxand syresult from combination of(7), (8), (4) and (5)

sx= cx irx+ RrLrcx

irxvsliry 1Lr

vrx (11)

sy= cy iry+RrLr

cy

iry+vslirx1

Lrvry (12)

which reveal that the DFIG is of rst-order relative degreeduring synchronisation. Accordingly, it may be commanded

by applying 1-SMC or 2-SMC [16]. As already mentioned,a structure based on 1-SMC would however lead to avariable switching frequency of the RSC transistors,therefore complicating the design of the power converteritself, as well as that of the grid-side AC lter. Aiming to

overcome that drawback, a PWM 2-SMC realisation,known as the super-twisting algorithm (STA), is adopted [20].Consequently, the voltage to be applied to the rotor is

computed according to control law

vrx= vrxST+vrxeq; vry= vryST+vryeq (13)

where the terms with subscript ST are calculated, throughapplication of the STA, as

vrxST=lx|sx |

sgn sx

+wx sgn sx dt (14)vryST=ly |sy | sgn(sy ) +wy sgn(sy )dt (15)

withx,wx,yandwybeing positive parameters to be tuned.The addends with subscript eqin (13), which correspond toequivalent control terms, are derived by letting sx= sy=0,as dened in [19]. As a result

vrxeq=Lr cx irx+RrLr

cx

irxvsliry

(16)

vryeq=Lr cy iry+RrLr

cy

iry+vslirx

(17)

The sliding regime in manifold sx= sy= sx= sy=0 canalso be attained by applying only the STA control terms in(13). Accordingly, the equivalent control terms in (16) and(17) are not strictly necessary. However, if incorporated, themore accurately they are estimated, the lower is the controleffort let to be done by the STA. In conclusion, althoughuncertainty in Lrand Rrleads to inaccurate vrxeqand vryeq,robustness of the overall control algorithm in (13) is not putin jeopardy, as robustness does actually lie on the STAcontrol terms given by (14) and (15).

Equivalent control terms are hence included not only toimprove the system transient response [21], but also to easeselection of constants cx and cy, as well as tuning of the

STA x, y, wx and wy gains. Owing to the analogyexisting between the procedures for tuning the algorithmscommanding irx and iry, only that related to irx will be

presented in detail.

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3.2.2 Tuning: Substitution of control law (13) into (11)produces

sx= 1

Lrlx

|sx |

sgn sx

+wx sgn sx dt

(18)

Now, given that sgn(sx) =sx/|sx|, taking the time derivativeof (18) leads to

sx= 1

Lr

lx

2|sx | sx+wx

sx

|sx |

(19)

Let us assume that, because of the rst term in the STA, thereaching phase is satisfactorily completed and the slidingregime is entered. From that moment on, |sx| x, with xclose to zero. Considering the most unfavourable case, inwhich |sx| = x, and using the denition of sx in (7), the

following expression can be worked out from (19)

ex+lx

2Lrdx

+cx

a2 cx ,lx( )

ex+1

Lrdx

lx cx2

+ wxdx

a1 cx ,lx ,wx( )

ex

+ wx cxLrdx

a0 cx ,wx( )

ex dt=0 (20)

where ex= irxirx . Taking the time derivative of (20),the following differential equation reecting the ex errordynamics while in sliding regime is obtained

ex+a2ex+a1ex+a0ex=0 (21)

Hence, once x is xed, adequate selection of cx, xand wx allows attaining certain target error dynamics,established through the third-order characteristic equationgiven next

p2 +2jxvnxp+v2nx p+ajxvnx =p3+(2+a)jxvnx

a2t

p2 + 1+2aj2x

v2nxa1t

p+ajxv3nxa0t

=0

(22)

which, provided that is selected high enough 10 gives rise to a pair of dominant poles with respect to a thirdone placed at p= xnx. As a result, it can beconsidered that target error dynamics are entirely denedvia x damping coefcient and nx natural frequency.Those designer-dened error dynamics would theoretically

be achieved just by tuning cx, x and wx so that a2 = a2t,a1 =a1tand a0= a0tare simultaneously fullled.

