Pitch Control of Wind Turbine Blades Using Fractional Particle Swarm Optimization

Ci a ion: Ka ami-Mollaee, A.;
Ba ambones, O. Pi ch Con ol o
Wind Tu bine Blades Using
F ac ional Pa icle Swa m
Op imiza ion. Axioms 2023,12, 25.
h ps://doi.o g/10.3390/
axioms12010025
Academic Edi o : Fe ie Valdez
Recei ed: 20 No embe 2022
Re ised: 20 Decembe 2022
Accep ed: 22 Decembe 2022
Published: 26 Decembe 2022
Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
axioms
A icle
Pi ch Con ol o Wind Tu bine Blades Using F ac ional Pa icle
Swa m Op imiza ion
Ali Ka ami-Mollaee 1and Osca Ba ambones 2,*
1Facul y o Elec ical and Compu e Enginee ing, Hakim Sabze a i Uni e si y, Sabze a 9617976487, I an
2Au oma ic Con ol and Sys em Enginee ing Depa men , Uni e si y o he Basque Coun y, UPV/EHU,
Nie es Cano 12, 48940 Vi o ia, Spain
*Co espondence: osca [email p o ec ed]; Tel.: +34-945013235; Fax: +34-945013270
Abs ac :
To achie e he maximum powe om wind in a iable-speed egions o wind u bines
(WTs), a sui able con ol signal should be applied o he pi ch angle o he blades. Howe e , he
a ailable unce ain y in he modeling o WTs complica es calcula ions o hese signals. To cope
wi h his p oblem, an op imal con olle is sui able, such as pa icle swa m op imiza ion (PSO). To
imp o e he pe o mance o he con olle , ac ional o de PSO (FPSO) is p oposed and implemen ed.
In o de o cons uc his app oach o a wo-mass WT, we p opose a new s a e eedback, which
was i s applied o he u bine. The idea behind his s a e eedback was based on he Taylo se ies.
Then, a linea model wi h unce ain y was ob ained wi h a new inpu con ol signal. The ea e , he
con en ional PSO (CPSO) and FPSO we e used as op imal con olle s o he esul ing linea model.
Finally, a compa ison was pe o med be ween CPSO and FPSO and he uzzy Takagi–Sugeno–Kang
(TSK) in e ence sys em. The p o ided compa ison demons a es he ad an ages o he Taylo se ies
wi h combina ion o hese con olle s. No ably, wi hou he s a e eedback, CPSO, FPSO, and TSK
uzzy sys ems canno s abilize WTs in acking he desi ed ajec o y.
Keywo ds:
wind u bine; pi ch angle con ol; ac ional pa icle swa m op imiza ion; uzzy in e ence
sys em; Taylo se ies
MSC: 93D15
1. In oduc ion
Sola o wind, as clean enewable iable ene gies, a e accessible wo ldwide and a e
clean. Howe e , due o economic easons, he use o wind ene gy and wind u bines (WTs)
is popula . The e a e wo kinds o WT, ixed speed WTs (FWTs) [
1
,
2
] and a iable-speed
WTs (VWTs) [
3
]. I is no capable o FWTs o wo k such ha he maximum powe o wind
can be ha nessed [
4
]. The e o e, VWTs ha e ecen ly been de eloped and cons uc ed. To
cap u e he maximum powe o wind in VWTs, i s ope a ion egions a e di ided in o ou
impo an sec ions using cu -ou , a ed, and cu -in bounda ies [
5
]. Below he cu -ou wind
speed, VWTs will be shu down, o balance economic pe o mance be ween he cu -ou and
a ed wind speeds by con olling he gene a o o que [
5
]. Mo eo e , be ween he a ed
and cu -in bounda ies o wind speeds, he pi ch angle o u bine blades is used as he inpu
con ol [
6
]. Finally, abo e he cu -in bounda y, he VWT will be shu down again o p o ec
i om a igue damage [7].
