Optimization of Wind Farm Layout for Maximum Energy Production by Stochastic Fractal Search

NGUYEN, K. D. e al. VOLUME: 23 |NUMBER: 1 |2025 |MARCH
Resea ch A icle
OPTIMIZATION OF WIND FARM LAYOUT FOR
MAXIMUM ENERGY PRODUCTION BY
STOCHASTIC FRACTAL SEARCH
Khoa Dang NGUYEN1,2, Tinh T ung TRAN2, Dieu Ngoc VO1,3,∗
1Depa men o Powe Sys ems, Ho Chi Minh Ci y Uni e si y o Technology (HCMUT),
268 Ly Thuong Kie S ee , Dis ic 10, Ho Chi Minh Ci y, Vie nam
2College o Enginee ing, Can Tho Uni e si y, Can Tho Ci y, Vie nam
3Vie nam Na ional Uni e si y Ho Chi Minh Ci y, Linh T ung Wa d, Thu Duc Ci y, Ho Chi Minh Ci y,
Vie nam
[email p o ec ed], [email p o ec ed], ndieu@hcmu .edu. n
∗Co esponding au ho : Dieu Ngoc VO; ndieu@hcmu .edu. n
DOI: 10.15598/aeee. 23i1.240404
A icle his o y: Recei ed Ap 13, 2024; Re ised May 26, 2024; Accep ed Jul 26, 2024; Published Ma 31, 2025.
This is an open access a icle unde he BY-CC license.
Abs ac . The wind powe plan designs a e di e en
om he design o o he con en ional powe plan s such
as hyd opowe plan s, he mal powe plan s, and nu-
clea powe plan s because he inpu uel o hese ypes
o powe plan s is con ollable. Wind powe plan s de-
pend on he speed o wind ene gy. The e o e, he p ob-
lem o op imizing he loca ion o u bines in a wind
a m o achie e maximum annual ene gy ou pu (AEP)
is o g ea in e es . In his pape , he S ochas ic F ac-
al Sea ch (SFS) algo i hm is p oposed o op imize he
a angemen o u bines in he wind a m o minimize
he wake e ec so ha he wind a m achie es he max-
imum gene a ing capaci y and he highes powe ac o
(CF). SFS ep esen s a signi ican ad ancemen in op-
imiza ion echniques, o e ing obus , adap able, and
e icien solu ions o complex p oblems like wind a m
layou op imiza ion. I s inno a i e use o ac ional
dynamics and s ochas ic p ocesses dis inguishes i om
adi ional me hods, p o iding supe io pe o mance in
many scena ios. The p oposed me hod was es ed on
a s anda d case wi h h ee ypes o u bines wi h di -
e en capaci ies o 850kW, 1000kW, and 1500kW o
con i m he sui abili y o he algo i hm and selec he
mos app op ia e u bine ype. The esul s o AEP and
wake loss calcula ed by he SFS algo i hm we e supe io
compa ed o hose ob ained by he PSO algo i hm o
hese h ee u bine ypes. The u bine wi h he highes
CF will be selec ed o applica ion in he wind a m.
The e o e, he p oposed SFS algo i hm can be a po en-
ial me hod o deal wi h he p oblem o op imiza ion o
wind a m layou .
Keywo ds
S ochas ic F ac al Sea ch Algo i hm, Wake e -
ec , WAsP so wa e, Wind a m layou op i-
miza ion, windPRO so wa e.
1. In oduc ion
The p ima y uel sou ces o powe plan s such as coal,
oil, and gas a e g adually deple ed. To ensu e a s able
powe sou ce and minimize he impac on he en i on-
men , coun ies a e in e es ed in de eloping sus ainable
ene gy. Using enewable ene gy sou ces such as wind
ene gy wi h g ea po en ial in many coun ies. In 2021,
wind powe capaci y ose by 93.6 GW, and he o al
global wind powe capaci y imp o ed 837 GW, an im-
p o emen o 12%o e he p e ious yea . NZE2050’s
goal in he nex 5 yea s, he wo ld needs mo e han
86 GW o wind powe annually and he o al global
wind powe capaci y is abou 469 GW. By 2050, he
wo ld is expec ed o ins all 2TW o wind powe and
o sho e wind will each 19%by 2024 [1]. To design a
wind a m, he e a e a numbe o issues wo h conside -
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ing and mos o hem ha e been ex ensi ely s udied in
dis inc ly indi idual ways such as u bine placemen ,
esea ch on wind cha ac e is ics, analyze he in e ac-
ion be ween wind u bines (wake e ec ), design an-
cilla y i ems ( u bine anspo ou es, elec ical cable
sys ems, u bine ounda ions), eliabili y, economic is-
sues, en i onmen al impac assessmen [2]. The wake
e ec is a complex and in e es ing p oblem in sol ing
he p oblem o wind u bine layou o achie e maxi-
mum powe [3]. he mos popula models buil by N.O.
