tageoT

oridvee tas

of the RSC into account. To execute the analysis, a simplied DFIG model with decoupled stator and rotor uxes is presented, and

www.ietdl.orgthe low-voltage ride through (LVRT) problem can be formulated as an optimisation problem, which intends to suppress the rotorwinding currents with voltage constraints. The Pontryagins minimum principle is employed to solve the optimisation problemand the results can identify the control limit of the RSC. A case study based on a typical 1.5 MW DFIG-based WECS undervarious grid voltage dips is carried out to validate the analytical method. The proposed method is also further veried byexperimental tests on a scaled 3 KW DFIG system. The results are expected to help the manufacturers to assess and improvetheir RSC controllers or LVRT measures.

1 Introduction

Owing to the rapid growth of wind power penetration, manycountries have revised their grid codes in order to ensure thestable operation of the power system. In the new grid codes,the wind energy conversion system (WECS) is required toremain connected to the grid even with grid faults, which isknown as the so-called low-voltage ride through (LVRT)capability [1].The doubly-fed induction generator (DFIG)-based WECS

is widely used in the world and its typical conguration isshown in Fig. 1. In such concept, the stator is connectedto the grid directly while its rotor is integrated into thegrid via a back-to-back converter. When grid fault occurs,the stator ux would contain dc and negative sequencecomponents, which can induce large electromotive force(EMF) in the rotor circuit. Without proper protectionscheme, the generator rotor will suffer from overcurrents,which may even destroy the rotor-side converter (RSC) [2].To avoid such problem, the crowbar circuits werecommonly used to short circuit the rotor windings andbypass the RSC during the grid faults [35]. This kind ofmethod can work well even under severe grid faults.However, the DFIG will lose controllability and absorblarge amount of reactive power from the grid when thecrowbar is activated. Such property can even aggravate thegrid faults. At the same time, the electromagnetic torque of

the DFIG oscillates dramatically, which will pose greatpress on the drive train [6, 7]. Therefore for less severefaults, instead of activating the crowbar, it is preferred toride through the faults with advanced control of the RSC [8].As analysed in [9], the short-circuit currents of the DFIG

can reach its peak value in 1/43/4 grid period after thegrid fault occurs. During this crucial period, the RSC mustbe controlled properly, so that the rotor-side short-circuitcurrents can be suppressed effectively. However, suchobjective cannot always be achieved. Usually, thecommercial converter for the DFIG is voltage source basedand the output voltage of the RSC is restricted by thedc-link voltage. If the voltage dips are severe, the EMFinduced in the rotor winding can be too large to becounteracted by the output voltage of the RSC. In suchcase, the overcurrent cannot be limited with the control ofRSC. Therefore there is a theoretical control limit for theRSC under voltage dips.As presented in [818], several improved RSC control

methods have been proposed for the LVRT of DFIG-basedWECS. However, the control limit of RSC is not clear andhas not been explored completely. This paper proposes amethod to nd out the control limit of RSC under voltagedips. First, the control limit calculation is formulated to bean optimisation problem with practical constraints. Then,such optimisation problem is solved with Pontryaginsminimum principle (PMP) and the control limit can bePublished in IET Renewable Power GenerationReceived on 26th December 2011Revised on 23rd October 2012Accepted on 14th November 2012doi: 10.1049/iet-rpg.2011.0348

Analysis of the control limiof doubly fed induction genconversion system under vShuai Xiao1, Hua Geng1, Honglin Zhou2, Gen1Department of Automation, Tsinghua University, Beijing, P2DEC Central Academy, Intelligent Equipments and ControlE-mail: xiaos09@mails.tsinghua.edu.cn

