Int ern at i onal  Journ al of  P ower   El ectr on i cs a n d  Drive  S ystem s   ( IJ PEDS )   Vo l.   12 ,  No.   1 ,  M a r 202 1 , p p.  58 5 ~ 59 6   IS S N:  20 88 - 8694 ,  DOI: 10 .11 591/ ij peds . v12.i 1 . pp58 5 - 59 6       585       Journ al h om e page :  http: // ij pe ds .i aescore.c om   Augmen ted   robu st   T - S   f uzzy   cont ro l   bas ed   PM SG   wind   turbin e   improve d   wi th   H ∞   pe rform ance       Naoual   Ti dja ni 1 ,   Abder rez ak   Guess ou m 2   1 Djil ali   Boun aama   Khe mi s   Mil i ana   Univ ersit y ,   Khemi s Mil ia na ,   Alger i a   2 LATSI,   Univ er sity   of   B li da ,   B lida ,   Alg eria       Art ic le   In f o     ABSTR A CT     Art ic le   hist or y:   Re cei ved   A ug   2 7 ,   20 20   Re vised   Jan  12 , 2021   Accepte d   Ja n   30 ,   20 21       In   thi s   pap er,   an   im prove d   aug me nt ed   T aka gi - Sugeno   fuz zy   c ontrol   design   appl i ed   to   the   s ystem   of   conve r ti ng   wind   turbi n e   ene rgy   was   pr oposed.   The   wind   gen erator   used   is   base d   on   a   p erm an ent   ma gne t   synch ro nous   wind   power   gene r at or   (PM SG )   under   var ying   op erati on   of   the   wind   spee d.   The   proposed   T - S   fuz zy   con trol   str ategy   aims   to   m a xim ize   wind   en erg y   in   low   wind   spee d .   A   par t   of   our   con t ribut ion   lies   in   the   li m itati on   of   the   power   output   of   the   wi nd   gene r at or   in   high   wind   spee d .   Through   the   co nce pt   of   th e   virt ual   desire d   var ia b le s,   the   d esign   of   the   ou tput   tr ac king   c ontrol ler   is   ac hi eve d .   In   li g ht   of   thi s   con c ept ,   the   dev el o ped   T - S   fu zz y   cont rol   was   designe d   v ia   p ara l le l - d istri but e d   com p ensa ti on   (PD C)   appr oa ch   with   H ∞   per forma n ce.   Suffici en t   cond itions   for   the   st ab il it y   of   th e   c lose d - loop   sys te m   a ffe cted   by   ext ern al   disturb anc es   ar e   p rove d   from   Ly apun ov’s   direct   m ethod   and   th e   fee dba ck   ga ins   of   the   cont rol ler   strat egy   are   d et er mi ned   by   linear   m at r ix   ine qua li ties   ( LMIs)   tool s.   Anoth er   cont ribu ti on   is   in   show ing   th e   robustness   of   the   T aka g i - Sugeno   bas ed   control   stra te gy ,   wi t h   a   foc us   on   a   s et   of   sys te m   par amete rs   with   model   un ce rt aintie s.   Th e   simu l at ion   result s   show   the   high   per forma n ce   of   the   proposed   cont rol le r   str a te gy   for   a   5MW   (PM SG )   obta in ed   through   simul at ion .   Ke yw or d s :   ∞   pe rformance   PM S G   Taka gi - S ug e no   f uzzy   co ntr ol   PD C   a ppr oach     Win d   tu r bin e   This   is   an   open   acc ess   arti cl e   un der   the   CC   BY - SA   l ic ense .     Corres pond in g   Aut h or :   Naoual   Ti djani ,   Djil al i   Bou naa ma   K hemis   M i li ana   U niv e rsity,   Bl ida ,   Algeri a   Emai l:   n. ti djan i@u niv - dbkm. dz       1.   INTROD U CTION   Nowa day s ,   ef f or ts   by   go vern ments   a nd   busi nesses   a re   f urt her   pro mo ti ng   to   pro pel   the   m ass   ad op ti on   of   re new a ble   e nerg y   in   orde r   to   ste er   t he   producti on   of   el ect rici ty   from   cl ean   ene r gy   s ources.   A mon g   these   energies,   t he   r enew a ble   wind   energies   a re   be coming   us e d   widely   in   this   con te xt,   because   it   is   cost - e ffec ti ve,   su sta ina ble,   cl e an,   a nd   do e s   not   pr oduce   gr e enho us e   gases .   Re centl y,   ma ny   wi nd   t urbine   te chnolo gies   hav e   bee n   devel op e d,   wh e re   the   main   goal   is   to   ens ure   a   good   qu al it y   of   el ect rical   po wer,   a nd   an   op ti miza ti on   of   t he   e nerg y   re source   that   is   e xt racted   from   the   wind.   Ther e f or e,   the   ch oice   of   the   mo st   a ppr opri at e   el ect rical   gen erat or   f or   a   wind   tu rb i ne   is   an   im porta nt   ta sk.   Appro pr ia te ly,   the   m os t   co m monly   us ed   ge ner at or s   in   w ind   po wer   generati on,   a re   ba sed   on   a   pe rm anent   mag net   s yn c hrono us   generato r   ( PMSG ),   f or   its   man y   a dv a nt ages   c ompare   to   ot her   mac hin es   [1].   The   P M S G   do e s   not   requir e   el ect rical   e xc it at ion ,   as   t he   mag netic   fiel d   is   pro duced   by   pe rma ne nt   ma gn et s ,   rathe r   t ha n   by   the   coil.   He nc e,   PMSG ’s   does   not   nee d   sli p   ri ng s   nor   brushes,   wh ic h   r edu ce s   the   we igh t,   im pleme ntati on   costs,   mai nten ance   an d   no   fiel d   co pper   l os ses.   Furthe r more,   the re   is   the   possibil it y   to   a vo i d   ge arbo x   connecti on   to   t he   tur bin e .   It   will   res ult   in   hi gh   dynamic   pe rformance ,   an d   hi gh   powe r   with   a   wide   op erati ng   Evaluation Warning : The document was created with Spire.PDF for Python.