Jensen [4] and imp o ed by Ka ic [5]. Jensen ea ed
he in luence a ea behind he u bines as a wind dis-
u bance and igno es he eddy e ec [6], which a ec ed
only he egion nea he u bines. Two ypes o wake e -
ec models ha e been p esen ed: Compu a ional luid
dynamics (CFD) model [7, 8] and analy ical model
[9, 10, 11, 12]. Recen ly, se e al me hods ha e been
p oposed o calcula e he wake e ec such as he bi-
na y ma ix me hod based on he Jensen model [13],
he wake e ec model combined wi h he mul i- u bine
e ec has been p oposed o ene gy loss analysis [14].
Many wo ks ha e p esen ed di e en me hods o
sol e p oblems ela ed o he op imal posi ion o wind
u bines such as Changshui e al. [15] ha e p oposed
he “lazy g eed” algo i hm used o op imize he wind
u bines. Zhang Changshui e al. [16] p esen ed a
"submodule" na u e o u bine placemen in wind
a ms based on he Jensen wake model, Se ano e
al. [17] applied by i e a i e me hod o inc ease he
dis ance be ween u bines in o sho e wind a m in o -
de o decline he wake e ec . In addi ion, he e a e
a numbe o s udies on de e mining he op imal loca-
ion o ins all powe sys ems o wind a ms such as
ins alling subs a ions and cable sys ems [18, 19]. The
s udy o wind speed educ ion h ough he u bine is
also a complex p ocess in ol ing he de e mina ion o
he u bine loca ion, wind condi ions, and wind u -
bine con ol me hods [20]. The ini ial da a o calcu-
la ing he u bine powe is he measu ed wind speed
and i is s a ically ep esen ed by he Weibull dis ibu-
ion [21, 22, 23, 24].
Cu en ly, he e a e comme cial so wa e o wind
ene gy e iciency assessmen and wind a m design, he
mos popula one being WAsP [25]. The main unc ion
o his so wa e is he assessmen o wind esou ces a -
e analyzing he measu ed wind da a se . WAsP an-
alyzes wind esou ces by analyzing wind lows using
a CFD model. In addi ion, he WAsP so wa e p o-
ides a ious ools o design he wind a m, such as
an assessmen o wind powe p oduc ion conside ing
he wake e ec , analyzing wind speed, wind dis o ion,
and wind u bulence. The windPRO so wa e in ol es
op imizing he wind a m’s u bine layou o maxi-
mum powe [26]. On he o he hand, his so wa e
also has ools o en i onmen al impac assessmen and
layou o u bines o espond o noise. Recen ly, me a-
heu is ic op imiza ion algo i hms ha e been inc eas-
ingly applied o enginee ing p oblems such as E olu-
iona y S a egy, Gene ic Algo i hms, Dolphin Echolo-
ca ion, Cuckoo Op imiza ion Algo i hm, A i icial Bee
Colony, Ray Op imiza ion, G ay Wol Op imize , Col-
liding Bodies Op imiza ion, and Chao ic Swa ming o
Pa icles [27]. These algo i hms ha e p o en hem-
sel es o be e y compe i i e compa ed o mode n
hype -simula ion algo i hms as well as o he con en-
ional me hods.
In his pape , an op imal sea ch algo i hm is p o-
posed, which is based on a andom ac al sea ch o
sol e he p oblem o op imizing wind a ms o achie e
maximum powe ene gy. The ma hema ical model o
he p oblem includes a i ness unc ion wi h he goal o
ob aining maximum ene gy and he cons ain s o he
u bine. To check he easibili y and e iciency o he
p oposed algo i hm, he esul s calcula ed by SFS will
be compa ed wi h he esul s calcula ed by PSO and
simula ion esul s by windPRO so wa e.
2. P oblem and Fo mula ion
2.1. Assump ions
To de elop a gene al model o he p oblem o op i-
miza ion o wind a m layou , a se o assump ions is
conside ed in his pape .
1. The u bines ha e he same cha ac e is ics.
2. The same numbe o u bines o he case s ud-
ies 3. The u bines a e a anged onsho e in wo-
dimensional (x,y).
4. Wind speed ( ) ollows Weibull dis ibu ion [20,
23, 24, 28].