Abstract: For the grid-connected doubly-fed induction generatimproved control algorithms have been developed for the rotor-sunder voltage dips. However, such objective can hardly be achieRSCs output voltage. An analysis tool is proposed to estimate thcircuit rotor currents during grid faults in this study. The tool is bIET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348ISSN 1752-1416

for rotor-side convertererator-based wind energyrious voltage dipsYang1

ples Republic of Chinaechnology Institute, Chengdu, Peoples Republic of China

(DFIG)-based wind energy conversion system (WECS), manye converter (RSC) to suppress the overcurrents in the rotor-sided under severe grid fault conditions because of the limitation ofhe theoretical control limit of the RSC in suppressing the short-ed on the optimisation theory and takes the practical constraints71& The Institution of Engineering and Technology 2013

www.ietdl.orgobtained. The analysis results of the control limit can serve asa guideline to assess the performance of different LVRTmethods, optimise the capacity of the RSC or determine theactivation time of the crowbar for the turbine manufacturersetc.This paper is organised as follows. First, a simplied ux

model of DFIG is derived for the subsequent transientanalysis. Based on the model, the analysis of the controllimit of RSC under voltage dips is then formulated as anoptimisation problem with constraints. Afterwards, theprocedure to solve the optimisation problem utilising PMPis discussed. Then, a simulation case study of the typical1.5 WM DFIG-based WECS is presented to evaluate thecontrol limits for different types of grid faults. Finally,experimental tests are carried out to further verify theproposed analysis.

2 Simplified flux model of DFIG for transientstudy

Selecting the stator and rotor uxes as state variables, andfollowing the generation convention, the ux model ofDFIG in synchronous dq-reference frame can be expressed as

dc/dt = Ac+ ui = L1c

{(1)

where c = [csd , csq, crd , crq]T, u = [usd , usq, urd , urq]T,i = [isd , isq, ird , irq]T are the ux, voltage and current ofDFIG, respectively. The superscript T denotes transpose.The primed variables represent the quantity referred to thestator-side throughout this paper. The matrix A is given by

A =vs vs vsm 0vs vs 0 vsmvrm 0 vr vsr0 vrm vsr vr

(2)

where s and sr are the stator and the slip angular frequency.vs and v

r reect the damping speed of the dc component of

the stator and rotor uxes, respectively, and they are denedas

vs = RsLr/ LsLr L2m( )

vr = RrLs/ LsLr L2m( ) (3)

where Rs, Rr, Ls, L

r, Lm are the stator resistance, rotor

resistance, stator inductance, rotor inductance and mutualinductance, respectively. vsm and v

rm reect the coupling

strength between rotor ux and stator ux, and they are

Fig. 1 Conguration of DFIG-based WECS72& The Institution of Engineering and Technology 2013dened as

vsm = RsLm/ LsLr L2m( )

vrm = RrLm/ LsLr L2m( ) (4)

The inductance matrix L is given by

L =Ls 0 Lm 00 Ls 0 LmLm 0 L

r 0

0 Lm 0 Lr

(5)

L1 = 1LsL

r L2m

Lr 0 Lm 00 Lr 0 Lm

Lm 0 Ls 00 Lm 0 Ls

(6)

The mechanical equation is ignored since the speed of therotor of DFIG changes little during LVRT because of thelarge inertia of the mechanical system [3].Since Rs and R

r are very small, the couplings between

stator and rotor ux, vsm and vrm are very weak [19],

which will be ignored in the analysis. Therefore matrix Acan be approximately expressed as

Avs vs 0 0vs vs 0 00 0 vr vsr0 0 vsr vr

= As 00 Ar

[ ](7)

The poles of the system, that is, the eigenvalues of A are givenby ps1,2 = vs+ jvs and pr1,2 = vr+ jvsr, whose realparts reect the damping speed of the dc component of theuxes, and the imagine parts reect the angular frequencyof the uxes. Note that the matrix with block diagonalimplies that the ux responses of the stator and rotor aredecoupled, that is, similar to that of two independentsecond-order systems. Therefore the state-variable model(1) can be simplied as

dcs/dt = Ascs + usdcr/dt = Arcr + uris = k rcs + kmcrir = kmcs kscr

(8)

where cs, cr, us, u

r, is, i

r are the stator ux, rotor ux,

stator voltage, rotor voltage, stator current and rotor current,respectively. The parameters ks, km, k

rare dened as

ks = Ls/(LsLr L2m)km = Lm/(LsLr L2m)k r = Lr/(LsLr L2m)

(9)

The block diagram of the simplied model is illustrated inFig. 2, where the dashed lines indicate that the couplingsbetween the stator and rotor uxes are removed. TheIET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348

www.ietdl.orgsimplied model is of high precision during the rstfundament circle after the faults [19, 20].