            IS S N :   2088 - 8 694   In t J   P ow  Ele c  &  D ri   S ys t,   V ol .  12 , N o.   1 ,  Ma rch  20 21   :   58 5   –   59 6   586   sp ee d   ra ng e   [2 ] ,   [ 3].   As   is   well   known ,   t he   co mp le x   dyna mic   be ha vio r   desc ribing   wind   tu rb i ne   s ys te ms,   su ggest s   r obus t   co ntr ol   strat eg ie s   to   ac hieve   t he   desire d   perf ormance   an d   e ns ure   high   sta bi li ty   [4 ] ,   [ 5].   In   previ ous   fin dings,   ma ny   co ntr ol   meth ods   are   propose d   i nclu ding   li nea r   co ntr ollers   th a t   are   li mit ed   in   the   abili ty,   to   achie ve   acc eptable   c ontrol   pe rformance s.   N on li nea r   co ntr ollers   a re   be tt er   ada pted   to   the   wind   e ne rgy   c onve rsion   sy st em   (PMS G - W ECS)   un der   di f fer e nt   wind   spe ed   va riat ion s   [6].   M a kha d   et   al .   [ 7]   desig n   a   novel   integral   bac ks te pp i ng   strat eg y,   to   opti mize   wind   e nergy   in   the   eve nt   of   a bru pt   cha ng e s   in   wi n d   sp ee d.   Da hbi   et   al .   [8]   cl a rified   the   wi nd   tur bin e   syst em   by   co mb i ning   maxim um   powe r   point   tra ckin g   (MPPT ) - pitch   ang le   co ntr ol,   us in g   arti fici al   neural   netw ork s   to   e nab le   the   netw ork   to   be   su ppli ed   with   s ta ble   and   e ff ic ie nt   el ect rical   ener gy .   In   t he   po wer   sy ste m,   t he   wi nd   farm   is   mor e   sensiti ve   to   disturba nces.   In   this   con te xt,   Che n   et   al .   [9]   de sig ned   MPPT   c ontr ol   sche me   ba sed   on   nonlin ear   ada ptive   c on t ro l   to   est imat e   the   lump e d   per t urbati on   te rm   ac cordin g   to   a   hi gh - gai n   pe rtu r bation   obser ve r.   M ora di   et   al .   [10]   desi gn   a   H ∞     con t ro ll er   to   i mpro ve   the   wi nd   tu r bin e   po wer   wit h   rob ust   sta bili ty   a nd   a   le ss   os ci ll at ory   beh a vior   c ompa red   to   the   cl assic al   PI D   co ntr oller.     M ore ov e r,   fu z zy   lo gic   is   wel l   su it ed   to   m od el li ng   a   no nlin ear   syst em   acc ordin g   to   [ 11 ] - [ 14] .   In   this   con te xt,   Allo uc he   et   al .   [ 15]   de sig ned   a   T akag i -   Suge no   re fer e nce   f uz zy   model,   in   order   to   ge nerat e   the   op ti mal   traj ect ory   co rr es pond ing   to   onl y   t he   ma xim um   po wer.   H ow e ver,   in   that   w ork,   wh il e   al s o   ope rati ng   unde r   ti me   va r ying   wind   s pee d,   the   case   of   s tro ng   wind   is   not   co ns i der e d,   des pite   the   fa ct   that   t he   T - S   fu zz y   con t ro l   ca n   pro vid e   a   wi de   la r ge   of   co ntr ol   ga in   va riat ion.   Ba sed   on   t he   pri nciples   of   the   T - S   fu zz y   co nt ro ll er,   w hich   c an   pro vid e   an   eff ect ive   re pre sentat ion   of   com plex   a nd   nonlinea r   s ys t ems,   i nvolv i ng   s ys te m   unce rtai nties   an d   e xter nal   distu r ba nces,   the   s ug gested   so luti on   in   thi s   work   is   to   a tt enu at e   i nf l ue nce   of   exter na l   distu r ban ces .   In   this   arti cl e   the   trac king   c on t ro l   desig n   base d   on   t he   au gm e nt ed   T – S   f uzzy   model   by   def i ning   a   l umped   pe rtu rbat ion,   is   descr i bed   us i ng   H ∞   performa nce   wh e re   the   trac king   er r or   must   be   e qu al   or   le ss   than   a   prescri bed   at te nuat ion   le vel.   In   this   appr oach,   the   direct - dr i ve   pe rma nen t   mag ne t   wind   tu r bine   sy ste m   is   de scribe d   by   an   aggre gation   of   li near   models,   interc onnected   t hro ugh   mem ber s hi p   f un ct i on s .   