The Weibull dis ibu ion is commonly used in wind
ene gy analysis o model wind speed da a because i
can p o ide a good i o he wind speed p obabili y
dis ibu ion. The Weibull dis ibu ion is cha ac e ized
by wo pa ame e s: shape (k) and scale (c). These pa-
ame e s de e mine he shape and scale o he dis i-
bu ion, espec i ely. The p obabili y densi y unc ion
o he Weibull dis ibu ion is gi en by:
( , k, c) = k
c
ck−1e−(
c)k(1)
whe e, is he wind speed; kis he shape pa ame e ;
cis he scale pa ame e .
2.2. Wake e ec model
The wind speed changes a e passing h ough he up-
s eam wind u bines, which a ec s he downs eam
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Fig. 1: A wind u bine inside he cone o ano he u bine [29].
wind u bines due o educed wind speed and inc eased
u bulence and is called he wake e ec . This e ec will
a ec he ope a ion and ene gy p oduc ion o wind u -
bines in he u bulence zone. The e o e, modeling he
wake e ec plays an impo an ole in de e mining he
loca ion o u bines in a wind a m. The wake e ec
becomes mo e signi ican when he wind a m has mul-
iple u bines. A conside ed u bine may be a ec ed
by wake e ec s om many o he wind u bines [3].
The wake e ec model is a c ucial componen in wind
a m layou op imiza ion. I simula es he in e ac ion
be ween wind u bines in a wind a m, accoun ing o
he educ ion in wind speed and u bulence caused by
he wake o ups eam u bines. A ai ly simple wake
e ec model wi h ex ensi e linea assump ions and a
decaying wind speed ha depends only on he dis ance
behind he u bine was de eloped by N.O. Jensen [3].
Jensen ea s he pos - u bine in luence as a wind dis-
u bance and igno es he eddy e ec , which a ec s only
he egion nea he u bine.
The angle βij,(0 ≤β≤π), be ween he ec o o ig-
ina ing om he op o he hypo he ical cone o he i h
u bine and he j h u bine, is calcula ed as [29]:
βi,j =cos−1



(xi−xj) cos θ+ (yi−yj) sin θ+R/κ
qxi−xj+R
κcos θ2+yi−yj+R
κsin θ2


(2)
The wind u bine j h is inside he wake o u bine i h,
i u bin j h is inside he cone. The dis ance be ween
u bine i h and j h p ojec ed on he wind di ec ion
θ, dij, is exp essed as ollows [29]:
di,j =|(xi−xj) cos θ+ (yi−yj) sin θ|(3)
The all in wind speed a a ce ain loca ion dis [29]:
Vde = 1 −Vdown
Vup
=1−√1−C
1 + κdi,j
R2(4)
whe e, C is he h us coe icien o u bine; d is he
dis ance be ween u bine i and u bine j as a p ojec-
ion along wi h wind di ec ion; kis he en ainmen
cons an (decay coe icien ) [30], which is empi ically
calcula ed as:
k=0.5
ln H
z0(5)
whe e, His he hub heigh , and z0 ep esen s he su -
ace oughness o he e ain. kis 0.075 o land a eas
and 0.04 o o sho e a eas [31]; di,j is he dis ance be-
hind he u bine conside ing wind di ec ion θ.
Due o he wake e ec (when a u bine is a ec ed by
mul iple u bines in on ) he wind speed is educed
[29]:
Vde i =
u
u
N
X
j=1,j=i,βi,j <α "1−√1−C
(1 + κdi,j/R)2#(6)
I is easy o obse e ha Vde i is a unc ion o wind
di ec ion (θ) and all u bine posi ions. I is shown ha
only he scaling pa ame e co he Weibull dis ibu ion
will be a ec ed by he wake loss [6]. The wake e ec is
s a is ically desc ibed as ollows [32]:
c′(θ) = c(θ).(1 −Vde i)(7)
2.3. Wind u bine cha ac e is ics
The exac pa e n o u bine cha ac e is ics is e y im-
po an in guessing wind powe ene gy. The e ha e
been many app oaches o he in oduc ion o wind u -
bines, including app oxima e polynomials [33]. In his
a icle, he 9 h deg ee polynomial model is applied o
calcula e he u bine cha ac e is ic model because his
is he mos sui able obse ed model.
( ) = p0+p1 +p2 2+p3 3+p4 4+p5 5+
+p6 6+p7 7+p8 8+p9 9(8)
The s anda d case is applied o he p oblem wi h
u bine capaci ies o 850kW, 1000kW and 1500kW as
shown in Table 1.
Figu es 2, 3, and 4 show he compa ison be ween he
polynomial model and he ac ual wind u bine cha ac-
e is ics p oposed in his pape . The cha ac e is ics o
he model a e obse ed o be e y simila o hose o
he ac ual u bine.
The wind u bine cha ac e is ics a e es a ed as ol-
lows:
(ν) = 


0, i< cu in, i> cu ou
(x)in Eq.(8), cu in ≤ i≤ cu ou
P a ed, a ed ≤ i≤ cu ou (9)
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Tab. 1: Types o u bines p oposed o wind a ms.
Type o Tu bine. GAMESA G52/850 NORDEX N-54/1000 VESTAS V63/1500
Gene al da a
Model: G52/850 N54/1000 V63/1500
Ra ed powe 850 kW 1,000 1,500
Ro o diame e 52 m 54 63.6
Swep a ea 2,124 m²2,291 3,177
Speci ic a ea 2.5 m²/kW 2.3 2.12
Numbe o blades: 3 3 3
Ro o
Minimum o o speed 19,44 14 d/min -
Maximum o o speed 30,8 d/min 21,5 d/min 22,9 d/min
Cu -in wind speed 4 m/s 3,5 m/s 4 m/s
Ra ed wind speed 16 m/s 15,5 m/s 16 m/s
Cu -o wind speed 25 m/s 25 m/s 25 m/s
Gene a o
Type ASYNC ASYNC ASYNC
Numbe 1 1 1
Maximum speed 1900 d/min 1513 d/min 1650 d/min
Vol age 690 V 690 V 690 V
Fig. 2: GAMESA G52/850 u bine cha ac e is ics.
Fig. 3: NORDEX N-54/1000 u bine cha ac e is ics.
2.4. Wind powe model
1) Wind Model
Wind model becomes e y impo an in es ima ing
wind powe p oduc ion. In wind pa e n, wind speed
and wind di ec ion a e wo pa ame e s ha need o
be ca e ully conside ed because hey a ec he powe
ou pu o he wind u bine. The wind speed is usually
desc ibed by he Weibull dis ibu ion and he wind di-
ec ion is exp essed by he p obabili y o each sec o o
he wind ose [34].
This s udy p oposes o use a 12-sec o wind ose be-
cause i is widely used o wind a m design.
Fig. 4: VESTAS V63/1500 u bine cha ac e is ics.
2) Wind Powe Ou pu
The ene gy p oduc ion o he u bine is shown in (10)
[35]:
E(P, θ) =
∞
Z
0
( )p( , c(θ), k(θ))d (10)
whe e p( , c(θ), k(θ)) is he Weibull p obabili y densi y
unc ion o wind speed.
Calcula ion o he ene gy p oduced by a u bine o
wind di ec ion om 0o o 360ois p esen ed as ollows
[29]:
E(P) =
360
Z
0
p(θ)dθ
∞
Z
0
( )p( , c(θ), k(θ))d (11)
The nume ical in eg a ion me hod will be applied o
calcula e he wind powe ou pu o he wind a m. The
wind powe ou pu in each wind di ec ion θis combined
as ollows [29]:
E(P) =
h
P
i=1
i(θ)
∞
R0
( )ki(θ)
c′i(θ)
c′i(θ)(ki(θ)−1)e−
c′i(θ)ki(θ)
d
(12)
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3) Objec i e Func ions
This a icle p esen s a me hod o op imizing he a -
angemen o u bines in a wind a m o maximize an-
nual ene gy ou pu :
Obj = max hXE(P)i(13)
3. Me hodology
The SFS algo i hm is a me aheu is ic op imiza ion al-
go i hm inspi ed by he p inciples o ac al geome y
and andomness. I ’s designed o sol ing complex op-
imiza ion p oblems and is pa icula ly use ul o global
op imiza ion. SFS combines andom sampling wi h
sel -simila i y, c ea ing a ich sea ch landscape o ind-
ing he global op imum, This is based on he simula ion
o a dielec ic b eakdown p ocess, so i becomes a sui -
able sea ch engine o sol ing op imiza ion p oblems
a a gene al le el. The p ocedu e o he algo i hm is
di ided in o wo p ocesses as di usi e and upda e [36].
In he i s phase, o inc ease sea ch chances, each
poin will di use a ound he cu en loca ion in e-
sponse o g ow h cha ac e is ics. In he second phase,
he simula ion algo i hm will wo k o an indi idual o
upda e i s loca ion based on he loca ion o o he in-
di iduals, he second phase uses some andom me hod
such as upda e p ocesses.
An ou line o he S ochas ic F ac al Sea ch Algo-
i hm wo ks:
S ep 1. Ini ializa ion: Ini ialize he sea ch space
and c ea e an ini ial solu ion poin wi hin he de ined
bounds.
S ep 2. Sel -Simila i y: The algo i hm gene a es new
solu ion poin s by pe u bing he cu en solu ion using
a andom ec o wi h a speci ic s uc u e based on a
ac al pa e n.
S ep 3. E alua ion: E alua e he objec i e unc ion
o each gene a ed solu ion poin .
S ep 4. Selec ion: Choose he bes solu ion poin
among he cu en one and he newly gene a ed ones
based on he objec i e unc ion alues. The selec ed
solu ion becomes he cu en one.
S ep 5. Te mina ion: The algo i hm epea s s eps
2-4 o a speci ied numbe o i e a ions o un il con e -
gence c i e ia a e me .
S ep 6. Global Op imum: The algo i hm aims o
con e ge o he global op imum by explo ing he en-
i e sea ch space h ough i s sel -simila i y and an-
domness.
3.1. F ac als
Some common me hods a e used such as an i e a i e
unc ional sys em, L sys em, ini e di ision p inciple,
and andom c acking o gene a e ac al shapes. These
me a-heu is ic algo i hms a e based on ac al ea u es
as a sea ch algo i hm ha achie es good esul s bo h
in e ms o accu acy and con e gence ime [36].
1) Random ac als
Random p ocesses such as Gaussian walk, ac al s uc-
u e, Le y ligh , osmo ic clus e , B own ee and
B ownian mo ion ajec o ies a e used o educe he
numbe o i e a ions o he algo i hm and gene a e he
s ochas ic ac al. Fo simplici y, conside o ming a
sequence wi h he ini ial e m loca ed a a andom po-
si ion. Then andom pa icles a e o med a ound he
o iginal pa icle and cause di usion. The andom walk
algo i hm is applied o simula e he di use . The pa -
icles p oduced by di usion s ick o he pa icles ha
make i up and o m a g oup o pa icles. Du ing he
o ma ion, he p obabili y o pa icles being pulled o
he edge is g ea e han ha o pa icles en e ing he
middle. Because o his p ope y, i leads o a b anched
clus e as shown in Figu e 6 [36].
2) Dielec ic b eakdown
Resea ch on dielec ic b eakdown cha ac e is ics ound
ha complex models can be applied o simula e he
b anching endency o dielec ic punc u e. Examples
a e lasho e and ligh ning. Niemeye e al. [32] in o-
duced dielec ic b eakdown using a andom model and
showed ha b anch discha ge pa e ns ollow ac al
cha ac e is ics. This model is ela i ely simila o he
new Di usion Limi ed Agg ega ion (DLA) model.
3.2. F ac al sea ch
1) Me hodology o SFS
The co e me hodology o he SFS algo i hm e ol es
a ound wo main componen s: ac ional calculus and
s ochas ic sea ch.
2) F ac ional Calculus
- F ac ional O de : U ilizes non-in ege o de s o di -
e en ia ion and in eg a ion, p o iding a lexible ame-
wo k o cap u e sys em dynamics wi h memo y e ec s.
- Memo y E ec : Helps in e aining his o ical sea ch
in o ma ion, which guides he cu en sea ch p ocess
mo e e ec i ely.
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- F ac al sea ch applies he ollowing h ee hypo he-
ses:
- Each poin will ha e an elec ic po en ial ene gy.
- Each poin will di use and andomly gene a e a
numbe o poin s.
- In each gene a ion only keep some o he bes
poin s.
Assume P(1 ≤P≤20) is he numbe o pa icles
examined. Ini ially, each Pipa icle is andomly placed
in he sea ch a ea wi h he same ene gy Eias ollows:
Ei=E
P(14)
whe e, he maximum ene gy is E.
To op imize he ission, each pa icle will be di used
in each gene a ion and gene a e a limi by he Le y
ligh . In his model, use Le y ligh o he DLA de-
elopmen model. The Lé y ligh is desc ibed by (13)
as ollows:
L(x) = 1
π
∞
Z
0e(−αq−β)cos(qx)dx (15)
whe e, αis he dis ibu ion coe icien ; βis he dis i-
bu ion index, 0< β ≤2.
Figu e 7 shows he di usion p ocess ha c ea es new
pa icles a ound he o iginal pa icle wi h andom po-
si ions.
As a esul o he di usion p ocess, a numbe o pa -
icles a e p oduced q(1≤q≤ he maximum di usion
numbe (MDN)). To gene a e each o hese pa icles,
bo h he Le y ligh and Gaussian a e applied as o -
mulas (12) and (13):
xq
i=xi+αq
i⊗Le y(λ)(16)
xq
i=xi+β×G(17)
whe e G=Gaussian(Pi,|BP|− (γ×BP −γ′×Pi);
β=log(ge)
ge;geis he numbe o gene a ions; BP(Bes
Poin ) is he bes sco e; γand γ′∈[0,1].
To ge a good sco e o bo h he Lé y ligh me hod
and he Gaussian dis ibu ion, he ac al sea ch
me hod andomly uses bo h me hods. Because he
Lé y dis ibu ion gi es a as con e gence algo i hm,
and he Gaussian dis ibu ion gi es be e esul s.
Since he app oaches all depend on s ochas ic p o-
cesses, as abso p ion canno be gua an eed. The e-
o e, αis an impo an pa ame e o as con e gence.
Two o mulas a e conside ed o α, one o a b oade
sea ch, and he o he o a highe p ecision sea ch:
αi=U−L
(ge×log (Ei))ε(18)
whe e, Eiis he ene gy o he Pi;Uis uppe bound
and Lis lowe bound o he sea ch a ea; εis usually
aken as 3/2.
A e he di usion has de e mined he posi ion o he
pa icles, he ene gy dis ibu ion be ween he pa icles
is c ea ed. Pa icles wi h be e a ge alues will ha e
a highe ene gy dis ibu ion. Each di use pa icle has
a a ge alue Fiwhe e i= 1,2, . . . , q. The ene gy is
dis ibu ed o he poin s as ollows:
Ej
i= Fi
Fi+Pq
k=1 Fk×Ei(19)
whe e, Fiis he poin be o e di usion.
Because o he complex di usion, only some o he
bes pa icles will be selec ed o he nex . The ene gy
o he disca ded pa icles will be dis ibu ed o he se-
lec ed pa icles and new pa icles will be c ea ed.
The o al ene gy o he emo ed poin s is Φ;µis he
a io o he ene gy dis ibu ion be ween he selec ed
poin s and he newly c ea ed poin s. The ene gy dis-
ibu ed o he emaining poin s is as ollows:
E +1 =E F /
ξ
X
k=1
Fk!×Φ!×µ(20)
whe e E +1 and E is he ene gy o he h poin a e
and be o e he ene gy dis ibu ion, ξis he o al o
numbe poin s in he i e a ion. Fo each di use poin ,
he numbe o newly o med and andomly posi ioned
poin s in he sea ch space is calcula ed as ollows:
υ=log(Ne)
log (MDN) (21)
whe e, Neis he numbe o elimina ed poin s.
The ene gy dis ibu ion o each p oduced poin is
equal:
E′
c=Φ (1 −µ)
υ, c = 1,2, ..., υ (22)
3.3. S ochas ic F ac al Sea ch
algo i hm
SFS is a mode n op imiza ion algo i hm designed o
add ess complex, mul i-dimensional op imiza ion p ob-
lems ha adi ional me hods o en s uggle o sol e
e ec i ely. These p oblems a e p e alen in a ious
ields, including enginee ing, inance, and a i icial in-
elligence, whe e inding he global op imum in a highly
nonlinea and mul imodal landscape is c ucial. SFS is
inspi ed by he na u al p ocess o ac al g ow h, cha -
ac e ized by sel -simila i y and ecu si e pa e n gen-
e a ion. The me hodology in ol es wo main phases:
di usion and upda e [36].
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Gene a e candida e solu ions by c ea ing andom
walks in luenced by he ac al dimension. Because
F ac al Sea ch does no exchange in o ma ion be ween
pa icles. The e o e, SFS adds an upda e p ocess o
balance he accu acy and con e gence ime o he al-
go i hm.
Lé y ligh and Gaussian walking me hod ha e been
applied o gene a e new pa icles by di usion p ocess
[27]. The Gaussian s eps in ol ed in he di usion phase
a e calcula ed as ollows:
Gaussian_walk1=G(µBP , σ)+(ε×BP −ε′×Pi)
(23)
Gaussian_walk2=G(µP, σ)(24)
whe e, εand ε′a e andomly dis ibu ed; µBP and σ
a e Gaussian pa ame e s; µPand σa e he second wo
Gaussian pa ame e s.
The s anda d de ia ion is as ollows:
σ=|β×(Pi−BP)|(25)
The single sea ch me hod is applied o educe he
size o he Gaussian walk log(ge)/ge, so he con e gence
ime is as e .
A ini ializa ion, poin s a e andomly ini ialized
based on uppe and lowe bound cons ain s. Ini ialize
he j h poin , Pjas ollows:
Pj=L+ε(U−L)(26)
whe e, εis a andom dis ibu ion and is limi ed o
he in e al [0,1]. The poin s will p obe a ound he
cu en posi ion o conside he sea ch space in he
di usion p ocess. Besides, wo s a is ical p ocesses a e
pe o med o a be e spa ial sea ch. The i s s a is ic
is pe o med on each ec o , and hen he nex s a is ic
is applied on all poin s.
The i s s a is ical p ocess, ank all poin s by (27),
Nis he o al numbe o poin s in g oup. Then each
i h poin is assigned a p obabili y alue as ollows:
Pai= ank(Pi)
N(27)
Equa ion (27) means ha he be e he sco e, he
g ea e he p obabili y o ha sco e. I helps he bad
poin s inc ease he chance o change posi ion. The
chances o inding a be e solu ion will imp o e in he
nex gene a ion.
P′
i(j) = P (j)−ε×(P (j)−Pi(j)) (28)
whe e, P′
iis he new poin ; P and P a e andomly
chosen poin s om he g oup.
(a) The lowcha o he SFS algo i hm.
(b) The di usion p ocess o he SFS algo i hm.
Fig. 5: The SFS algo i hm lowcha and Di usion p ocess al-
go i hm lowcha [36].
This p ope y is in ended o be e explo e and sa -
is y he di e se na u e o he algo i hm based on wo
s a is ical p ocesses [36]. All he poin s ob ained om
he p e ious p ocess will be e- anked acco ding o (27)
be o e he second p ocess is pe o med. Simila o he
i s p ocess, i he P′
isa is ies he condi ion Pai< ε,
he posi ion is changed acco ding o (29).
Pi′′ =Pi′−ˆε(P ′−BP)|ε′≤0.5
Pi′′ =Pi′−ˆε(P ′−P ′)|ε′>0.5(29)
whe e, P′
and P′
a e andomly chosen poin s om he
i s p ocess. The P”iis new poin i i s objec i e unc-
ion alue is be e han P′
i.
S ep 1: Ini ialize popula ion size: N,MDN,
maxi e , and G.
S ep 2: Find he bes sco e (BP) by calcula ing he
i ness unc ion.
S ep 3: Check condi ion:
- I G≤maxi e : Ou pu esul s.
- I G>maxi e : Call Di usion P ocess.
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S ep 4: Di usion P ocess
- Se MDN.
- Gene a e new pa icles, i MDN is eached.
- The new poin will eplace he cu en poin .
- Iden i y he BP o his phase.
S ep 5: Upda ing P ocess
The Fi s Upda ing P ocess:
Rank poin using Eq. (24).
- Gene a e a new poin P′
i(i Pai< ε), using Eq.
(28).
- The new poin (P′
i) will eplace he cu en poin
(Pi). - Iden i y he BP o his p ocess.
The Second Upda ing P ocess:
Rank pa icles using Eq. (27).
- Gene a e a new poin P′′
i(i Pai< ε), using Eq.
(29). - The new poin (P′′
i) will eplace he cu en
poin (P′
i). - Iden i y he BP.
S ep 6: Check p ocess s op condi ion:
The BP is he op imal solu ion i he maxi e is
eached.
O he wise, go o S ep 3:.
Upda e Phase: Imp o e candida e solu ions by em-
ploying andom walks ha allow o bo h local and
global sea ch capabili ies.
Selec ion: E alua e he i ness o each candida e so-
lu ion and selec he bes ones o o m he new popu-
la ion.
Con e gence Check: Repea he di usion and up-
da e phases un il con e gence c i e ia, such as a p e-
de ined numbe o i e a ions o an accep able i ness
le el, a e me .
Explo a ion and Exploi a ion Balance: SFS e -
ec i ely balances explo a ion (global sea ch) and
exploi a ion (local sea ch) h ough i s dual-phase
me hodology, enhancing i s abili y o ind he global
op imum. SFS can handle la ge-scale op imiza ion
p oblems due o i s inhe en scalabili y and adap abil-
i y o di e en p oblem sizes.
The i e a i e na u e and ex ensi e explo a ion mech-
anisms, like Le y ligh s, can be compu a ionally ex-
pensi e, especially o e y la ge p oblems. The pe o -
mance o SFS can be sensi i e o i s pa ame e se ings,
such as he s ep size in Le y ligh s and he ac al di-
mension, equi ing ca e ul uning. Implemen ing SFS
can be mo e complex compa ed o adi ional op imiza-
ion algo i hms, necessi a ing a deepe unde s anding
o ac al ma hema ics and s ochas ic p ocesses.
(a) The i s upda e p ocess.
(b) The second upda e p ocess.
Fig. 6: The upda e p ocesses o he SFS algo i hm [36].
4. Nume ical Resul s
Wind a m da a is e y impo an o he p oblem
o de e mining he op imal loca ion o u bines in
a wind a m o educe wake-up e ec s and achie e
maximum gene a ing capaci y. The wind a m p o-
posed in his pape is an onsho e wind a m, e e -
enced om he WAsP wo kspace sample, ilename Ve -
sion8Wind a m.wh [25]. The wind a m is a complex
e ain wi h ele a ions anging om 146.7m o 350m.
The a e age wind speed o he p ojec is 7.25m/s and
he a e age wind ene gy densi y is 388 W/m2. The
goal o he p oblem is he op imiza ion o wind a m
layou o each p oposed u bine ype so ha he en-
e gy powe ou pu and he powe ac o o he wind
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a m is he bes , he eby p oposing he selec ion o he
app op ia e u bine.
Tab. 2: Inpu pa ame e s.
Inpu pa ame e s
Roughness (Z0)0.083
Wind eloci y in ee low (V0)7.25m/s
Hub heigh (h) 50m
Wind Fa m dimension 3200m x 4200m
Th us coe icien (C )0.75
Wind densi y 388W/m2
This pape p oposes o selec 3 ypes o wind u -
bines a anged on he same wind a m o de e mine
he op imal layou and selec he app op ia e ype o
u bine. Case s udies a e summa ized in he ollowing
Table 3.
Among hese pa ame e s, SFS is sensi i e o MDN.
The esea ch esul s show ha he di usion numbe
can a ec he pe o mance o he p oblem, and de-
pends on he op imiza ion p oblem. Some esea ch
esul s show ha some unc ions o he p oblem a e
signi ican ly imp o ed when inc easing MDN. Howe e ,
inc easing MDN will a ec he con e gence ime and
con e gence speed o he p oblem, so i needs o be
conside [36].
S.Walk is a di use walk and an op ional pa ame e .
(S.Walk = 0 o he i s Gaussian walk and simple
p oblems. S.Walk = 1 o he second Gaussian walk
and ha d p oblems).
The con ol pa ame e s o he men ioned algo i hm
used in case s udies a e gi en acco ding o Table 3.
The ene gy p oduced by he wind a m is g ea ly
a ec ed by wind speed and wind di ec ion. Speci y
he wind speed le el o shu down he u bine as below
3m/s o abo e 25m/s [37]. The di ec ion o he u bine
mus be pe pendicula o he wind di ec ion o ecei e
maximum wind ene gy [38]. The wind a las in he
p ojec a ea is calcula ed based on long- e m co ec ed
wind measu emen s aking in o accoun he e ec s o
obs acles, oughness, and e ain ele a ion. Below is
he wind da a o he p ojec a ea e e enced om he
WAsP so wa e [25] as shown in Figu e 7.
Figu e 7 shows he p e ailing wind di ec ion o he
p ojec a ea wi h an angle om 2700 o 3000. A sim-
pli ied wind ose is used, which is di ided in o 2 sec o s
(sec o s 10 and 11). The highes p obabili y occu s a
a wind speed o 7.25m/s, accoun ing o abou 12%.
The case s udies using he same da a sou ce om he
WAsP so wa e lib a y. The e ain is complex and he
al i ude a ies om 146.7m o 350 m.
Fig. 7: The wind ose and wind speed dis ibu ion.
4.1. Case s udy 1
Case s udy 1 conside s he op imal a angemen o 11
wind u bines, u bine capaci y is 0.85MW. The esul s
o he op imal a angemen o u bines in he wind
a m by he SFS algo i hm will be compa ed wi h he
esul s calcula ed by he PSO algo i hm and he sim-
ula ion esul s by he windPRO so wa e.
A compa ison o he con e gence cu es o he bes
i ness alues ob ained om he SFS and PSO algo-
i hm is shown in Figu e 8. This g aph gi es in o ma-
ion he con e gence cu e o SFS and PSO algo i hms.
I is clea ha he SFS algo i hm has eached s abili y
Fig. 8: The con e gence cu es o SFS and PSO.
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