Considering the expressions fora2, a1and a0provided in(20), as well as those for a2t, a1t and a0t reected in (22),

the latter requirements lead to the following tuning equations

c3

x(2+a)jxvnxa2t

c2

x+ 1+2aj2x

v2nx

a1t

cxajxv3nxa0t

=0

(23)

lx=2Lrdx

(2+a)jxvnxcx

(24)

wx= Lrdxajxv3nx

cx(25)

It is important to note that the coefcients in (23) coincidewith those of target characteristic equation (22), except forthe signs of the squared and independent terms, which arenegative. It therefore turns out that the three possible valuesfor cx are equal to the roots poles of targetcharacteristic equation (22), although their real parts haveopposite signs. The real parts of the desired poles mustnecessarily be negative to ensure stability, which impliesthat the real parts of the three possible values for cx willalways be positive. As a result, since (23) is third-order, it

is guaranteed that at least one of the three solutions for cxwill be both real and positive, as required. If (23) has morethan one feasible solution forcx, then the {cx, x, wx} setleading to the best performance may be identied throughnumerical simulation.

3.2.3 Implementation: The functional diagram providedin Fig. 3 illustrates the procedure for implementing the

proposed synchronisation algorithm. Any variable presentin a given layer of the diagram is also available to thelayers inside, which implies that the algorithm must beimplemented from the outer to the inner layer labelled,respectively, as 1st Step and 3rd Step at their bottom

left-hand corners.In addition to the irx andiry set-points provided in (6), the

actual values ofirxandiryare required to compute switchingfunctions sx and sy, as well as equivalent control terms(16) and (17). Knowledge ofr is also needed to calculatethe sl slip angular frequency present in (16) and (17). In

accordance with Figs. 1 and 2, the Parks ej rsur( )

transform is used to bring rotor current ir and ircomponents to the xy reference frame, where ir and irare derived by applying the Clarkes transform to directlymeasured three-phase rotor current. Based on Fig. 2, anglesis calculated as

arctan

vgQ

vgD 90 (26)

while r, as well as r, can either be measured using anincremental encoder or estimated by an observer, if asensorless scheme is to be implemented.

Similarly, rotor voltage vr and vr natural-framecomponents are obtained through application of the inverseej r

sur( ) Parks transform to vrx and vry control signals. The

inverse Clarkes transform is then applied to vrand vr inorder to achieve the three-phase rotor voltage waveforms to

be supplied as reference signals to the PWM algorithm.

3.3 Power controlDFIG connected to the grid

3.3.1 Design: Considering that|cs|in (1) turns out to benegligible when grid connected, combination of (9) and(10) with (1)(3) leads to the time derivatives of switching

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variablessPand sQgiven next

sP= Ps

3

2

LmLs

|vg|RrLr

cP

iry+vsl irx+Lm

LsLr

|cs|

+cPPs+3

2

LmLsL

r

|vg|vry(27)

sQ= Qs

3

2

|vg|Ls

cQ|cs| +RrLr

cQ

LmirxvslLmiry

+cQQs+3

2

LmLsL

r

|vg|vrx(28)

which prove that the DFIG is also ofrst-order relative degreewhen connected to the grid. Therefore the design and tuningof power controllers is analogous to that presented inthe preceding section. Thereby, the rotor is fed with the

followingxyaxes-referred voltage

vrx=vrxST+vrxeq; vry=vryST+vryeq (29)

The STA terms are then obtained as

vrxST= lQ|sQ|

sgn(sQ) wQ

sgn(sQ)dt (30)

vryST= lP|sP|

sgn(sP) wP

sgn(sP)dt (31)

whereQ, wQ, Pand wPare positive constants to be tuned.

On the other hand, the equivalent control terms in (29) arecomputed by zeroing (27) and (28), which yields

vrxeq= 2LsL

r

3Lm|vg|Q

s+cQ QsQs

+RrirxLrvsliry(32)

vryeq= 2LsL

r

3Lm|vg|P

s+cP PsPs

+Rriry+vsl Lrirx+

LmLs

|cs| (33)

3.3.2 Tuning: Given that the algorithms governing Psand Qs are adjusted by following the same method, onlytuning of that controlling the active power will be dealt

Fig. 3 Functional diagram of the proposed synchronisation algorithm

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with. Substitution of control law (29) in (27) leads to

sP= 3

2

LmLsL

r

|vg| lP|sP|

sgn sP

+wP sgn sP dt

(34)

expression which turns out to be identical to that in (18), once

factor (3Lm|vg|)/(2LsL

r) and subscript

P

are, respectively,replaced by 1/(Lr) and x. Consequently, tuning equations(23)(25) remain valid forcP, Pand wPjust by substituting

Lrand subscript xwith (2LsLr)/(3Lm|vg|) and P. Therefore

c3P(2+a)jPvnPc2P+ 1+2aj2P

v2nPcPajPv3nP=0(35)

lP=4LsL

r

dP

(2+a)jPvnPcP

3Lm|vg|(36)

wP=2LsL

rdPajPv

3nP

3Lm|vg

|cP

(37)

3.3.3 Implementation: Fig. 4 displays the functionaldiagram corresponding to the proposed power controlscheme. Given that it is analogous to that described inSection 3.2, only the main differences will be highlightedhere.