On he o he hand, mechanical s esses a e ano he impo an challenge, which e-
qui e powe ul op imal o adap i e app oaches o p o ec WTs [
7
]. The e o e, some pi ch
angle con olle s ha e been p oposed o WT blades be ween cu -ou and a ed wind
speeds
[6–14]
. In [
6
], a digi al con olle was designed; classical con olle s such as PID
(p opo ional–in eg al–de i a i e) a e p oposed in [
7
,
8
]; a PID con olle wi h an adap i e
sel - uning egula o (STR) was cons uc ed in [
9
]; a gain-scheduled PID con olle was
designed in [
10
]; a PI con olle scheme is shown in [
11
]; and a combina ion o adap i e
Axioms 2023,12, 25. h ps://doi.o g/10.3390/axioms12010025 h ps://www.mdpi.com/jou nal/axioms
Axioms 2023,12, 25 2 o 16
and PI con olle s is p esen ed in [
12
]. Some simple nonlinea eedback con olle s a e p o-
posed in [
13
,
14
]. Finally, o imp o e he pe o mance o WTs, a iable equency con e e
con ols o egula e he o o speed we e also used in [3,15].
Among hese app oaches, ac ional con olle s can ha e be e pe o mance [
16
,
17
]
because hey can p ecisely desc ibe he beha io o many dynamical sys ems in physical,
ma hema ical, and enginee ing ields [
18
–
20
]. Hence, many s udies ha e ocused on
ac ional subjec s o de elop hei heo ies [
21
,
22
]. The e o e, he ac ional calcula ions
ha e p og essed in a ious phenomena due o hei applica ions in dynamic sys ems [
17
].
In he o he hand, pa icle swa m op imiza ion (PSO) is a powe ool o op imiza-
ion [
23
] and con olle s [
24
]. The e o e, based on he ad anced p ope ies o ac ional
calculus and PSO, we imp o ed he pe o mance o con en ional PSO (CPSO) using a
combina ion o ac ional and PSO. The p oposed app oach is ac ional PSO (FPSO), and
was applied o WTs o pi ch angle con ol. Ini ially, s a e eedback was applied o he
WT model; hen, FPSO o ced he WT o o angula eloci y o ack i s e e ence while
he pi ch angle o he blades was egula ed. To demons a e he ad anced pe o mance o
FPSO, compa ison was pe o med wi h CPSO and he Takagi–Sugeno–Kang (TSK) uzzy
sys em wi h simila pa ame e s [25].
Hence, he p oposed con olle is demons a ed in i e sec ions. Fi s , he WT model
and hei subsys ems a e explained in Sec ion 2. Then, he con olle de ails, consis ing o
s a e eedback, wi h e e ence o o o angula eloci y, PSO, and TSK sys em, a e p o ided
in Sec ion 3. The simula ion esul s and compa isons o FPSO, CPSO, and TSK sys ems a e
p esen ed in Sec ion 4. Sec ion 5p esen s he conclusion.
2. Wind Tu bine (WT) Model
The gene a o and d i e ain a e wo WT subsys ems, hei elec ical and mechan-
ical sec ions, espec i ely [
16
]. O he impo an subsys em o a WT is he ae odynamic
sec ion [16]. These subsys ems a e depic ed in Figu e 1.
Axioms 2023, 12, x FOR PEER REVIEW 2 o 18
i e sel - uning egula o (STR) was cons uc ed in [9]; a gain-scheduled PID con olle
was designed in [10]; a PI con olle scheme is shown in [11]; and a combina ion o
adap i e and PI con olle s is p esen ed in [12]. Some simple nonlinea eedback con-
olle s a e p oposed in [13,14]. Finally, o imp o e he pe o mance o WTs, a iable
equency con e e con ols o egula e he o o speed we e also used in [3,15].
Among hese app oaches, ac ional con olle s can ha e be e pe o mance [16,17]
because hey can p ecisely desc ibe he beha io o many dynamical sys ems in physical,
ma hema ical, and enginee ing ields [18–20]. Hence, many s udies ha e ocused on ac-
ional subjec s o de elop hei heo ies [21,22]. The e o e, he ac ional calcula ions ha e
p og essed in a ious phenomena due o hei applica ions in dynamic sys ems [17].