3 Control limit analysis of RSC under voltagedips

3.1 Optimisation problem formulationof the control limit analysis

To realise LVRT for DFIG-based WECS, it is critical toreduce the short-circuit rotor current during the rst fewcycles after voltage dips. To achieve such objective, theoutput voltage of RSC has to be properly adjusted tocounteract the large EMF induced in the rotor circuit.However, since the dc-link voltage of RSC is kept almostconstant during the process of LVRT [4, 21], the rotorvoltage is restricted because of the buck inherence ofconverter. The maximum amplitude of rotor voltage can beexpressed as Urmax = kmaxUdc, where kmax = 1/2 for SPWMmodulation, and kmax =

NameMeNameMe3

/3 for SVPWM (kmax is slightly

larger if overmodulation is considered). If the grid faultis so severe that the induced EMF is too large to becounteracted by the RSC, overcurrent will occur regardlessof which the specic RSC control method is used.Therefore it is understandable that there exists a controllimit for RSC in suppressing the short-circuit rotor currentunder voltage dips.Intuitively, the analysis of the control limit is an

optimisation problem in fact, that is, how to minimise therotor current with the restricted rotor voltage during gridfaults. Therefore the analysis of the control limit is thenformulated as a rotor current suppression optimisationproblem with rotor voltage constraints, as

minur Ulim

J ur[ ] = tf

0ir 2 dt (10)

where J[] is the cost function with rotor voltage as variables,|||| is the Euclidean norm of a vector, Ulim is the constraintof the amplitude of rotor voltage and [0, tf] is the interestedtime interval. The optimisation problem means that, duringthe interested time interval, the rotor voltage is optimallyexplored within the constraint range to minimise theamplitude of the short-circuit rotor current.

3.2 Mathematic expression of the optimisationproblem

The expression of stator ux has to be derived rst in orderto obtain the mathematic expression of the optimisation

Fig. 2 Block diagram of the simplied model of DFIGIET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348problem. From (8), the stator ux equation of DFIG isrewritten as

dcs/dt = Ascs + us (11)

It shows that the stator ux is decided by stator voltage. Whenvoltage dips occur, regardless of the types of voltage dips, thestator terminal voltage can be decomposed intopositive-sequence, negative-sequence and zero-sequencecomponents according to the symmetrical componentstheory [22]. Assuming that a fault happens at t = 0, thestator voltage space vector oriented on positive-sequencestator voltage can be given in the form of the sum ofsymmetrical components as

us(t) = 10[ ]

Us1 + cos(2vst + wu0) sin(2vst + wu0)[ ]

Us2 (12)

where t is the time, us(t) is the stator voltage space vector ofDFIG, Us1 is the amplitude of positive-sequence vector andUs2 is the amplitude of negative-sequence vector, which iszero for symmetrical faults. ju0 represents the initial spaceangle between positive- and negative-sequence voltagevectors, and its value is variable in the range of [0, 2)depending on the fault occurrence time.The positive- and negative-sequence stator voltages create

positive- and negative-sequence stator ux vectors,respectively, and zero-sequence stator voltage creates noux vector. For asymmetrical faults, different values of ju0lead to different values of j0, which is the initial apaceangle between positive- and negative-sequence stator uxvectors. Neglecting the stator resistance, the positive- andnegative-sequence stator ux vectors s1, s2 in thesteady-state condition can be expressed as

cs1 =us1jvs

cs2 = us2jvs

(13)

where us1, us2 are the positive- and negative-sequence statorvoltage vectors. So, the relation between ju0 and j0 canbe obtained

wc0 = wu0 + p (14)