On the one hand, it is essential to note that the actual valuesofPsand Qsare computed as

Ps=3

2 vgDisD+vgQisQ

; Qs=3

2 vgQisDvgDisQ

(38)

hence avoiding the use of the DFIG parameter-dependentexpressions in (3), which are only to be used for controllerdesign. Regarding the reference values for active andreactive powers, a maximum power point tracking (MPPT)strategy provides Ps as a function of r, while Q

s is

established directly or based on Ps and on the demandedcos j* power factor.

On the other hand, anglesis required instead ofsto carryout the Parks transform and inverse transform reected in thediagram of Fig.4. In this case, as evidenced by Fig.1, siscalculated as

rs=arctancsQ

csD(39)

wheresDand sQare in turn estimated as

csD=

vgDRsisD

dt; csQ=

vgQRsisQ

dt

(40)

Fig. 4 Functional diagram of the proposed power control scheme

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Aiming at avoiding drift, bandpass lters are used rather thanpure integrators to implement (40) [22]. Accordingly, |s| inequivalent control term (33) is computed from sD and sQas follows

|cs| =c2sD+c2sQ

(41)

3.4 Bumpless transfer

So far, the separate performance of each of the two DFIGoperating states disconnected from and connected to thegrid has only been considered, but an undesirable

phenomenon may appear if the switch between the twocontrollers is not properly carried out. If a direct transfer isaccomplished, discontinuities arise in the control signals atthe instant of connection, owing to the magnitude mismatch

between the rotor voltage components generated by the twocontrollers. This effect produces high stator current values,leading the machine to exchange excessive power with the

grid at the instant of connection. Aiming to attenuate oreven avoid this bump, it is required to apply the samevalues of the control signals previous to and just after thetransfer, that is

vrx=vrx ; vry=vry (42)

Setting the focus on the DFIG rotor voltage components whenconnected to the grid, appropriate combination of (29)(33)

produces

vrx= lQ |sQ| sgn(sQ) wQ sgn(sQ)dt 2LsL

r

3Lm|vg|Q

s+cQ QsQs

+RrirxLrvsliry(43)

vry= lP|sP|

sgn sP

wP sgn(sP)dt 2LsL

r

3Lm|vg|P

s+cP PsPs

+Rriry+vsl Lrirx+LmLs

|cs| (44)

Condition (42) can be satised by carefully selecting theinitial values for integrals

sgn(sQ)dt and

sgn(sP)dt at

the instant of connection. Thus, substitution of (43) and(44) into (42) allows deriving those initial values as

sgn(sQ)dtt=0=

RrirxLrvslirylQ|sQ| sgn(sQ) vrx

wQ

2LsLrQ

s+cQ QsQs 3Lm|vg|wQ

(45)

sgn(sP)dtt=0=

Rriry+LrvslirxlP|sP| sgn sP vry

wP

2LsLr P

s+cP PsPs

3Lm|vg|wP

+Lmvsl|cs|LswP

(46)

4 Experimental validation

4.1 Description of the experimental rig

Experimental validation of the global control schemepresented throughout Section 3 is carried out on the 7 kWDFIG test bench shown in Fig. 5. The electric parametersof the DFIG under consideration are collected in Table 1.The values of resistances and inductances were identied ina recent study reported in [23]. The DFIG is driven by a15 kW armature-controlled dc motor, whose rotationalspeed can be commanded at will via a commerciallyavailable controller. The eMEGAsim OP4500 F11-13

platform by Opal-RT is used to perform rapid controlprototyping of the synchronisation and power controlalgorithms reected, respectively, in the functional diagramsof Figs. 3 and 4. The state of the synchronisation stage isdisplayed by means of a conventional lamp synchronoscope.

The sample rate for the global control algorithm, as wellas the frequency for the triangular carrier of the PWM

Table 1 7 kW DFIG electric parameters

Parameter Value

|vg| 4002/3

V

|is|R 16

2

A

|ir|R 24

2

A

Rs 370 mRr 145.8541 mLls 4.86 mHLlr 1.2138 mHLm 37.6812 mH

N 2.001Ls = Lls + nLm 80.2601 mHLr = Llr + Lm/n 20.045 mHP 2

Fig. 5 Experimental setup

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modulator, are xed to 5 kHz. In addition, the techniqueproposed in [24] is implemented in order to compensate the2.3s dead time inherent to the RSC. The integral termsincluded in both the switching functions and the STA itselfare digitally implemented based on Tustins trapezoidalmethod. Yet, with the aim of avoiding derivative ringing

[25], Eulers rectangular method is adopted to discretise thetime derivatives of Qs and P

s included in equivalent

control terms (32) and (33).Tuning equations (23)(25) and (35)(37) have been

directly applied to derive the values of the global 2-SMCalgorithm parameters reected in Table2. x, y and nx, yhave been selected so as to achieve synchronisation ofstator voltage to that of the grid in 105 ms, showing noovershoots. Given thatx, y = 1, two values are possible forcx, y; namely, nx, y and nx, y. However, numericalsimulation revealed that the {cx, y, x, y, wx, y} parameterset leading to the best performance was that resulting fromselecting cx, y = nx, y. Similarly, as a result of the valuesxed for

P, Qand

nP, Q, possible errors arising in active

and reactive powers would theoretically vanish in 50 ms,with no overshoots. As for the previous case, cP, Q arechosen to be equal to nP, Q.

For simplicity, the power signal feedback method isapplied in this case to dene the maximum power point to

be attained, Ps . Nevertheless, it is to be noted that theproposed 2-SMC power control algorithm is independent ofthe MPPT strategy adopted to establish the time-varyingreference value for Ps. As a consequence, tip-speed ratio orhill-climb searching strategies could equally be used todenePs [26].

In particular, the optimum power curve of a high-powerDFIG installed in an actual wind turbine is taken as starting

point. It was obtained by the manufacturer via off-lineexperimentation and consists in an empirical functionexpressing the total output power to be generated, Pt, as acubic function of the rotor speed, rm. That function is thenscaled in power level to derive a plausible optimum powercurve for the 7 kW DFIG under consideration.

Bearing in mind that the following relation holds for aDFIG

Pt=Ps+Pr (47)

where [18]

Pr sPs= vsvr

vsPs=

vsPvrmvs

Ps (48)

it turns out that

Psvs

PvrmPt (49)

Accordingly, based on the optimum power curve providingP

t, the rotor speed-dependent reference value for the

stator-side active power is computed as

Ps=

vsP

Ptvrm

(50)

thus evidencing that the resultingPs is a quadratic function ofthe rotor speed, asPtis, in turn, described as a cubic functionofrm.

In order to avoid setting unachievable reference values forPsandQs, it is required to observe the feasibility region of theDFIG, represented by the shaded area of Fig. 6. The DFIGfeasibility region is obtained by intersection of twosemicircles. The dashed line delimiting one of those

semicircles displays the restriction because of the rated rotorcurrent, and its analytical expression is found to be [2]

P2s+ Qs

3|vg|22Lsvs

2= 3

2

LmLs

|vg||ir|R 2

(51)

On the other hand, the analytical expression for thesemicircumference in solid line is given by

P2s+Q2s=3

2|vg||is|R

2(52)

which corresponds to the constraint imposed by the ratedvalue of the stator current. Examination of the feasibilityregion reveals the limited ability of this particular DFIG togenerate reactive power. In effect, Qs can be injected intothe grid only when generated Psis below 4.547 kW.

4.2 Experimental results

Aiming at showing some of the most illustrative results of theglobal 2-SMC scheme put forward, a test is conducted wherethe DFIG is subjected to the sequence of operation modesreected in Table 3. Additionally, the 15 kW dc motor iscommanded so that, during the test, the DFIG rotor speed

evolves with time as shown in Fig.7.The time progress of the DFIG rotor speed has been

conceived with the aim of evaluating the proposedsynchronisation and power control algorithms under

Table 2 Specifications set to tune the 2-SMC arrangement,and resulting parameter values

Specification Value Parameter Value

x, y 0.05 A cx,cy 55.2381x, y 1 x, y 5.4469nx, ny 55.2381 rad/s wx,wy 30.5811 10

P 35 W cP,cQ 116Q 15 VAr P 1.5453 10

1

P,Q 1 Q 1.0116 10 1

nP,nQ 116 rad/s wP 48.2032 10 wQ 20.6585 Fig. 6 Feasibility region for the 7 kW DFIG

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realistic conditions. In an actual wind turbine, rotor speedvariations are expected to be inappreciable duringsynchronisation, as it only takes a few cycles 105 ms inthis particular case. Considering this, rotor speed isregulated around 1220 rpm when testing the synchronisationscheme. However, it should be noted that thesynchronisation algorithm described in Section 3.2.1 isdesigned to operate with no restriction on rotor speed.

Regarding the second part of the test, devoted to analyse

the presented power control algorithm, rotor speed is variedin order that a signicant part of the optimum power curveis to be tracked both up and downwards. For that purpose,starting at second 9, rotational speed is accelerated from1220 to 1620 rpm, and subsequently decelerated back to1220 rpm from second 16 onwards. The acceleration anddeceleration displayed in Fig. 7emulate, respectively, thoseexperienced by a DFIG installed in a real wind turbine of

H= 0.9346 s inertia constant while undergoing a suddenincrease followed by a sudden decrease of wind speed.

The initial rotor positioning stage is activated manually atinstant p, and it lasts 0.5 s, until instant s. The purpose ofthis operation mode is to obtain an accurate estimate of the

rotor position at instant s, r0. From instant s on, theencoder measures the increment of the rotor electrical anglewith respect to r0, r. Thus, the absolute rotor positionrequired for synchronisation and power control is derived asr= r0 + r. The procedure described in [14] is followedfor initial rotor positioning.

Grid synchronisation is automatically launched at instants.Fig.8zooms at the beginning of the synchronisation process.In particular, Fig. 8a displays the transient response of thevoltage induced at the terminals of the DFIG open stator,showing that synchronisation with grid voltage isaccomplished at approximately the specied settling time of105 ms. For clarity, only voltages corresponding to phaseAare represented. The vra, vrb and vrc voltage components

shown in Fig. 8b are those supplied as inputs to thePWM modulator driving the RSC transistors duringsynchronisation. The resulting three-phase rotor current isdisplayed in Fig.8c.

After the lamps of the synchronoscope turn off, indicatingthat grid synchronisation has been achieved, the DFIG statoris manually connected to the grid at instantc. As a result ofthe bumpless transfer between the 2-SMC algorithms incharge of synchronisation and power control, gridconnection at zero power turns out to be smooth. This issubstantiated by Figs. 9a and b, which zoom, respectively,at the active and reactive powers interchanged between theDFIG stator and the grid around instant c. As expected,

they are both equal to zero prior to connection, and they arekept within the ranges of 80100 W and 200250 VAjust after connecting the DFIG stator to the grid. Thethree-phase rotor voltage and current leading to such a

Table 3 Operation modes of the DFIG during the test

Operation mode Time range, s

global control algorithm is idle 0pinitial rotor positioning ps;s= p+ 0.5grid synchronisation scconnected to the grid with no power exchange c5linear increase of generated power up to3.752 kW

56

generation according to optimum power curve 923

Fig. 8 Grid synchronisation

aGrid and DFIG open-stator voltagesb Reference values of the three-phase rotor voltage fed into the PWMmodulatorcThree-phase rotor currentFig. 7 DFIG rotational speed

Fig. 9 Smooth connection at zero power

aActive powerbReactive powerc Reference values of the three-phase rotor voltage fed into the PWMmodulatordThree-phase rotor current

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smooth connection are those shown in Figs. 9c and d,respectively. If the manoeuvre of connecting the DFIGstator to the grid at zero power was ideal, both the rotorvoltage and current previous to instant c would remainunchanged after connection.

As evidenced by Fig. 10, power generation is started atsecond 5. Between seconds 5 and 6, Ps is raised linearlyfrom 0 to 3.752 kW value which, according to (50),

corresponds to the optimum stator-side power for a 1220rpm rotational speed and established as dictated by (50)for the remaining time. In view of the feasibility regiondepicted in Fig.6, Qs is set in order that the DFIG operateswith a 0.81 lagging inductive power factor allthrough the test. Figs. 10a and b evidence the excellent

performances of Ps and Qs when tracking their respectivetime-varying reference values. In the worst case, whileoperating at supersynchronous speeds, chatter in reactive

power approximately reaches 3% of the 7 kW ratedpower. However, during the rest of the experiment, chatterpresent inPsand Qs represents < 1.5%.

Aiming at further assessing the dynamic performance ofthe proposed power control scheme, the experimentdiscussed above is repeated, but dening Qs as thesuccession of 1 kVAr steps displayed in Fig. 11b. Thedifferent set-points for Qs are carefully selected so thatthe resulting (Ps, Qs) operating points fall inside the DFIGfeasibility region shaded in Fig. 6. Fig. 11b conrms theoutstanding dynamic response of Qs when reacting to

sudden set-point changes. Moreover, as shown by Fig.11a,tracking ofPs is not affected by the steps in reactive power.The reection of those reactive power steps as abruptvariations in rotor current is clearly observable byexamining Figs.11band d.

In order to evaluate the robustness of the proposed globalcontrol scheme, the two tests above are repeated once x, y,wx, y, P, Qand wP, Qare re-tuned by deliberately assumingincorrect values for Lm, Ls and Lr. In particular, Lm isconsidered to be a 50% higher than its actual value inTable 1, which in turn leads to adopt incorrect values for

Fig. 10 Tracking of optimum stator-side power and Qs


Fig. 11 Tracking of optimum stator-side power and reactive power

steps


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both Ls and Lr see their Lm-dependent expressions inTable 1. Moreover, not only those inductances but also Rris deliberately assumed to be incorrect a 50% lower thanits actual value when computing equivalent control terms(16), (17), (32) and (33).

The main results of the rst robustness experiment areshown in Figs.12 and 13. The grid synchronisation detailed

by Fig. 12a, as well as the active and reactive power

tracking displayed in Figs. 13a and b, remain highlysatisfactory in spite of the aforementioned 50% mismatchesin Lm and Rr, hence supporting the robustness of the

proposed global 2-SMC scheme. In addition, the low-powerow between the DFIG stator and the grid reected byFigs.12b and c at instantc corroborates that smoothness of

connection is also preserved. Rotor voltage and currentsignals have not been provided to avoid reiteration with nosignicant added information.

To conclude, the second robustness experiment consists inrepeating the test whose results are provided in Fig.11underthe afore-cited parameter mismatches. Its most representativeresults are shown in Fig.14, where a still excellent tracking of

both the optimum stator-side power and the reactive powersteps can be observed.

5 Conclusions

A PWM-based 2-SMC scheme both for grid synchronisationand power control of a DFIG has been presented. Itallows keeping the tracking accuracy and robustnessfeatures characteristic of standard SMC, while leading to axed switching frequency of the RSC transistors.Experimentation conducted on a 7 kW DFIG test bench

proves that high dynamic performance control and superiorrobustness against DFIG parameter variations are achievedwhen applying the proposed global 2-SMC scheme. Inaddition, bumpless transfer between the gridsynchronisation and power control operating regimes isguaranteed, which results in smooth connection of theDFIG stator to the grid.

A systematic methodology to tune all the parametersinvolved in the presented 2-SMC realisation is also

provided. Its main benet lies in deriving the values ofthose parameters via direct application of tuning equations,hence eluding the generally time-consuming task oftrial-and-error adjustment suggested in [16], which is stillthe most usual practice. The satisfactory experimentalresults obtained support the effectiveness of the proposedtuning method.

6 Acknowledgments

The authors thank Giovanna Santamara, from Jema Energy,for her helpful advice. We are also grateful to ManuLarramendi and J. Ignacio Susperregui for their help in

Fig. 12 Synchronisation and grid connection under 50%

mismatches in Lmand Rr

aSynchronisationbActive powercReactive power

Fig. 13 Tracking of optimum stator-side power and Qs under 50%mismatches in Lmand Rr

aActive powerbReactive power

Fig. 14 Tracking of optimum stator-side power and reactive power

steps under 50% mismatches in Lmand Rr

aActive powerbReactive power

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modifying the 7 kW DFIG test bench. This work has beendeveloped within the Intelligent Systems and Energy (SI+E)research group of the University of the Basque Country UPV/EHU, and has been funded by the Spanish Ministry ofEconomy and Competitiveness, under project codeDPI2012-37363-C02-01 and grant code BES-2008-002563,as well as by the Basque Government (Spain), under researchgrant IT677-13, and the UPV/EHU, under unit of formation

and research UFI11/28.

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