In he o he hand, pa icle swa m op imiza ion (PSO) is a powe ool o op imiza-
ion [23] and con olle s [24]. The e o e, based on he ad anced p ope ies o ac ional
calculus and PSO, we imp o ed he pe o mance o con en ional PSO (CPSO) using a
combina ion o ac ional and PSO. The p oposed app oach is ac ional PSO (FPSO), and
was applied o WTs o pi ch angle con ol. Ini ially, s a e eedback was applied o he
WT model; hen, FPSO o ced he WT o o angula eloci y o ack i s e e ence while
he pi ch angle o he blades was egula ed. To demons a e he ad anced pe o mance o
FPSO, compa ison was pe o med wi h CPSO and he Takagi–Sugeno–Kang (TSK) uzzy
sys em wi h simila pa ame e s [25].
Hence, he p oposed con olle is demons a ed in i e sec ions. Fi s , he WT model
and hei subsys ems a e explained in Sec ion 2. Then, he con olle de ails, consis ing o
s a e eedback, wi h e e ence o o o angula eloci y, PSO, and TSK sys em, a e p o-
ided in Sec ion 3. The simula ion esul s and compa isons o FPSO, CPSO, and TSK
sys ems a e p esen ed in Sec ion 4. Sec ion 5 p esen s he conclusion.
2. Wind Tu bine (WT) Model
The gene a o and d i e ain a e wo WT subsys ems, hei elec ical and mechani-
cal sec ions, espec i ely [16]. O he impo an subsys em o a WT is he ae odynamic
sec ion [16]. These subsys ems a e depic ed in Figu e 1.
Figu e 1. The WT subsys ems.
2.1. The Ae odynamic Subsys em
Conside ing a WT wi h blade leng h
, powe coe icien p
C, he cap u ed powe
can calcula ed using he ollowing equa ion:
),(C
2
Pp
22
aλβ
ρπ
= (1)
whe e
ρ
is he ai densi y and ) ( is he wind speed, which a e dependen on en i-
onmen condi ions. The powe coe icien is dependen on he ip speed a io,
λ
, and
he pi ch blades,
β
[5], which a e de ined as ollows:
ω
=λ (2)
Figu e 1. The WT subsys ems.
2.1. The Ae odynamic Subsys em
Conside ing a WT wi h blade leng h , powe coe icien Cp, he cap u ed powe can
calcula ed using he ollowing equa ion:
Pa=ρπ 2 2
2Cp(β,λ)(1)
whe e
ρ
is he ai densi y and
( )
is he wind speed, which a e dependen on en i onmen
condi ions. The powe coe icien is dependen on he ip speed a io,
λ
, and he pi ch
blades, β[5], which a e de ined as ollows:
λ= ω
(2)
Cp(β,λ) = d1d2
λi−d1d3β−d1d4e−d5
λi+d6λ
1
λi=1
λ+0.08β−0.035
β3+1
(3)
Axioms 2023,12, 25 3 o 16
whe e ω is he o o side angula eloci y o he u bine blades, and:
d1=0.5176, d2=116, d3=0.4, d4=5, d5=21, d6=0.0068 (4)
Then, he gene a ed o o o que is desc ibed by:
Ta=Pa
ω
=ρπ 3 2
2λCp(β,λ)(5)
2.2. The D i e ain Subsys em
The wo-mass mechanical d i e ain, which shows he ansien esponse and s eady-
s a e esponse in he p esence o he con olle , is desc ibed by he ollowing equa ions and
is depic ed in Figu e 2[26].
J
.
ω =−K ω +Ta−Tls
Jg
.
ωg=−Kgωg−Tg+Ths (6)
Axioms 2023, 12, x FOR PEER REVIEW 3 o 18
1
035.0
08.0
11
dedddd
dd
),(C
3
i
6
d
4131
i
21
p
i
5
+β
−
β+λ
=
λ
λ+







−β−
λ
=λβ λ
−
(3)
whe e
ω
is he o o side angula eloci y o he u bine blades, and:
0068.0d,21d,5d,4.0d,116d,5176.0d
654321
====== (4)
Then, he gene a ed o o o que is desc ibed by:
),(C
2
P
Tp
23
a
aλβ
λ
ρπ
=
ω
= (5)
2.2. The D i e ain Subsys em
The wo-mass mechanical d i e ain, which shows he ansien esponse and
s eady-s a e esponse in he p esence o he con olle , is desc ibed by he ollowing
equa ions and is depic ed in Figu e 2 [26].
hsggggg
lsa
TTKJ
TTKJ
+−ω−=ω
−+
ω
−=
ω


(6)
Figu e 2. The d i e ain s uc u e.