T hen,   local   li ne ar   co mp e ns at ors   f or   eac h   s ubsy ste m   to   achie ve   t he   desire d   ob je ct i ve   determi ne   t he   global   co ntr oller   s ys te m.     Unde r   a   wi de   r ang e   of   the   wind   s pee d,   t wo   diff e re nt   opera ti on s   m us t   be   c on si der e d.   The   first   one   is   a   maxim um   po wer   poin t   trac ki ng   ( M PPT )   c ontr oller,   est a blishe d   to   opti mize   the   gen e rato r   s peed.   In   t his   way,   the   capt ur e d   e nerg y   is   set   at   its   maxim um   le vel,   unde r   r at ed   wi nd   s pe ed.   On   the   ot he r   hand,   pitch   an gle   op e rati on   c on t ro l   st rateg y   is   f ocused   in   li mit ing   the   t urbine   outp ut   powe r   a nd   t he   generato r   sp ee d   resp ect ivel y,   e xceed i ng   the   w ind   s peed   rate d   val ue.   F ur t hermo re,   in   our   a ppr oac h,   the   m odel   base d   T - S   f uzzy   con t ro ll er   is   de sign e d   via   pa rall el - distrib ut ed   c ompe ns at ion   ( PD C ),   to   determi ne   t he   sta te   feedbac k   f uzzy   con t ro ll er,   from   the   pro pose d   a ugmente d   T - S   f uzzy   m odel s.   So,   to   imp rove   t he   s ugge ste d   c ontro ll er,   as   a   con t rib ution,   a   sta bili ty   crit e rio n   is   der i ve d   from   L ya punov’s   direct   me thod   a nd,   t he   H ∞   trackin g   c ontr ol   performa nce   c an   the n   be   gu aran te e d   at   the   same   ti me.   In   this   co ntext,   the   fee dbac k   ga ins   are   so l ve d   very   eff ic ie ntly   by   conve x   opti miza ti on   li near   matri x   ine qu al it ie s   (LMIs ).   Also ,   in   orde r   to   pr ov i de   opti mal   dynamic   performa nce   in   te r m   of   c onve rg e nce   an d   r obus t ness,   the   pro pose d   meth od   t akes   al so   the   sy ste m   par a mete r   unc ertai nties   of   t he   mathe mati cal   mo del   int o   acco unt   dur ing   the   desig n   proce ss   as   ano t her   sign ific a nt   c ontribu ti on.     In   t he   f ollo wing,   sect io n   2   i nc lud es   a   T - S   f uzzy   lo gic   des cripti on   f or   the   wind   tu r bin e   mod el   base d   on   the   P MSG .   In   s ect ion   3,   a   trackin g   a ugm ented   T - S   f uzz y   c on tr oller   is   desig ne d   via   t he   par al le l   distri bu te d   com pensat ion   (P DC )   sc heme   an d   the   L yapuno v   sta bili ty   crit erion   is   pr e sented   f or   the   cl os ed - lo op   s yst em   to   pro vid e   e ff ic ie nt   power   c onver si on.   T he n   the   r obus t   t r ackin g   c ontr ol   pe rformanc e   H ∞   is   desc ribe d,   to   gu a ra ntee   the   sta bili ty   of   the   sy ste m   a nd   at te nu at e   t he   dis tur ban ces ,   res pe ct ively.   Linea r   mat rix   i nequ al ity   (L M I )   te ch nique   is   ad opte d   to   so l ve   for   the   fee db ac k   gains   of   the   f uzzy   co ntr oller.   Sect io n   4,   pr ese nts   simulat ion   res ults   ap plied   for   a   5   MW   permane nt   ma gne t   sy nc hro nous   gen e rato r.   The   co nclusi on   an d   f ut ure   sco pe   cl os e   the   arti cl e.       2.   WIN D   E NER GY   CON VER SION   S YS TE M   2.1.   Wind   ener gy   mod el   The   mecha nica l   powe r   harnes sed   by   a   horizo ntal - axis   wind   tur bin e   ca n   be   expresse d   i n  ( 1)   [ 15]:     = 1 2 ( , )  2 3   (1)     wh e re   R t   is   the   r otor   ra diu s ,   ρ   is   the   ai r   de ns it y,   V w   is   the   wind   sp ee d   an d   C p ( λ , β )   is   a   powe r   coe ffi ci ent   pr ese nted   as   a   no nlinear   fun ct ion   that   de pe nd s   on   tip   s pe ed   rati o     an d   t he   pitch   a ng le   of   the   blades   β   as   sh ow n   in   Fi gur e   1.   Evaluation Warning : The document was created with Spire.PDF for Python.