With regard to a certain type of asymmetrical grid faults, if thedepths of voltage dips are the same, the amplitudes of theinduced positive- and negative-sequence stator ux vectorwill keep constant. However, the dc component of statorux vector is variable because of the difference of j0. Nodc component is induced if j0 = 0, whereas the dccomponent is of largest amplitude if j0 = [2, 23].Substituting (12) into (11), the expression of stator ux can

be obtained as

cs(t) = T1(t)cs(0)+ T2(t)U s1(0+ )+ T3(t)U s2(0+ ) (15)

and

T1(t) = evst cos(vst) sin(vst)sin(vst) cos(vst)[ ]73& The Institution of Engineering and Technology 2013

www.ietdl.orgT2(t) =1

v2s + v2s(1 evst cos(vst))

vsvs

[ ](

+ evst sin(vst)vs

vs

[ ])

T3(t) = 1+A2s4v2s

( )11

2vs

sin(2vst + wu0)cos(2vst + wu0)

[ ]((

evst sin(vst + wu0)cos(vst + wu0)

[ ])

+ As4v2s

cos(2vst + wu0)sin(2vst + wu0)

[ ](

+ evst cos(vst + wu0)sin(vst + wu0)[ ]))

where s(0) is the pre-fault value of the stator ux,Us1(0+), Us2(0+) are the amplitudes of positive- andnegative-sequence stator voltage vector, respectively.It can be seen that the expression of stator ux vector

consists of three parts: the rst part T1(t)s(0) representsthe zero-input response, and it is related to the pre-faultvalue of stator ux s(0); the second part T2(t)Us1(0+)represents the zero-state response of the positive-sequencestator voltage, and it depends on the amplitude of positive-sequence stator voltage; the third part T3(t)Us2(0+)represents the zero-state response of the negative-sequencestator voltage, and it is not only decided by the amplitudeof negative-sequence stator voltage, but also by the initialspace angle ju0. For symmetrical faults, the third part is zero.During the transient analysis for LVRT, the following

assumptions are made:

1. No extra protection (e.g. crowbar) is activated.2. The dc-link voltage of RSC is well controlled by thecoordination of grid-side converter (GSC) and the dc-linkchopper during faults [4, 21].3. The peak value appears in the rst fundament circle afterthe instant of faults occurrence [3, 19]; so tf = 2/s.

Substitute (8) and (15) into (10), the mathematic expressionof the optimisation problem can be nally expressed as

minur Ulim

J [ur]=tf0

k2scTr c

r 2kskmcTscr+ k2mcTscs

( )dt

(16)

s.t.

cs(t)= T1(t)cs(0)+T2(t)U s1(0+)+T3(t)U s2(0+)dcr/dt=Arcr+urcr(0

+)=cr(0)tf = 2p/vs

3.3 Solution of the optimisation problem

PMP [24] is a very effective tool to cope with optimisationproblem with constraints. With respect to the optimisationproblem (16), a form of PMP that intends to solve thenon-autonomous optimisation problem with integral cost,74& The Institution of Engineering and Technology 2013constrained controls, xed terminal time and free terminalstate is adopted [25].The optimal solutions of u(t) and cr(t) are denoted by the

optimal variable u(t) and the optimal state trajectory cr (t)(t will be omitted in the following part for simplicity).According to (31), the Hamiltonian for this optimisationproblem can be expressed as

H = k2scTr cr 2kskmcTscr + k2mcTscs + lT Arcr + ur( )

(17)

where is a covariate. According to the minimum condition(33), ur satises

H(t, cr , ur , l

) = minur Ulim

H (t, cr , ur, l

){ }

(18)

Note that this optimisation problem is with only one optimalvariable. Substituting (17) into (18), it can be found thatsolving (18) is equal to the solution of the followingconvex optimisation problem

argminur

lTur

s.t. ur Ulim

(19)

Applying the CauchySchwartz inequality and incorporatingthe constraints condition, it can be obtained that

lTur l ur Ulim l (20)

Then, we can obtain

lTur Ulim l (21)

This equality is satised if and only if ur points to theopposite direction of * and ||ur|| =Ulim. Therefore thesolution of (19) can be given by

ur = Uliml

NameMeNameMeNameMeNameMeNameMeNameMeNameMelTl

(22)

Substituting this equation into (17) to eliminate ur, theHamiltonian can be further expressed as

H = k2scTrcr 2kskmcTscr + k2mcTscs + lTArcr Ulim

NameMeNameMeNameMeNameMeNameMeNameMeNameMelTl

(23)

Using the canonical equation (30), there is

dl/dt = ATr l+ 2ks(kmcs kscr)dcr/dt = Arcr Uliml/

NameMeNameMeNameMeNameMeNameMelTl

{

(24)IET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348

www.ietdl.orgwith the boundary conditions as

l(2p/vs) = 0cr(0) = cr(0)

{(25)

By solving the differential equations (24) with the boundaryconditions (25), the optimal state trajectory cr and theoptimal covariate * can be gained. Then substitute thevalue of * into (22), the optimal variable ur can beachieved. The minimised rotor current ir can be obtainedby substituting the value of cr and s into (8). Then thecontrol limit of RSC can be analysed based on the obtainedresults. So far, the theoretical derivation of the proposedanalytical method is completed, and the owchart isillustrated in Fig. 3. In the next section, a case study of atypical 1.5 MW DFIG-based wind turbine is carried out.

4 Case study

In this section, a case study of a typical 1.5 MW DFIG-basedWECS is carried out in MATLAB based on the proposedanalytical method using optimisation theory. Theparameters of the system are listed in the Appendix. Thecontrol limits under various voltage dips, including bothsymmetrical and asymmetrical voltage dips, are evaluated,and further the safe operation regions are depicted. Finally,a brief comparison of the situations under various voltagedips is shown.

4.1 Control limit analysis for symmetrical faults

Initially, the rotor speed of the turbine r is 1.2 pu, and themechanical power produced by the turbine is 0.67 pu,whereas the reactive power output of the stator of DFIG is0. Assuming that a three-phase symmetrical voltage dipwith 60% depth happens at t = 0.1 s. Applying the proposed

Fig. 3 Flowchart of the control limit analysisIET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348analytical method, the calculated optimal variable ur isshown in Fig. 4, and the amplitude of the correspondingrotor current irm is shown in Fig. 5. Note that u

rd , u

rq

are the optimal values of urd , urq, and u

rm is the amplitude

of the optimal rotor voltage. To validate the analysedresults, the three-phase rotor current response of thefull-order DFIG system under the optimal rotor voltage uris shown in Fig. 6.Fig. 4 shows that the amplitude of rotor voltage is always

kept at its constraint during the interested time interval,which implies that the rotor voltage constraint is optimallyused to suppress the rotor current. As a result, the rotorcurrent is well suppressed as shown in Fig. 5, and theamplitude of peak value is a little < 2 pu. If the threshold,namely the maximum current allowed by RSC is 2 pu, itmeans that the fault is possible to be ride through withthe proper control of RSC. Otherwise, some additionalprotection circuits, such as crowbar, have to be activated.From Fig. 6, it can be seen that the time-domain responseof the full-order DFIG system coincides well with thecalculated result as shown in Fig. 5, which validates theanalysed results.The pre-fault states, including the input mechanical power

from the turbine PWT and the output reactive power of statorQs, affect the dynamical response of DFIG during faults.Usually, when the grid is normal, the DFIG is under MPPT

Fig. 4 Optimal value of rotor voltage

Fig. 5 Calculated amplitude of the corresponding rotor current75& The Institution of Engineering and Technology 2013

www.ietdl.orgcontrol, and the reactive power is controlled to regulatethe grid voltage or the power factor [26]. In the case study,the MPPT curve of the system is shown in Fig. 7, and thereactive power output is assumed to vary between 0.2 and0.2 pu. Once the pre-fault states are known, the uniqueoperating point can be determined. Then applying theproposed optimisation method, the rotor current undercertain depth of voltage dip can be calculated. In each case,the amplitude of the maximum rotor current irm max can beextracted. Finally, the control limit of the RSC isrepresented by a collection of maximum rotor currentsurfaces as shown in Fig. 8, in which a surface correspondsto a certain depth of voltage dip. The percentage labelled inthis gure is the depth of the voltage dip. The guresindicate that the control limit tends to be larger withincreased initial active and reactive power output, whichmeans that it becomes more difcult for the RSC toride-through the fault. This is because that increased activepower output corresponds to higher speed with MPPTcontrol, and the induced EMF during faults becomes largerat higher speed. Moreover, increased active and reactivepower output mean larger pre-fault rotor current, whichmakes overcurrent more prone to appearing in the presenceof grid faults.Further, with the control limits achieved, the safe operation

regions by the control of RSC can be depicted. If thethreshold is 2 pu, the safe operation regions are shown in

Fig. 6 Rotor current response of the full-order DFIG system underthe optimal rotor voltage

Fig. 7 MPPT curve of the DFIG76& The Institution of Engineering and Technology 2013Fig. 9. The white coloured regions depict situations whenthe DFIG may successfully ride through the grid faults withthe proper control of RSC, and the grey coloured regionsmean that LVRT cannot be realised with the control ofRSC. For the latter cases, some other protection measures,for example, triggering the crowbar circuit, have to betaken. If the capacity of RSC is improved to 1.5 times, asFig. 10 shows, the safe operation regions become largersignicantly, which means that the RSC are able tocope with more severe faults. With this information,manufactures can make a good compromise between theLVRT performances and the cost according to local gridcodes.

4.2 Control limit analysis for asymmetrical faults

In this section, three types of asymmetrical faults areanalysed, including single-phase-to-ground faults, phase-to-phase faults and two-phase-to-ground faults. Different fromsymmetrical faults, the stator ux would contain not onlydc component, but also negative-sequence componentduring asymmetrical faults. First single-phase faults areconsidered. When single-phase faults occur, voltage dipswill appear in one phase, and phase A is taken for example.

Fig. 8 Control limits for three-phase faults

Fig. 9 Safe operation regions of three-phase faults (2 puthreshold)IET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348

www.ietdl.orgAssuming that positive- and negative-sequence networkshave the same impedance, the voltages of phases B and Cremain unchanged. Then the phase voltage of phases ACare given by

Ua = U (1 p)Ub = Ua2

U c = Ua(26)

where U is the amplitude of pre-fault phase voltage, p isthe depth of voltage dips, a = 1120 = ej(2/3). Then thepositive-, negative- and zero-sequence components of statorvoltage are expressed as

U1U2U0

= 1

3

1 a a2

1 a2 a1 1 1

U (1 p)Ua2

Ua

= U 1 p/3p/3

p/3

(27)

By virtue of (11), the stator voltage space vector undersingle-phase faults can be expressed in dq synchronousreference frame as

us(t) =1

0

[ ]Us(1 p/3) +

cos(2vst + wu0) sin(2vst + wu0)

[ ]Us(p/3)

(28)

As mentioned in Section 2, for asymmetrical faults, thedynamical response of DFIG is related to j0. To comparethe dynamical response of DFIG with different j0, agroup of simulations is done, as shown in Fig. 11. In thissimulation, the rotor voltage is kept unchanged during thefaults. The results show that if j0 = , the rotor overcurrentis largest, whereas if j0 = 0, the rotor overcurrent issmallest. This is well consistent with the theoretical analysis.To ensure that LVRT is feasible for cases with different

j0, only the worst situation j0 = is consideredthroughout the remainder of the analysis. For a single-phasevoltage dip with a 60% depth, using the proposed

Fig. 10 Safe operation regions of three-phase faults (3 puthreshold)IET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348optimisation method, the calculated optimal variable ur isshown in Fig. 12, and the corresponding irm is shown inFig. 13. Similar to the symmetrical faults, the amplitude ofrotor voltage is kept at its constraint to minimise the rotorcurrent. The time-domain response of the full-order DFIGsystem under the optimal rotor voltage is shown in Fig. 14,which also agrees well with the analysed result.

Fig. 11 Dynamical response of DFIG under single-phase faultswith different j0

Fig. 13 Calculated amplitude of the corresponding rotor current

Fig. 12 Optimal value of rotor voltage77& The Institution of Engineering and Technology 2013

www.ietdl.org

The obtained control limits for single-phase faults are

shown in Fig. 15, and the safe operation regions withthreshold 2 pu are shown in Fig. 16. The meanings of thegures are identical with the symmetrical case, and areomitted here for brevity. The similar conclusions to thesymmetrical case can be derived, which are not repeated here.Using the similar method, the cases of phasephase and

two-phase faults can also be analysed. The detailed processis not presented here for brevity. The control limits forphasephase faults and two-phase-to-ground faults areshown in Figs. 17 and 18, respectively.

4.3 Comparison of different grid faults

In this section, a brief comparison of the four different typesof faults is carried out. Applying the proposed optimisationmethod, the amplitudes of the minimised rotor currentunder different types of voltage dips with 60% depth areshown in Fig. 19. Also, only the worst situation j0 = isconsidered for asymmetrical faults. It can be seen that, thepeak value of short-circuit rotor current is largest underthree-phase faults, the second is phasephase faults, thethird is two-phase faults and the last one is single-phasefaults. It implies that for the typical 1.5 MW DFIG-basedWECS, the three-phase faults are the most difcult for the

Fig. 14 Rotor current response of the full-order DFIG systemunder the optimal rotor voltage

Fig. 15 Control limits for single-phase faults78& The Institution of Engineering and Technology 2013RSC to ride through, whereas the single-phase faults arethe easiest. This conclusion can also be derived by acomparison of the calculated control limits under differentfaults obtained before in this section.

5 Experiment verification

From the analysis in Section 3, it is known that, since thecontrol limit calculated by the proposed analytical methodis obtained using optimisation method, it should be the bestresult that can be achieved for the specic cost functionwith the rotor voltage constraint. In this section, to furtherverify the proposed analytical method, the calculatedcontrol limits is compared with the experiment results ofthe ux linkage tracking control strategy as presented in[17] on a scaled 3 KW DFIG system. The ux linkagetracking control strategy has been proven to be effective tosuppress the rotor current under various voltage dips bycontrolling the RSC [17]. The experiment system has beenintroduced in detail in [17], and it is not repeated here forbrevity.Initially, the DFIG is under the VC control, with 0.2 pu

active power output and 0 pu reactive power output. Whengrid faults occur, the control strategy switches to the uxlinkage tracking method immediately on detecting the

Fig. 17 Control limits for phasephase faults

Fig 16 Safe operation regions of single-phase faultsIET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348

www.ietdl.orgfaults. The slip of the DFIG is assumed constant and xed at0.2 by the prime mover controller during the faults. Bothsymmetrical and asymmetrical faults are examined and onlysingle-phase faults are studied here representatively forasymmetrical faults.The experimental results for a symmetrical fault and a

single-phase fault both with a dip depth of 90% are shownin Fig. 20. It is shown that the short-circuit rotor currentcan be reduced effectively with the ux linkage trackingcontrol strategy. The control limits of the 3 KW DFIGsystem calculated by the proposed method are alsoindicated in Fig. 20. The calculated control limit for thesymmetrical fault is 1.23 pu, and for single-phase fault is1.40 pu. It can be seen that, although the ux linkagetracking control strategy is effective to suppress theshort-circuit rotor current, the maximum amplitude of therotor current response can still not be smaller than thecalculated control limit. This result supports the proposedanalytical theory, since the calculated control limit is theoptimised result. It is also partly because of the delayexisting in the real control system, which may degrade thecontrol performance slightly. On the other hand, the controllimit can be used to evaluate the performance of the controlstrategy. In Fig. 20, it is also shown that the maximumamplitude of the rotor current response is very close to thecontrol limit, and it can be derived that the control

Fig. 19 Amplitudes of the minimised rotor currents under differenttypes of voltage dips

Fig. 18 Control limits for two-phase faultsIET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348performance of the ux linkage tracking method ispretty good.

6 Conclusions

For DFIG-based WECS, there is a theoretical control limit forthe RSC in suppressing the short-circuit rotor current becauseof the limitation of dc-link voltage. To analyse the controllimit, the rotor current suppression problem is formulated asan optimisation problem with rotor voltage constraint. PMPis successfully applied to solve the optimisation problem.A case study of a typical 1.5 MW DFIG-based WECS iscarried out to validate the analytical method quantitatively.Applying the proposed method, the control limit of RSCunder various voltage dips can be evaluated and nallyrepresented by a collection of maximum short-circuitcurrent surfaces over the operation area. With the result, thesafe operation regions can be readily worked out, thusallowing different LVRT strategies to be designedpredictably. Experimental tests are also carried out tofurther verify the proposed analytical method. The proposedmethod can be used to evaluate the performance of existingcontrol systems, and is also useful to optimise the design ofRSC capability and LVRT controllers.

7 Acknowledgments

The authors acknowledge the support from the NationalNatural Science Foundation of China (Grant numbers60974130 and 61273045) and the Power ElectronicsScience and Education Development Programme of DeltaEnvironmental and Educational Foundation.

Fig. 20 Experiment results of the ux linkage tracking controlstrategy and the calculated control limits

a 90% symmetrical faultb 90% single-phase fault79& The Institution of Engineering and Technology 2013

www.ietdl.org

8 References

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17 Xiao, S., Yang, G., Zhou, H., Geng, H.: A LVRT control strategy basedon ux linkage tracking for DFIG-based WECS, IEEE Trans. Ind.Electron. accepted

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9 Appendix

9.1 Pontryagins minimum principle

For the following problem

minu(t)[V

J[u] =tft0

L(t, x(t), u(t)) dt

s.t. dx(t)/dt = f (x(t), u(t))x(t0) = x0t [ [0, tf], with fixed tf

(29)

where x Rn is the state variable, u Rm is the control variable,, Rm is the class of admissible controls, assume that

1. f (x, u) is continuous with respect to x, u.2. f (x, u) has a continuous derivative with respect to x andlocally Lipschitz in u.

If u*(t) is the optimal variable, then there exists a non-zero,absolutely continuous co-state function *(t) such that foralmost all t [0, tf], the optimal variable u*(t), the optimalstate trajectory x*(t) and *(t) satises the following conditions:

1. Canonical equation

dl(t)

dt= H(t, x(t), u(t), l(t))

x(t)dx(t)

dt= H(t, x(t), u(t), l(t))

l(t)

(30)

where the Hamiltonian H(t, x(t), u(t), (t)) is dened as

H(t, x(t), u(t), l(t)) = L(t, x(t), u(t))+ lT(t)f (x(t), u(t))(31)

2. Boundary conditions

l(tf) = 0x(t0) = x0

{(32)

3. Pontryagin minimum condition

H(t, x(t), u(t), l(t)) = minu(t)[V

H(t, x(t), u(t), l(t)){ }

(33)

4. The Hamiltonian is constant over the optimal trajectory

H (t, x(t), u(t), l(t)) = H(t, x(tf), u(tf), l(tf))= const (34)

9.2 System parameters

System frequency: 50 Hz;Nominal capacity of DFIG: 1.67 MVA;Rated stator voltage: 690 V;Pairs of poles: 2;IET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.0348

Turn ratio: 1: 3;Stator resistance: Rs = 0.007 pu, linkage inductance:Lls = 0.171 pu;Rotor resistance: Rs = 0.007 pu, linkage inductanceLls = 0.156 pu;

Excitation inductance: Lm = 2.9;Rotational inertia: H = 4.5s;Stator-rated current (base value): 1105 A;Rotor-rated current (base value): 476 A;dc-link rated voltage: 1200 V.

www.ietdl.orgIET Renew. Power Gener., 2013, Vol. 7, Iss. 1, pp. 7181doi: 10.1049/iet-rpg.2011.034881& The Institution of Engineering and Technology 2013