In Figu e 2, g
ω
and
ω
a e angula eloci y, g
J and
J a e ine ia, g
K and
K
a e he ex e nal dapping, g
T and
a
T a e he o que ou pu , on he gene a o and o o
side, espec i ely, and inally, hs
T and ls
T a e he b aking o que in he high-speed and
low-speed sha . The a io o gea box is de ined as:
g
g
nω
ω
= (7)
Using he second pa o Equa ion (6) esul s in:
() ()








+−ω−=ω
g
ls
g gg gg n
T
TnKnJ  (8)
o :
Figu e 2. The d i e ain s uc u e.
In Figu e 2,
ωg
and
ω
a e angula eloci y,
Jg
and
J
a e ine ia,
Kg
and
K
a e
he ex e nal dapping,
Tg
and
Ta
a e he o que ou pu , on he gene a o and o o side,
espec i ely, and inally,
Ths
and
Tls
a e he b aking o que in he high-speed and low-speed
sha . The a io o gea box is de ined as:
ng=ωg
ω (7)
Using he second pa o Equa ion (6) esul s in:
Jgng
.
ω =−Kgngω −Tg+Tls
ng(8)
o :
ng2Jg
.
ω =−ng2Kgω −ngTg+Tls (9)
Finally, adding his equa ion o he i s pa o Equa ion (6) esul s in [5]:
J
.
ω =−K ω +Ta−ngTg(10)
Such ha
J =J +ng2Jg
and
K =K +ng2Kg
. In ac , all he pa ame e s a e ans e ed o
a low-speed sha [5].
Axioms 2023,12, 25 4 o 16
2.3. The Gene a o Subsys em
The
Tg
, i.e., he ou pu o que o he gene a o , can be modeled using he i s -o de
dynamic, whe e T e is he e e ence o que and τg=15 s is he gene a o ime cons an .
.
Tg=T e −Tg
τg(11)
We ocused on pi ch con ol; hus, he gene a o o que e e ence was se as
T e =T e ed
. Mo eo e , he p oduced ou pu powe deli e ed o he g id can be w i en as
Pg=ηgωgTg, whe e he e iciency o he gene a o is ηg.
3. The Op imal Con olle Design
In his sec ion, we i s used s a e eedback and hen calcula ed he desi ed o o
angula eloci y. The CPSO and FPSO op imal con olle s a e also desc ibed.
3.1. S a e Feedback
Ini ially, we calcula ed he de i a i e o he powe coe icien o Equa ion (3) wi h
espec o he pi ch angle o he blades.
dCp
dβ=∂Cp
∂λi
∂λi
∂β+∂Cp
∂β
=−c1c2
λi2+c1c2c5
λi3−c1c3c5β
λi2−c1c4c5
λi2−0.08
(λ+0.08β)2+0.035
(β3+1)2e−c5
λi−c1c3e−c5
λi
(12)
The e o e, he Taylo se ies o Equa ion (5) a ound i s op imal ope a ing poin s
βop
and
λop would be:
Ta=ρπ 3 2
2λop
dCp
dββ=βop
λ=λop
(β−βop ) + HOT (13)
whe e HOT is used o deno e highe -o de e ms; hus:
.
ω =−K
J
ω −ng
J
Tg+Ta
J
=−K
J
ω −ng
J
Tg+ρπ 3 2
2J λop
dCp
dββ=βop
λ=λop
(β−βop ) + ∆(14)
Due o he con e gence o he Taylo se ies a ound he ope a ing poin s, he unknown
unce ain y
∆=HOT
J
is bounded, i.e.,
|∆|≤η
. Then, he ollowing s a e eedback wi h he
new inpu signal, u, and he a bi a y pa ame e , a, can be used.
β=ng
J Tg+K
J −aω +u
ρπ 3 2
2J λop
dCp
dββ=βop
λ=λop
+βop (15)
Then, sys em Equa ion (14) can be ew i en as ollows:
.
ω =−aω +u+∆(16)
In which
∆
is an unknown unce ain unc ion. We aimed o design an op imal app oach
such ha in his linea sys em, he o o angula eloci y,
ω
, acked he desi ed
signal, ω d
.