In t J  P ow Elec   & Dri S ys t   IS S N: 20 88 - 8694       Au gm e nted  rob us t T - S f uz zy  c on tr ol  ba se d P MSG wi nd tu r bin e i mprove d  …   ( Na oual  Tid jan i )   587   The   em pirical   and   nonline ar   equ at io n   of   ( , )   ba sed   on   t he   tu r bin e   c har act e risti cs   are   giv e n   by   the   ex pressi on   in   ( 2)   [ 16]:     ( , ) = 0 . 73 ( 151 − 0 . 58 − 0 . 02 2 . 14 − 13 . 2 ) − 18 . 4   (2)   = 1 1 − 0 . 02 − 0 . 003 3 + 1       The   tip   sp ee d   r at io   is   de fine d   as   ( 3) :     = Ω   (3)     wh e re   Ω m   is   the   t urbi ne   s peed.   The   t yp ic al   va riat ion s   of   C p ( λ , β )   from   ( 2 )   with   t he   tip - sp ee d   r at io   λ   f or   va rio us   values   of   pi tc h   ang le   β   are   il lustrate d   in   Fi gure   1.         Figure   1.   Coe ffi ci ent   of   pe rfo r mance   as   a   fun ct ion   of   Tip   Spee d   Ra ti o       The   P M SG   s pe ed   c ontrol   ba sed   on   M P PT   (ma xim um   powe r   point   tra ckin g)   co ntr ol   co ns ist s   in   maximizi ng   t he   powe r   c onve rted.   Th us,   the   op ti mal   s pee d   of   the   t urbine   mu st   be   a dju st ed   as   ( 4)   [ 17 ] ,   [ 18]:     Ω  = λ    (4)     for   eac h   wind   s peed   with   ma xi mu m   = 0 . 44   an d   opti mal   λ = 6 . 89   as   s how n   in   Fig ure   1.   The   pur po se   of   the   MPPT   ca n   there fore   be   pro vid e d   by   set ti ng   t he   el ect romag netic   powe r   ext racte d   from   t he   wind   at   the   ma xim um   val ue   giv e n   by   ( 5)   [ 15]:     P  = 1 2  5 C  3 Ω 3   ( 5 )     Wh e n   the   wind   vel ociti es   are   higher   tha n   rated,   t he   blad es   are   r otate d   by   a   c on t ro l   de vice   base d   on   pitch   con t ro l   as   s hown   in   Fig ur e   2,   to   re du ce   the   aerod yn a mic   powe r   ca pture d   by   the   wi nd   an d   m ai ntain   t he   ou t put   powe r   of   PMS G   at   its   rate d   va lue   [19 ] ,   [ 20].       Evaluation Warning : The document was created with Spire.PDF for Python.
            IS S N :   2088 - 8 694   In t J   P ow  Ele c  &  D ri   S ys t,   V ol .  12 , N o.   1 ,  Ma rch  20 21   :   58 5   –   59 6   588       Figure   2.   Pit ch   an gle   co ntr oller       2.2.   PMS G   m od el     Fo r   a   non - sal ie nt   PM   mac hin e ,   the   dyna mic   model   of   the   P M S G   with   r ota ti on al   to po l ogy   is   giv e n   in   the   d – q   s ynch r onous   fr a me   usi ng   t he   Pa r k’ s   trans f or mati on   as   de fine d   in   ( 6)   [ 16]:     {   = +  − = +  + ( + )   (6)       Wh e re   ,     are   the   sta tor   vo lt ag es   in   the   d - q   a xis,   ,     are   the   c urre nts   in   the   d - q   a xis,   =   is   the   el ect rical   ro ta ti on   s pee d,     is   the   sta tor   resist ance,     the   flu x   li nk age   of   pe r mane nt   ma gn e ts   and   p   is   the   numb e r   of   pole   pairs .   T he   dyna mic   eq uatio n   of   the   wind   t urbine   is   gi ven   by   (7)   [16] :     Ω  = − − Ω   (7)     J   is   eq ual   to   t he   su m mati on   of   inerti a   m om e nts   of   the   tu rb i ne   a nd   the   generator ,     is   the   f rict ion   c oeffici ent ,     an d     prense nts   res petivel y   t he   el ect ro ma gnet ic   and   aer odyn amic   tor que .   The   e quat ion   of   the   el ect roma gn et ic   to r qu e   pro duced   by   t he   machi ne   is   de f ined   as   (8) :       = 3 2   (8)     Using   ( 6 )   an d   ( 7 ) ,   the   dy nam ic   mo de l   of   t he   PMSG - WT   in   d - q   re fer e nce   fr ame   can   be   descr i bed   in   the   f ollo wing   nonlinea r   sta te   s pace   form   ( 9)   [ 24]:     { ̇ ( ) = ( ( ) , ( ) ) + ( ( ) ) ( ) ( ) = ( ( ) )   (9)     wh e re  ∈   is   the   sta te   vecto r,     ∈   is   t he   c on t ro l   i nput   vecto r,   ( )   de no t es   the   e xter nal   disturba nce,   ∈   is   the   mesu re d   outp ut,   ( ) , ( )   an d   ( ) ar e   nonlinea r   with   a ppr opriat e   di mensions.     In   orde r   to   e xpress   t he   nonlin ear   m odel   of   t he   machi ne   as   a   T - S   f uzzy   m odel ,   with   t he   me asur a ble   par a m et ers   as   decisi on   va r ia bles,   ( 6 )   is   de scribe d   in   the   f ollow i ng   no nlinear   sta te   sp ac e   f or m   ( 10)   [ 24 ] ,   [ 25 ]:     { ̇   ( ) = ( Ω ) ( ) +   ( ) +   ( ) ( ) =   ( )   (10)     wh e re     ( ) = [ ( )   ( )   Ω ( ) ]   (11)     ( Ω ) = [         − Ω   0 − Ω   − − Φ 0   3 2 Φ − ]         , = [       1 0 0 1 0 0 ]       , = [ 0 0 − 1 ] , = [ 0 0 1 ] , ( ) = [ ( ) ( ) ]   (12)     Evaluation Warning : The document was created with Spire.PDF for Python.
In t J  P ow Elec   & Dri S ys t   IS S N: 20 88 - 8694       Au gm e nted  rob us t T - S f uz zy  c on tr ol  ba se d P MSG wi nd tu r bin e i mprove d  …   ( Na oual  Tid jan i )   589   2.3.   Takagi - Su geno   (T - S)   f uzzy   mod el   The   T - S   f uzz y   model   of   P MSG   is   est ablis he d   ba sed   on   the   dynamic   model   giv e n   by   ( 9 ) .   Th us,   f or   ens ur in g   the   c on t ro l   of   the   s ta tor   c onve rter   side   gen e rato r,   the   s ys te m   is   a pproximat e d   by   the   T - S   fu zz y   dynamic   m od e l   com posed   of   r   r ules   in   ( 11 ) .   The   i th   r ule   of   t he   f uzzy   m od el   is   as   ( 13)   [ 22 ] ,   [ 23]:        1 ( )    1    …      ( )          ̇ ( ) =   ( ) +   ( ) +  ( )   (13)     = 1 , 2 , … , ;      de no te s   the   f uzzy   set s   a nd   r epr ese nt   the   gr ade   of   mem be r sh ip   of     1 ( )   in   1 .     is   the   num ber   of    … .    ru le   a nd     ( ) =   ( 1 ,   2 , … ,   ) ∈ ℝ are   the   pr e mise   va riables   [24]   A i   B i   are   the   local   s ubsyst em   mat rices.   Using   t he   sin gl et on   f uzzifier,   the   resu lt in g   overall   fu z zy   sy st em   is   in ferred   as   ( 14)  a nd (1 5)     { ̇ ( ) = ∑ ( ( ) ) ( ( ) + ( ) + ( ) ) = 1 ( ) = ∑ ( ( ) )  ( ) = 1   (14)     ( ( ) ) = 1 ( ( ) ) ∑ 1 ( ( ) ) = 1   (15)     Fu rt hermo re,   it   shou l d   be   note d   that   t he   e xpr ession   ( ( ) )   for   al l   i   sat isfy   (16 ) :     ∑ ( ( ) ) ≥ 0 = 1   (16)     With   ∑ ( ( ) ) = 1 = 1   f or   al l   > 0   w her e   ( ( ) ) ≥ 0   for   = 1 , 2 , … ,   Con si der i ng   t he   ro t or   sp ee d   c ho s en   as   prem ise   var ia ble,   t wo   r ules   are   de du ce d   a nd   the   sub - matri ces   can   be   wr it te n   as   ( 17) :     1 = [         − Ω    0 − Ω    − − Φ 0   3 2 Φ − ]         , 2 = [         − Ω    0 − Ω    − − Φ 0   3 2 Φ − ]         ,   (17)     1 = 2 = [       1 0 0 1 0 0 ]       ,   1 = 2 = [ 0 0 − 1 ]       Figure  3  il lustr at es  cl early   th e  pri nci ple  of   the  pro posed   f uz zy  c ontr oller  wh ic h  c orres ponds   to   the   maxim um   ene r gy capt ur e d fro m the win d  a nd  it s li mit at ion   at  the nomi nal  value.   In this  reg a rd, t o  sim plify t he  tracki ng contr ol le r  desi gn, a se t of ne w o ptimal  traje ct ory  is  (18) :      ( ) = [   ( )     ( )   Ω     ( ) ]   (18)   ( ) −  ( ) → 0      → 0     ̃ ( ) = ( ) −  ( )       ̃ ( )   is de f ine d   as  the t ra cki ng   err or   a nd  it s  ti m e  d eri v at iv e  is g ive n   by   (19):     ̃ ̇ ( ) = ̇ ( ) − ̇  ( ) = ∑ ( ( ) ) ( ̃ ( ) + ( ) + ( ) ) − ̇  ( ) = 1   (19)     The ne w  f uzz y co ntr oller  τ ( t ) is sub sti tute d  in  (1 4) is de fine d b y(2 0) :     ∑ ( ( ) ) ( ) = = 1 ∑ ( ( ) ) (  ( ) + ( ) ) − ̇  ( ) = 1   (20)     (19) is re w ritt en  in  the  f ollow i ng compact   f orm (2 1) :     ̃ ̇ ( ) = ∑ ( ( ) ) ( ̃ ( ) + ( ) + ( ) ) = 1   (21)       Evaluation Warning : The document was created with Spire.PDF for Python.
            IS S N :   2088 - 8 694   In t J   P ow  Ele c  &  D ri   S ys t,   V ol .  12 , N o.   1 ,  Ma rch  20 21   :   58 5   –   59 6   590         Figure   3.   Ge ne rator - side   co ntr ol   sc heme       3.   ROBUST   AUGME NTED   T - S   F UZZY   C ONTROL LE R   DESIG N   In   the   ne xt   sect ion ,   this   c ontr ol le r   is   desi gned   f ollow i ng   the   Parall el   distrib uted   co mp e ns a ti on   (PDC)   te chn iq ue   em plo ye d   f or   t he   T - S   f uzz y   m odel   ( 21 )   [25 ] ,   [ 27].     Hen ce ,   the   i ℎ   ru le   f uzzy   co ntr oller   ca n   be   descri bed   as   ( 22) :     :      1 ( )    1    …      ( )           ( ) = − ̃ ( ) ,    = 1 , 2 , … ,   (22)       Wh e re     ∈ ℝ ∗   is   the   l ocal   sta te   feedback   vecto r.   T hen,   the   ove rall   fu zzy   PD C   c on t ro ll er   ( 22 )   is   inferre d   as   ( 23)   [26 ]:       ( ) = − ∑ ( ( ) ) = 1 ̃ ( )   (23)     Substi tuti ng   ( 23 )   into   ( 21 )   yie lds   the   cl os e d - l oop   f uzzy   syst em   as   (24 ) :     ̃ ̇ ( ) = ∑ ∑ ( ( ) ) = 1 ( ( ) ) = 1 ( − ) ̃ ( ) + ( )   (24)     le t   us   denote   the   integ ral   sta te   error   vect or   as   (25) :     ( ) = ∫ ̃ ( )  0   (25)     The   a ugme nted   s ys te m   c onta ining   the   ne w   f uzzy   co ntr oller   is   prese nted   as   ( 26) :     ̃ ̅ ̇   ( ) = ∑ ( ( ) ) ( ̅ ̃ ̅   ( t ) + ̅ ̅ ( ) + ̅ ̅ ( ) ) = 1   (26)     Wh e re   t he   new   fee db ac k   c ontrolle r   is   infe rre d   as   ( 27) :       τ ̅   ( ) = − ∑ ( ( ) ) ̃ ( ) = 1 − ∑ ( ( ) ) ( ) = 1   (27)     Th us ,   ( 27 )   is   re wr it te n   as   ( 28) :       τ ̅   ( ) = − ∑ ( ( ) ) [   ] [ ̃ ( ) ( ) ] = 1     Evaluation Warning : The document was created with Spire.PDF for Python.
In t J  P ow Elec   & Dri S ys t   IS S N: 20 88 - 8694       Au gm e nted  rob us t T - S f uz zy  c on tr ol  ba se d P MSG wi nd tu r bin e i mprove d  …   ( Na oual  Tid jan i )   591   ̅ = [ 0 0 ] , ̅ = [ 0 ] , ̅ = [ 0 ] , ̅ ( ) = [ ( )   0 ]   (28)     Substi tuti ng   ( 28 )   into   ( 26 )   yie lds   the   au gm e nt ed   cl os e d - l oop   co ntinuo us   T - S   f uzz y   m odel   as   ( 29) :     ̃ ̅ ̇   ( ) = ∑ ∑ ( ( ) ) = 1 j ( ( ) ) ( G ̅  ̃ ̅   ( ) + ̅ ̅ ( ) ) = 1   (29)     3.1.   H ∞   tr acki ng   c ontr ol   per fo rm an ce   The   op ti mali ty   crit erio n   by   ∞   con t ro l   perform ance   c on sist s   to   s yn t hesize   a   con t ro ll er   su c h   that   the   equ il ib rium   of   the   cl ose d - loop   sy ste m   ( 29 )   is   sta ble   a nd   the   at te nuat ion   of   e xter nal   distu rb a nces   is   guar anteed   consi der i ng   the   ∞   pe rformance   as   ( 30)   [25 ] ,   [ 27 ]:     ∫ ̃ ̅    ̃ ̅ ≤ ̃ ̅ ( 0 )   ̃ ̅ 0   ( 0 ) + 2 ∫ ̅ 0 ( ) ̅ ( )    (30)     Wh e re     de no te s   the   final   ti mes   an d   δ   is   a   s pecified   distu rb a nc e   co ns ta nt.   In   orde r   to   e ns ure   t he   as ymptoti c   sta bili ty   of   the   au gme nted   cl os e d - lo op   m od el   ( 29 )   gu a ra nteei ng   the   ∞   trackin g   perf ormance   desc rib ed   ( 30 )   f or   al l   disturba nce,   t he   gain s   K i ̅ is   ob ta i ned   f rom   G ij ̅ ̅ ̅ ̅ .     Le mma.   The   a ugmente d   cl os e d - l oop   sy ste m   descr i be d   by   ( 29 )   is   asym pto ti cal ly   sta ble   with   the   pr esc ribe d   ∞   performa nce   i ne qu al it y   ( 30 ) ,   if   a nd   only   if   th ere   e xists   a   ma trix   = > 0   su c h   that   (31 )     [ ̅   +   ̅   − ̅ −   ̅ ̅ − 2 0 0 − ] < 0   (31)     Pro of .   Let   the   L ya punov   f unct ion   f or   the   a ugmente d   cl os e d - loop   s yst em   be   de fine d   as   ( 32)     = ̃ ̅ ̅ ̅ ̅   ( )     ̃   ̅ ( ) > 0   (32)     Takin g   t he   der i vative   V ̇ ( X ̃ ̅   ( t ) )   will   be   require d   to   sat isfy   ( 33)  [23 ] ,   [ 26]     ̃ ̅ ̅ ̅ ̅ ̇   ( )     ̃   ̅ ( ) +   ̃ ̅ ̅ ̅ ̅     ̃ ̅ ̅ ̅ ̅ ̇   ( )   < 0   (33)     ∞   ou t pu t   t rack i ng   ind e x     is   ac hieved   f or   the   res ulti ng   a ugme nt ed   cl ose d - lo op   fu zz y   if   s ys te m   ( 23 )   ho l ds     ̇ ( ̃ ̅   ( ) ) ≤ −   ̃ ̅ ̅ ̅ ̅ ̇   ( )   ̃   ̅ ( ) +   2 ̅ ( )   ̅ ( ) < 0   (34)     On ce   the   f ollo wing   mat rix   i ne qu al it ie s   ( for   al l   i)   are   sat isfi ed,   i.e .     ∑ ( ( ) ) = 1 ( G ̅  ̃ ̅   ( ) + ̅ ̅ ( ) )   ̃   ̅ ( ) + ̃   ̅ ( )   ( G ̅  ̃ ̅   ( ) + ̅ ̅ ( ) ) + ̃ ̅ ̅ ̅ ̅   ( ) ̃   ̅ ( ) −   2 ̅ ( )   ̅ ( ) < 0   (35)     ∑ ( ( ) ) ( = 1 ̃ ̅   ( ) [   ̅ ̅ ̅ + P   ̅ ] ̃   ̅ ( ) + ̅ ( ) [ ̅ ] ̃   ̅ ( ) +   ̅ ̅ ( ) ) −   2 ̅ ( )   ̅ ( ) < 0   (36)     [ ̃ ̅   ( )   ̅ ( ) ]   [ ∑ ( ( ) ) = 1   ̅ ̅ ̅ + P   ̅ +   ∑ ( ( ) ) ̅ = 1 ∑ ( ( ) ) ̅ = 1 −   2 ]   [ ̃ ̅   ( ) ̅ ( ) ] < 0   (37)     [ ∑ ( ( ) ) = 1 ( ̅ + P   ̅ ) +   ∑ ( ( ) ) ̅ = 1 ∑ ( ( ) ) ̅ = 1 −   2 ]   < 0   (38)     Evaluation Warning : The document was created with Spire.PDF for Python.
            IS S N :   2088 - 8 694   In t J   P ow  Ele c  &  D ri   S ys t,   V ol .  12 , N o.   1 ,  Ma rch  20 21   :   58 5   –   59 6   592   Hen ce ,   it   s uffices   to   c hec k   tha t     [           ∑ ( ( ) ) = 1 (   ̅ ̅ ̅ ̅ ̅ + P   ̅ )   ∑ ( ( ) ) ̅ = 1 ∑ ( ( ) ) ̅ = 1 −   2 ]           + [ 0 0 0 ] < 0         (39)     [           ∑ ( ( ) ) = 1 (   ̅ ̅ ̅ ̅ ̅ + P   ̅ )   ∑ ( ( ) ) ̅ = 1 ∑ ( ( ) ) ̅ = 1 −   2 ]           + [ 0 ] [ 0 ] < 0         (40)     [             ∑ ( ( ) ) = 1 ( ̅ + P   ̅ )   ∑ ( ( ) ) ̅ = 1 ∑ ( ( ) ) ̅ = 1 − 2 0 0 − ]             < 0       (41)     [ ̅ + P   ̅   ̅ ̅ − 2 0 0 − ] < 0       (42)     [ − 1 0 0 0 0 0 0 ]   [ ̅ + P   ̅   ̅ ̅ − 2 0 0 − ] [ − 1 0 0 0 0 0 0 ] < 0       (43)     Th us ,   ( 43 )   is   e qu i valent   to   ( 44)     [   − 1 ( ̅ ̅ ̅ ̅ + P   ̅ ) − 1 − 1   ̅ − 1 ̅ − 1 − 2 0 − 1 0 − ]   (44)     Let 's   co ns ide r   = − 1   and   = ̅ − 1   ;   the   c on t rol   gains   are   obta ining   by   so l ving   the   L M I   obta ined   by   ( 45 )     [     ̅ + A ̅ X − ̅   − ̅   ̅   ̅ − 2 0 0 − ]   (45)     3.2.   Virt ua l   disi red   vari abl es   system   By   us in g   the   fa ct   that   g ( t ) = ∑ ( ( ) ) = 1   ( 20 )   is   r ewr it te n   in   the   com pact   f orm   (46)   [ 25]:     ( ) ( ) = ( ) (  ( ) + ( ) ) −  ( )   ( ( ) ) ( ( ) − ( ) ) = ( ( ) ) ̇  ( ) +  ( )   ̇   (46)     wh e re     ( ( ) ) = ( ) , ( ( ) ) = ( )   (47)     Evaluation Warning : The document was created with Spire.PDF for Python.
In t J  P ow Elec   & Dri S ys t   IS S N: 20 88 - 8694       Au gm e nted  rob us t T - S f uz zy  c on tr ol  ba se d P MSG wi nd tu r bin e i mprove d  …   ( Na oual  Tid jan i )   593   [       1 0 0 1 0 0 ]       ( ( ) − ( ) ) = − [         − Ω   0 − Ω   − − Φ 0   3 2 Φ − ]         [  ( )  ( ) Ω    ( ) ] + [   ( ) ̇  ( ) ̇ Ω    ( ) ̇ ]   (48)     In   order   to   pro vid e   the   ma xi mu m   to r qu e   pe r   a mp e re        is   al way s   kep t   at   zer o.   And   f rom   the   m easu re d   op ti mal   s pee d,    ( )   can   be   deduce d   from   ( 48 ) .   Ther e f or e,   the   desire d   sta te   va riables   for   the   tracki ng   co ntr ol   are   set   to   be      ( ) = [ 0 2 Φ ( Ω ̇    ( ) + Ω    ( ) ) Ω    ( ) ]   (49)     The   re qu i red   c ompone nts   of   t he   recti fied   vo l ta ge   vecto r   a re   de rive d   f rom   t he   i nteg ral   c ontrolle rs   f or   eac h   d - q   current   c omp onent.   Co ns e quently,   from   ( 6 )   the   mac hin e   vo lt age s   are   desc ribe d   as   ( 50) :     { = − Ω   +  +   ̇ + = Φ Ω  +  + Ω  +   (50)       4.   RESU LT S   A ND   DI SCUS S ION   M at la b   simp ower   s ys te m   en vir onment   is   e mp lo ye d   to   va li date   the   el ab or at e d   model   a nd   c on t ro l   of   the   co nversi on   sy ste m   wind   e nerg y.   T he   P MSG   a nd   tu r bin e   pa rameters   are   li ste d   in   T able   1.       Table  1.   Sp eci f ic at ion  of  PMS G  a nd tu rb i ne  [ 14]   PMSG  p a ra m eters   Valu es   W in d  turb in e param ete rs   Valu es   Gen erator  rated po wer   5      Turb in e blad e r ad i u s   58    Gen erator  rated vo ltag e   3 .3       Op tim al tip sp eed   6 .89   Gen erator  rated cu rr en t   1 .51 5      Maximu m     v alu e   0 .44   Po le par nu m b er   75   The air  den sity   1 .22 5    / 2   Gen erator  stato r  re sis tan ce   6 .23  e - 0 0 3  Ω   Rated  wind  sp eed   1 2 .12   /   Gen erator  −   ax is in d u ctan ce   4 .22 9  e - 003        Perman en t m ag n et  f lu x   1 1 .14 6 4      Gen erator  in e rtia    2  e+5   . 2       The   e vo l ution   of   wind   s pee d   is   il lustrate d   in   Fig ure   4,   w hich   s hows   t ha t   the   sp ee d   va ries   ar ound   10m / s ± 30%   durin g   t he   ti me   inter val   0   to   60   s.   Dep e ndin g   on   the   wind   ve locit y   var ia ti on ,   two   op e rati ons   ca n   be   tri gg e re d   acc ord ing   to   the   appr opriat e   co ntr ol   strat eg y.           Figure   4.   Wi nd   Profile       Figure   5.   Ge ne rator   r otor   s pee d     Evaluation Warning : The document was created with Spire.PDF for Python.
            IS S N :   2088 - 8 694   In t J   P ow  Ele c  &  D ri   S ys t,   V ol .  12 , N o.   1 ,  Ma rch  20 21   :   58 5   –   59 6   594   Figure   5   s how s   that   bel ow   the   rate d   wind   s peed,   t he   ge ne rator   s pee d   is   a dju ste d   to   be   opti mal   wh e n   regulat ing   the   sy ste m   un der   t he   s pecifica ti on   of   ma xim um   po wer ,   a nd   is   li mit ed   at   the   r at ed   value   be yond   the   rated   wind   sp e ed   with out   ov ersho oting.   T hi s   sh ows   the   r obus t ness   of   t he   meth od   a ga inst   the   wi nd   sp ee d   disturba nce.     Figure s   6 ( a )  a nd   6 ( b )   s how   t hat   bel ow   t he   rated   wind   s pe ed,   the   ene r gy   captu red   from   the   wind   is   maximize d   f rom   MPPT   oper at ion ,   by   adj ust ing   the   opti m al   ro to r   s pee d   that   corres ponds   to   t he   ma xi mu m   powe r.   In   this   con te xt,   the   po wer   c oeffici ent   is   ke pt   c onsta nt   at   its   ma xi mu m   value   at   about   0.4 4,   a nd   t he   pitch   a ng le   β   is   pr eser ve d   at   ze r o   for   eac h   M P PT   op e rati on,   des pite   the   var i at ion   of   the   wi nd.   Wh e reas   beyo nd   t he   rate d   wind   s peed   va lue   = 12 . 12     / ,   the   t urbine   is   protect e d   agai ns t   dama ge   a nd   t he   bla des   a re   ro ta te d   by   adj us ti ng   the   pitch   an gle.   As   a   resu lt ,   β   increas es   with   s ubse qu e nt   degra dation   of   tip   s peed   rati o   λ   in   order   to   re duce   powe r   coe ff ic ie nt     to   a   va lue   le ss   tha n   0. 44,   as   show n   at   pitch   c on t ro l   operati on.           (a)   (b)     Figure   6.   (a )   P ow e r   c oeffici ent,   (b)   Ti p   s pee d   rati o   a nd   pitc h   a ng le   va riat ion       Figure   7   sho w s   cu rr e nt   res pons es   in       frame,   as   the   cu rr e nts   track   well   the   r efere nce   t raject ory.   In   this   co ntext,   t he   qu a drat ur e   current     ( Fi gure   7   ( a) )   va ries   acco rd i ng   to   the   gen e rato r   sp ee d,   ta king   into   account   the   two   wind   op e rati ng   m od e s   ( MP PT   a nd   pitch   a ng le ).   The   direct   cu rrent   id   ( Fig ur e   7   ( b ) )   trac ks   w el l   the   op ti mal   ref e ren ce   (at   z ero)   in   orde r   to   pro vid e   the   maxim um   t orq ue   per   am per e .         (a)     (b)     Figure   7.   Stat or   c urren t   in   d - q   fr a me , (a)   Q ua dr at ur e   sta tor   c urren t ,  ( b)  Di re ct   sta tor   c urre nt       Figure  8  de pic ts  the  sta tor  c urren ts  of   t he  PM S G  in   sin usoidal  f orm.  It   can  be   noti ced  that  sta to r  currents   hav e   a   simi la r   sh a pe   as  the   wind  s pe ed  duri ng  the   ti me  inter vals  [ 0s ,   5s ]   a nd  al s o  is   li mit ed,  be yond  the nomi nal  wind spee d as i t s how n durin g  t he  inter val [5s,   8s ] .   Evaluation Warning : The document was created with Spire.PDF for Python.