3.2. Re e ence o Ro o Angula Veloci y
As men ioned in he In oduc ion and based on Figu e 3, he VWT ope a ion modes
we e di ided in o ou egions using wind speed bounda ies o cu -in, a ed, and cu -ou .
The c i ical poin is he a ed wind speed, such ha below his poin , he pi ch o u bine
Axioms 2023,12, 25 5 o 16
blades is ixed and gene a o o que is con olled; hence, he o o speed is inc eased
o ha e he maximum o powe coe icien . Mo eo e , abo e he a ed wind speed, he
gene a o e e ence o que is ixed and is se o i s a ed alue. In his egion, he pi ch
angle would be inc eased o educe he o o speed. Finally, ou o he cu -in and cu -ou
wind speeds, he u bine would be shu down due o he economic c i e ion and a igue
damages, espec i ely [5].
Axioms 2023, 12, x FOR PEER REVIEW 5 o 18
op
p
op
23
g
g
op
op
d
dC
J2
ua
J
K
T
J
n
β+
βλ
ρπ
+ω







−+








=β
λ=λ
β=β
(15)
Then, sys em Equa ion (14) can be ew i en as ollows:
Δ++
ω
−=
ω
ua
 (16)
In which Δ is an unknown unce ain unc ion. We aimed o design an op imal app oach
such ha in his linea sys em, he o o angula eloci y,
ω
, acked he desi ed signal,
d
ω
.
3.2. Re e ence o Ro o Angula Veloci y
As men ioned in he In oduc ion and based on Figu e 3, he VWT ope a ion modes
we e di ided in o ou egions using wind speed bounda ies o cu -in, a ed, and cu -ou .
The c i ical poin is he a ed wind speed, such ha below his poin , he pi ch o u bine
blades is ixed and gene a o o que is con olled; hence, he o o speed is inc eased o
ha e he maximum o powe coe icien . Mo eo e , abo e he a ed wind speed, he
gene a o e e ence o que is ixed and is se o i s a ed alue. In his egion, he pi ch
angle would be inc eased o educe he o o speed. Finally, ou o he cu -in and cu -ou
wind speeds, he u bine would be shu down due o he economic c i e ion and a igue
damages, espec i ely [5].
Figu e 3. Ope a ion egions o he VWT.
In his s udy, we ocused on he pi ch angle con ol in egion h ee, whe eas he o-
o angula eloci y should be educed wi h he inc eased o wind speed. The e o e, he
e e ence o o o angula eloci y is as ollows:
a edou cu
a ed
a ed a ed d
−
−
ω−ω=ω
−
(17)
Based on Equa ion (2), one can conclude ha :
Figu e 3. Ope a ion egions o he VWT.
In his s udy, we ocused on he pi ch angle con ol in egion h ee, whe eas he o o
angula eloci y should be educed wi h he inc eased o wind speed. The e o e, he
e e ence o o o angula eloci y is as ollows:
ω d =ω a ed −ω a ed
− a ed
cu −ou − a ed
(17)
Based on Equa ion (2), one can conclude ha :
ω a ed =λop a ed
(18)
3.3. Pa icle Swa m Op imiza ion (PSO) and Con olle S uc u e
Acco ding o he p e ious sec ions, he aim was o de e mine he angula eloci y o
he o o , i.e.,
ω
acks he desi ed ajec o y,
ω d
. To his end, he e o ,
e= (ω −ω d)2
,
is applied o he PSO. PSO is applied o calcula e he inpu con ol signal,
u
, in Equa ion
(16), while he e o signal,
e
, con e ges o ze o. In CPSO, he eloci y o each pa icle is
upda ed as ollows [23]:
.
i( ) = c1φ1(pb−xi( ))+c2φ2pg−xi( ): i =1, 2, . . . , n (19)
whe e
n
is he numbe o pa icles,
i( )
is he eloci y o each pa icle,
φ1
and
φ2
a e
uni o mly andom unc ions be ween 0 and 1,
pb
is he bes posi ion o each pa icle,
pg
is
he global bes posi ion be ween all o he pa icles,
xi( )
is he cu en posi ion o he each
pa icle, and coe icien s
c1
and
c2
a e cons an numbe s. Then, he posi ion o any pa icle
is upda ed as ollows [23]: .
xi= i( )(20)
The e a e se e al de ini ions o ac ional di e en ia ions and in eg a ions, such as
G ünwald–Le niko , Riemann–Liou ille, and Capu o o mulae [
27
]. Among hem, he

Axioms 2023,12, 25 6 o 16
Capu o me hod is popula because ini ial condi ions a e conside ed [
17
,
27
]; hus, he
Capu o de ini ion was used in his s udy.
De ini ion 1.
Capu o q-o de in eg a ion and he di e en ia ion o a iable
( )
wi h espec o
ime, , is de ined as ollows [27]:
0Iq
( ) = 1
Γ(q)Z
0
( −τ)q−1 (τ)dτ(21)
0Dq
( ) = 1
Γ(1−q)Z
0
0(τ)
( −τ)qdτ(22)
He e,
> 0
and
0
is he ini ial ime; 0
<q<
1and
Γ(q) = R∞
0τq−1e−τdτ
is he
Gamma unc ion
.
Rema k 1.
In his s udy, we conside ed he ze o ini ial condi ion, i.e.,
0=
0. Mo eo e , o
simplici y, subsc ip
was also elimina ed; hence, we use
Dq ( )
ins ead o
0Dq
( )
and
Iq ( )
ins ead o 0Iq
( ).
To imp o e he pe o mance o he CPSO, we p opose FPSO, as ollows:
Dq i( ) = c1φ1(pb−xi( ))+c2φ2pg−xi( ): i =1, 2, . . . , n (23)
Fo a alid compa ison, bo h CPSO and FPSO we e implemen ed. The e o e, he imple-
men ed diag am o he p oposed app oach is illus a ed in Figu e 4. F om his igu e, one
can see he combina ion o wo eedbacks o nonlinea sys ems o Figu es 1and 2. The
i s s a e eedback o Equa ion (15) is based on he heo y o he Taylo se ies wi h a new
inpu con ol signal,
u( )
, o ob ain a linea sys em, as in Equa ion (16). Then, in he second
eedback, FPSO o CPSO we e applied o his linea sys em in o de o minimize he e o
signal. Mo eo e , we used he uzzy TSK sys ems.
Axioms 2023, 12, x FOR PEER REVIEW 7 o 18
inpu con ol signal, ) (u , o ob ain a linea sys em, as in Equa ion (16). Then, in he
second eedback, FPSO o CPSO we e applied o his linea sys em in o de o minimize
he e o signal. Mo eo e , we used he uzzy TSK sys ems.
Figu e 4. The implemen ed s uc u e o he p oposed con olle .
3.4. Takagi–Sugeno–Kang (TSK) Con olle S uc u e
The s uc u e o p oposed TSK con olle is shown in Figu e 4, wi h wo inpu s
)(e d
ω
−
ω
= and i s de i a i e and one ou pu ) (u . Fo each inpu and ou pu , i e
iangula membe ship unc ions a e de ined as nega i e-la ge (NL), nega i e-small
(NS), ze o (Z), posi i e-small (PS), and posi i e-la ge (PL). The ange o inpu s was se
be ween −2 and +2, and he ange o ou pu was also se o be ween −200 and 200.
The e o e, 25 ules wi h he agg ega ion de uzzi ica ion we e used.
4. Simula ions Resul s
We used he wo-mass 5 MW, VWT in Na ional Renewable Ene gy Labo a o y
(NREL) loca ed a Colo ado, wi h he ae odynamic pa ame e s in Table 1 and d i e ain
pa ame e s in Table 2 [28].
Table 1. The ae odynamic pa ame e s o VWT.
Pa ame e Value Uni
a ed
T
4
103094.4 × mN ×
incu
− 3 s/m
a ed
5.10 s/m
ou cu
− 25 s/m
op
β
0 deg
op
λ
Scala 55.7
Figu e 4. The implemen ed s uc u e o he p oposed con olle .
3.4. Takagi-Sugeno-Kang (TSK) Con olle S uc u e
The s uc u e o p oposed TSK con olle is shown in Figu e 4, wi h wo inpu s
e= (ω −ω d)
and i s de i a i e and one ou pu
u( )
. Fo each inpu and ou pu , i e
iangula membe ship unc ions a e de ined as nega i e-la ge (NL), nega i e-small (NS),
ze o (Z), posi i e-small (PS), and posi i e-la ge (PL). The ange o inpu s was se be ween
Axioms 2023,12, 25 7 o 16
−
2 and +2, and he ange o ou pu was also se o be ween
−
200 and 200. The e o e,
25 ules wi h he agg ega ion de uzzi ica ion we e used.
4. Simula ions Resul s
We used he wo-mass 5 MW, VWT in Na ional Renewable Ene gy Labo a o y (NREL)
loca ed a Colo ado, wi h he ae odynamic pa ame e s in Table 1and d i e ain pa ame e s
in Table 2[28].
Table 1. The ae odynamic pa ame e s o VWT.
Pa ame e Value Uni
T a ed 4.3094 ×104N×m
cu −in 3 m/s
a ed 10.5 m/s
cu −ou 25 m/s
βop 0 deg
λop Scala 7.55
Table 2. The mechanical d i e ain pa ame e s o VWTs.
No a ion Value Uni
21.62 m
ρ1.308 kg/m3
J 3.25 ×105kg ×m2
Jg34.4 kg ×m2
K 27.36 (N×m)/( ad/s)
Kg0.2 (N×m)/( ad/s)
Kls 9.5 ×103(N×m)/ ad
Bls 2.691 ×105(N×m)/( ad/s)
ng43.165 Scala
Fo a eliable compa ison, all o he simula ions we e pe o med using MATLAB
so wa e wi h a sample ime o 0.01. The wind speed is shown in Figu e 5, wi h a mean
alue 16 and maximum dis u bance o 5. No ably, his is be ween 11.4 and 25, i.e., in egion
3:
a ed =
11.4
< ( )< cu −ou =
25. In addi ion, he ini ial alue o he o o angula
eloci y is se o 3, i.e.,
ω (
0
) =
3, and he eedback pa ame e is se as
a=
2. Mo eo e ,
Figu e 6shows he e e ence o o o speed in egion 3 deno ed by Equa ion (17).
Axioms 2023, 12, x FOR PEER REVIEW 8 o 18
Table 2. The mechanical d i e ain pa ame e s o VWTs.
No a ion Value Uni
62.21
m
ρ
308.1 3
m/kg
J 5
1025.3 × 2
mkg ×
g
J 4.34 2
mkg×
K 36.27 )s/ ad/()mN( ×
g
K 2.0 )s/ ad/()mN( ×
ls
K 3
105.9 × ad/)mN( ×
ls
B 5
10691.2 × )s/ ad/()mN( ×
g
n 165.43 Scala
Fo a eliable compa ison, all o he simula ions we e pe o med using MATLAB
so wa e wi h a sample ime o 0.01. The wind speed is shown in Figu e 5, wi h a mean
alue 16 and maximum dis u bance o 5. No ably, his is be ween 11.4 and 25, i.e., in e-
gion 3: 25 ) ( 4.11
ou cu a ed
=<<=
−
. In addi ion, he ini ial alue o he o o an-
gula eloci y is se o 3, i.e., 3)0(
=
ω
, and he eedback pa ame e is se as 2a =.
Mo eo e , Figu e 6 shows he e e ence o o o speed in egion 3 deno ed by Equa ion
(17).
Figu e 5. The p o ile o ime se ies wind speed.
Figu e 5. The p o ile o ime se ies wind speed.
Axioms 2023,12, 25 8 o 16
Axioms 2023, 12, x FOR PEER REVIEW 9 o 18
Figu e 6. Re e ence o he o o speed in egion 3 o VWTs.
Example 1. The CPSO app oach.
As he i s esul , simula ions o he CPSO in Equa ion (19) a e shown in Figu e 7–
10. The pa ame e s o he PSO a e 5.1cc
21
== , wi h 20 pa icles and 4000 i e a ions.
Figu e 7. Con e gence o he CPSO.
Figu e 6. Re e ence o he o o speed in egion 3 o VWTs.
Example 1. The CPSO app oach.
As he i s esul , simula ions o he CPSO in Equa ion (19) a e shown in Figu es 7–10.
The pa ame e s o he PSO a e c1=c2=1.5, wi h 20 pa icles and 4000 i e a ions.
Axioms 2023, 12, x FOR PEER REVIEW 9 o 18
Figu e 6. Re e ence o he o o speed in egion 3 o VWTs.
Example 1. The CPSO app oach.
As he i s esul , simula ions o he CPSO in Equa ion (19) a e shown in Figu e 7–
10. The pa ame e s o he PSO a e 5.1cc
21
== , wi h 20 pa icles and 4000 i e a ions.
Figu e 7. Con e gence o he CPSO.
Figu e 7. Con e gence o he CPSO.
Axioms 2023,12, 25 9 o 16
Axioms 2023, 12, x FOR PEER REVIEW 10 o 18
Figu e 8. Angula eloci y o he o o in CPSO.
Figu e 9. The inpu con ol signal o s a e eedback in CPSO.
Figu e 8. Angula eloci y o he o o in CPSO.
Axioms 2023, 12, x FOR PEER REVIEW 10 o 18
Figu e 8. Angula eloci y o he o o in CPSO.
Figu e 9. The inpu con ol signal o s a e eedback in CPSO.
Figu e 9. The inpu con ol signal o s a e eedback in CPSO.
Axioms 2023,12, 25 16 o 16
18.
Kilbsa, A.A.; S i as a a, H.M.; T ujillo, J.J. Theo y and Applica ions o F ac ional Di e en ial Equa ions; Else ie : New Yo k,
NY, USA, 2006.
19.
Podlubny, I. F ac ional Di e en ial Equa ions: An In oduc ion o F ac ional De i a i es, F ac ional Di e en ial Equa ions, o Me hods o
Thei Solu ion and Some o Thei Applica ions; Academic P ess: San Diego, CA, USA, 1998.
20.
Qiu, F.; Liu, Z.; Liu, R.; Quan, X.; Tao, C.; Wang, Y. Fluid low signals p ocessing based on ac ional Fou ie ans o m in a s i ed
ank eac o . ISA T ans. 2019,90, 268–277. [C ossRe ]
21.
Li, Y.; Chen, Y.; Podlubny, I. S abili y o ac ional-o de nonlinea dynamic sys ems: Lyapuno di ec me hod and gene alized
Mi ag–Le le s abili y. Compu . Ma h. Appl. 2010,59, 1810–1821. [C ossRe ]
22.
Aguila-Camacho, N.; Dua e-Me moud, M.A.; Gallegos, J.A. Lyapuno unc ions o ac ional o de sys ems. Commun. Nonlinea
Sci. Nume . Simul. 2014,19, 2951–2957. [C ossRe ]
23.
Choi, Y.-P.; Ju, H.; Koo, D. Con e gence analysis o Pa icle Swa m Op imiza ion in one dimension. Appl. Ma h. Le .
2023
,
137, 108481. [C ossRe ]
24.
He, Z.; Liu, T.; Liu, H. Imp o ed pa icle swa m op imiza ion algo i hms o ae odynamic shape op imiza ion o high-speed ain.
Ad . Eng. So w. 2022,173, 103242. [C ossRe ]
25.
Ja a zadeh, S.; Fadali, M.S. On he S abili y and Con ol o Con inuous-Time TSK Fuzzy Sys ems. IEEE T ans. Cybe n.
2012
,43,
1073–1087. [C ossRe ] [PubMed]
26.
Boukhezza , B.; Sigue didjane, H. Nonlinea Con ol o a Va iable-Speed Wind Tu bine Using a Two-Mass Model. IEEE T ans.
Ene gy Con e s. 2010,26, 149–162. [C ossRe ]
27.
Shan anu, D. Func ional F ac ional Calculus o Sys em Iden i ica ion and Con ols; Sp inge -Ve lag:
Be lin/Heidelbe g, Ge many, 2008
.
28.
Bossanyi, E.A.; W igh , A.D.; Fleming, P.A. Con olle Field Tes s on he NREL CART2 Tu bine; No. NREL/TP-5000-49085; Na ional
Renewable Ene gy Lab (NREL): Golden, CO, USA, 2010.
Disclaime /Publishe ’s No e:
The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .