Ci a ion: Pandya, S.B.; Visuma hi, J.;
Mahdal, M.; Mahan a, T.K.; Jangi , P.
A No el MOGNDO Algo i hm o
Secu i y-Cons ained Op imal Powe
Flow P oblems. Elec onics 2022,11,
3825. h ps://doi.o g/10.3390/
elec onics11223825
Academic Edi o : Da io Di Ca a
Recei ed: 26 Sep embe 2022
Accep ed: 16 No embe 2022
Published: 21 No embe 2022
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elec onics
A icle
A No el MOGNDO Algo i hm o Secu i y-Cons ained
Op imal Powe Flow P oblems
Sunda am B. Pandya 1, James Visuma hi 2, Mi osla Mahdal 3,* , Tapan K. Mahan a 4and P adeep Jangi 5
1Depa men o Elec ical Enginee ing, Sh i K.J. Poly echnic, Bha uch 392 001, India
2Depa men o Compu e Science and Enginee ing, Vel Tech Ranga ajan D Sangun hala R&D Ins i u e o
Science and Technology, Chennai 600 062, India
3Depa men o Con ol Sys ems and Ins umen a ion, Facul y o Mechanical Enginee ing,
VSB-Technical Uni e si y o Os a a, 17. Lis opadu 2172/15, 708 00 Os a a, Czech Republic
4School o Mechanical Enginee ing, Vello e Ins i u e o Technology, Chennai 600 127, India
5Rajas han Rajya Vidyu P asa an Nigam, Losal, Sika 332 025, India
*Co espondence: mi osla [email p o ec ed]
Abs ac :
The cu en esea ch in es iga es a new and unique Mul i-Objec i e Gene alized No -
mal Dis ibu ion Op imiza ion (MOGNDO) algo i hm o sol ing la ge-scale Op imal Powe Flow
(OPF) p oblems o complex powe sys ems, including enewable ene gy sou ces and Flexible AC
T ansmission Sys ems (FACTS). A ecen ly epo ed single-objec i e gene alized no mal dis ibu ion
op imiza ion algo i hm is ans o med in o he MOGNDO algo i hm using he nondomina ed so ing
and c owding dis ancing mechanisms. The OPF p oblem ge s e en mo e challenging when sou ces
o enewable ene gy a e in eg a ed in o he g id sys em, which a e un eliable and luc ua ing. FACTS
de ices a e also being used mo e equen ly in con empo a y powe ne wo ks o assis in educing
ne wo k demand and conges ion. In his s udy, a s ochas ic wind powe sou ce was used wi h
di e en FACTS de ices, including a s a ic VAR compensa o , a hy is o - d i en se ies compensa o ,
and a hy is o —d i en phase shi e , oge he wi h an IEEE-30 bus sys em. Posi ions and a ings
o he FACTS de ices can be in ended o educe he sys em’s o e all uel cos . Weibull p obabili y
densi y cu es we e used o highligh he s ochas ic cha ac e o he wind ene gy sou ce. The bes
comp omise solu ions we e ob ained using a uzzy decision-making app oach. The esul s ob ained
on a modi ied IEEE-30 bus sys em we e compa ed wi h o he well-known op imiza ion algo i hms,
and he ob ained esul s p o ed ha MOGNDO has imp o ed con e gence, di e si y, and sp ead
beha io ac oss PFs.
Keywo ds:
FACTS con olle ; MO-OPF; me a-heu is ics; p obabili y densi y unc ion; s ochas ic; WTGS
1. In oduc ion
Cons ain -based op imiza ion p oblems wi h mul iple objec i es a e he mos p e a-
len ype. In con as o single-objec i e op imiza ion p oblems, mul i-objec i e op imiza-
ion p oblems ha e a wide a ie y o op imal solu ions. The PF is an asso men o pe ec
esponses [
1
,
2
]. A mul i-objec i e op imiza ion app oach mus be able o loca e solu ions
ha a e uni o m in he gene a ed PFs and a e wo kable op imum solu ions o add ess
mul i-objec i e p oblems [
3
]. Mul i-objec i e op imiza ion app oaches a e challenged by
he simul aneous achie emen o hese many objec i es [
4
]. MH algo i hms a e ypically
es ed on simple , well-known op imiza ion scena ios. Howe e , unlike classic sea ch p ob-
lems, enginee ing design asks can ha e di e en speci ica ions. Modi ying and de eloping
he algo i hm o hem is he mos e ec i e way o op imize o hem. The ealm o appli-
ca ion o mul i-objec i e op imiza ion algo i hms is qui e as , anging om machining
p ocesses [5,6], o ehicle ou ing [7], o op imizing AI sys ems [8].
Powe sys ems esea che s ha e been seeking solu ions o he OPF challenges o many
decades. One issue wi h managing powe sys ems and making plans o mode n elec ical
Elec onics 2022,11, 3825. h ps://doi.o g/10.3390/elec onics11223825 h ps://www.mdpi.com/jou nal/elec onics
Elec onics 2022,11, 3825 2 o 34
ene gy ne wo ks is wo king wi h sys ems ha use non-con en ional sou ces o ene gy.
Ullah e al. [
9
] de eloped a hyb id phaso pa icle swa m op imiza ion and g a i a ional
sea ch algo i hm o add ess he OPF p oblem in he wind and sola ene gy sys ems ha a e
connec ed o elec ical powe g ids, while accoun ing o he con ol a iables (PPSOGSA).
Fo he OPF p oblem wi h wind and sola sys ems, he de eloped PPSOGSA algo i hm
p oduced ou s anding and help ul esul s. In Ela a ’s esea ch, he OPF p oblem was
p incipally modelled ma hema ically using a combined hea and powe sys em wi h
s ochas ic wind ene gy. On an IEEE 30-bus es sys em unde a ious es si ua ions, he
sugges ed me hod was assessed. The o mula ion o he OPF p oblem co e ing ene gy
sou ces and he sugges ed app oach o sol e i p oduced e ec i e answe s in con as o p e-
exis ing algo i hms [
10
] ha we e used o add ess simila p oblems. Anongpun e al. [
11
]
used IEEE 30- and 118-bus es sys ems o s udy he use o enhanced pa icle swa m
op imiza ion (PSO) o sol e a mul i-objec i e OPF p oblem using a wind ene gy sys em ha
combined chao ic mu a ion and s ochas ic weigh s. When compa ed o he o he algo i hms
included in hei s udy, he sugges ed me hod p oduced be e esul s. Salku i [
12
] used he
glowwo m swa m op imiza ion me hod o o e a solu ion o a mul i-objec i e OPF p oblem
equi ing a mode n elec ical ene gy sys em ha u ilized wind ene gy. On IEEE 30- and
300-bus es sys ems, he me hodology was examined in many ope a ional scena ios. The
simula ion indings indica ed ha he p oposed me hodology migh o e an al e na i e.
A lowe pollina ion algo i hm was used by Ka hi a an e al. [
13
] o add ess he OPF
p oblem using coal-based, wind, and sola ene gy sys ems. In a a ie y o es si ua ions,
he au ho s used hei me hod o es sys ems o he IEEE 30-bus and Indian u ili y 30-bus.
Duman e al. [
14
] used di e en ial e olu iona y pa icle swa m op imiza ion o add ess
he OPF p oblem (DEEPSO) wi h manageable wind and sola (PV) ene gy sou ces. IEEE
30-, 57-, and 118-bus es sys ems we e used o e alua e he DEEPSO echnique o explo e
he issue unde a ious objec i e unc ions. DEEPSO p oduced be e simula ion esul s
when compa ed o he o he op imiza ion app oaches ha we e looked a [
14
]. They
ecommended using FACTS de ices such as a hy is o -con olled phase shi e (TCPS) and
a hy is o -con olled se ies capaci o (TCSC) o sol e he OPF p oblem. To accoun o
he unce ain ies associa ed wi h wind ene gy ins alla ion, hey used chao ic maps and a
modi ied e sion o he PSOGSA (pa icle swa m op imiza ion and g a i a ional sea ch
algo i hm). The me hod p esen ed [
15
] appea s o be a po en ial app oach o a solu ion
based on he indings o simula ions. Biswas e al. [
16
] sol ed he OPF challenge, which
inco po a ed coal-based, wind, sola , and small-hyd o ene gy sou ces coupled o IEEE 30-
bus es sys ems, by unning mul iple ounds o he mul i-objec i e e olu iona y algo i hm.
The cons ained mul i-objec i e popula ion ex emal op imiza ion (CMOPEO) echnique
was used o handle he wind and sola -in eg a ed OPF p oblem by Chen e al. [
17
], who
also es ed he me hod on an IEEE 30-bus o di e en scena ios. Addi ionally, esea ch
has been done on he e olu iona y pa icle swa m op imiza ion (EPSO) [
18
], he hyb id
di e en ial e olu ion and symbio ic o ganisms sea ch algo i hm (HMICA-SQP) [
19
], he
success-his o y-based adap a ion o di e en ial e olu ion wi h supe io i y o easible
solu ions (SHADE-SF) [
20
], and he hyb id modi ied impe ialis compe i i e algo i hm and
sequen ial quad a ic p og amming algo i hm (HMICA-SQP) [
21
]. Pandya and Ja iwala [
22
]
add essed single and mul i-objec i e OPF issues by in eg a ing wi h a ious sus ainable
ene gy sou ces using ecen ly de eloped me aheu is ics algo i hms. Biswas e al. [
23
]
analyzed he in eg a ion o h ee FACTS de ices, wind u bines, and coal- i ed powe
plan s. The success his o y-based adap i e di e en ial e olu ion (SHADE) me hod was
used o conduc he in es iga ion. Acco ding o he “No F ee Lunch (NFL)” heo em [
24
],
no me aheu is ic can sol e e e y issue ha occu s in eal-wo ld si ua ions. This heo em has
opened he doo o he c ea ion o bo h no el me aheu is ic echniques and imp o emen s
o exis ing ones.
When using he Gene alized No mal Dis ibu ion Op imiza ion me hod [
25
] in mul i-
objec i e op imiza ion scena ios, se e al hings need o be aken in o accoun . The ini ial
p oblem in mul i-objec i e gene alized no mal dis ibu ion op imiza ion is balancing con-
Elec onics 2022,11, 3825 3 o 34
e gence and di e gence a chi es. The algo i hm’s gene a ed solu ion se is il e ed ac-
co ding o a ce ain quali y me ic, and he non-domina ing solu ion se is kep in sepa a e,
ex e nal a chi es. In he li e a u e, he e a e nume ous ecommenda ions o cons uc ing
a chi es ha could be employed in he mul i-objec i e Gene alized No mal Dis ibu ion
Op imiza ion me hod. The p e-de ined maximum size a chi es a e widely employed, since
mo e non-domina ing solu ions can eme ge quickly. The algo i hm’s abili y o begin wi h
he s aigh o wa d de e mina ion o he equi ed popula ion size and he e minal condi-
ion is he mos no iceable ea u e o he GNDO. The loca ion o he pe son is au oma ically
changed by he gene alized no mal dis ibu ion unc ion (GNDO), which has a simple
cons uc ion. The bene i s and d awbacks o he GNDO algo i hm a e as ollows:
•
I o e s a as e and smoo he con e gence, especially o di icul p oblems, and i
s ikes he pe ec balance be ween explo a ion and exploi a ion.
•Local minima a e less likely o become en angled in elaxed con e gence.
•E o lessly simple, adap able, and simple o use
•
The adi ional GNDO may ha e issues wi h con e gence ends o become s uck in
na ow, decep i e op ima o challenging op imiza ion asks, such as high-dimensional
and mul imodal p oblems.
Cu en ly, bo h con en ional and non-con en ional ene gy sou ces equi e mo e s ud-
ies. The cu en body o esea ch ecommends using coal-based plus wind and FACTS
de ices, combined wi h single and mul i-objec i e op imum powe low (MOOPF) p ob-
lems. The con en ional IEEE 30-bus ne wo k has been al e ed o include non-con en ional
sou ces o esea ch pu poses. Using Weibull PDF, non-con en ional uni s’ s ochas ic be-
ha io s a e calcula ed. The gene a ing cos is sui ably adjus ed o accoun o ese e cos i
hese s ochas ic uni s a e o e -es ima ed and adjus ed o penal y cos in he case ha hey
a e unde es ima ed. Using he Gene alized No mal Dis ibu ion Op imiza ion me hod,
Pa e o solu ion clus e s a e disco e ed o he mul i-objec i e p oblem. The ollowing is a
lis o he con ibu ions made by his s udy:
1.
This wo k ocuses on he ma hema ical modelling o he single and mul iple-objec i e
OPF issue modelled, which akes in o accoun bo h con en ional uni s and non-
con en ional sou ces o ene gy uni s, as well as FACTS de ices.
2.
The app op ia e p obabili y densi y unc ions (PDFs) a e modeled in he second s age
o desc ibe he wind powe plan s’ andom beha io .
3.
S ochas ic non-con en ional sou ces o ene gy sou ces a e among he single and
mul iple-objec i e OPF issues o which he Non-Domina ed So ing Gene alized
No mal Dis ibu ion Op imiza ion (NSGNDO) echnique is used o de elop solu ions.
4.
S udies and pe o mance e alua ions o he MOGNDO algo i hm using empi ical
compa isons a e conduc ed.
The no ion o he ma hema ical models o coal-based powe , wind powe , and FACTS
de ices is p esen ed in Sec ion 2o he s udy. An explana ion o he objec i es ha need o
be op imized is included in Sec ion 3. Sec ion 4p o ides an explana ion and illus a ions o
he mul i-objec i e GNDO echnique. Sec ion 5p esen s nume ical esul s and discussion,
and Sec ion 6p o ides concluding ema ks.
2. Ma hema ical Rep esen a ions
The case s udies p esen ed he e es uc u e he o iginal IEEE 30-bus es appa a us.
The modi ied app oach inco po a es wind u bines and FACTS de ices, and is lis ed in
Table 1. The equipmen used o he analysis is depic ed in Figu e 1. The placemen
and a ings o FACTS de ices a e depic ed in he diag am wi h do ed lines because hey
ha e been op imized. The sec ion below p o ides in o ma ion on he cos s o adi ional
coal-based p oduc ion acili ies and plan s using non-con en ional sou ces o ene gy.
Elec onics 2022,11, 3825 4 o 34
Table 1. Tes sys em key ea u es o analysis.
Pa icula s Quan i y De ails
To al buses 30 [23]
To al b anches 41 [23]
Coal-based gene a o s (TG1; TG2; TG3; TG4) 4Buses: 1 (swing), 2, 8 and 13
Wind gene a o s (WG1; WG2) 2Bus-5 and Bus-11
Tap changing ans o me s 4B anches: 11, 12, 15 and 36
SVC 2 Op imal bus and a ing de i ed
TCSC 2 Op imal b anch posi ion and a ing de i ed
TCPS 2 Op imal placemen and a ing de i ed
Demand - 283.4 MW, 126.2 MVA
Elec onics 2022, 11, x FOR PEER REVIEW 4 o 37
2. Ma hema ical Rep esen a ions
The case s udies p esen ed he e es uc u e he o iginal IEEE 30-bus es appa a us.
The modi ied app oach inco po a es wind u bines and FACTS de ices, and is lis ed in
Table 1. The equipmen used o he analysis is depic ed in Figu e 1. The placemen and
a ings o FACTS de ices a e depic ed in he diag am wi h do ed lines because hey
ha e been op imized. The sec ion below p o ides in o ma ion on he cos s o adi ional
coal-based p oduc ion acili ies and plan s using non-con en ional sou ces o ene gy.
Table 1. Tes sys em key ea u es o analysis.
Pa icula s
Quan i y
De ails
To al buses
30
[23]
To al b anches
41
[23]
Coal-based gene a o s (TG1; TG2; TG3;
TG4)
4
Buses: 1 (swing), 2, 8 and 13
Wind gene a o s (WG1; WG2)
2
Bus-5 and Bus-11
Tap changing ans o me s
4
B anches: 11, 12, 15 and 36
SVC
2
Op imal bus and a ing de i ed
TCSC
2
Op imal b anch posi ion and a ing
de i ed
TCPS
2
Op imal placemen and a ing de i ed
Demand
-
283.4 MW, 126.2 MVA
Figu e 1. Adap ed IEEE 30-bus scheme wi h powe uni s and FACTS policies [23].
2.1. Cos o Coal-Based Powe Uni s
The gene alized quad a ic equa ion o he calcula ion o gene a ion cos is ex-
p essed in (1) in $/h [23]:
Figu e 1. Adap ed IEEE 30-bus scheme wi h powe uni s and FACTS policies [23].
2.1. Cos o Coal-Based Powe Uni s
The gene alized quad a ic equa ion o he calcula ion o gene a ion cos is exp essed
in (1) in $/h [23]:
CT0(PTG)=∑NTG
i=1ai+biPTGi +ciP2
TGi (1)
Fo a mo e p ac ical case, he al e poin e ec included:
CT(PTG)=∑NTG
i=1ai+biPTGi +ciP2
TGi +di×sinei×Pmin
TGi −PTGi(2)
The alues o bo h coal-based p ice cons an s and emana ion cons an s wi h a ious
scena ios a e shown in [23].
2.2. Toxic Gas Emana ion
Pollu ed gases a e eleased by using coal-based plan s. So, oxic gas emana ions in
ons pe hou can be de e mined as (in on/h):
F2 = ,E=∑NTG
i=1hαi+βiPTGi +γiP2
TGi×0.01 +ωie(µiPTGi)i(3)
Elec onics 2022,11, 3825 5 o 34
The oxic gas emana ion cons an s o coal-based powe plan s a e aken om [22].
2.3. Di ec Cos o S ochas ic Non-Con en ional Sou ces Plan s
I is pa icula ly challenging o in eg a e non-con en ional ene gy sou ces in o he
powe g id since hey a e s ochas ic. The independen sys em ope a o (ISO) is esponsible
o managing hese non-con en ional ene gy sou ces. Due o his, he p i a e ope a o
mus con ac wi h he g id o ISO o a speci ic quan i y o planned powe . The scheduled
elec ici y mus be main ained by he ISO scheduled powe . I hese non-con en ional
sou ces a e unable o main ain he planned powe , he ISO is liable o he absence o powe .
So, i a need a ises, he e a e spinning ese e equi emen s. This spinning ese e inc eases
cos s o he ISO, and his ci cums ance is known as an o e es ima ion o non-con en ional
sou ces. Con e sely, i non-con en ional sou ces o med mo e ene gy han was planned,
i migh go o was e, due o unde use. The e o e, he ISO mus accep he penal y cha ge.
The scheduled powe cos , he o e es ima ion cos caused by he spinning ese e, and he
penalized cos caused by he unde es ima ion a e he h ee cos s ela ed o elec ici y. The
di ec cos linked o wind a ms is demons a ed wi h he
Pws
scheduled powe om he
same sou ces as:
Cw(Pws)=gwPws (4)
2.4. Inde e mina e Non-Con en ional Sou ces o Wind Powe Cos
Due o he e a ic na u e o wind, he wind a m occasionally p oduces less ene gy
han expec ed. This means ha i demand inc eases, i needs he spinning ese e o
main ain he ag eed-upon amoun o scheduled powe . I is some imes easible ha he
eal powe gene a ed by wind a ms won’ be enough o mee demand and will ha e lowe
alues. Such powe is e e ed o as exagge a ed powe by an ambiguous esou ce. To
con ol his kind o unce ain y and p o ide end use s wi h a eliable powe sou ce, he
ne wo k ISO ope a es spinning ese es. The p ice o hi ing a backup gene a o o supply
he o e es ima ed powe is known as he ese e cos .
Rese e cos o he wind uni is o mula ed by:
CRw(Pws −Pwa )=KRw(Pws −Pwa )=KRw ZPws
0(Pws −pw) w(pw)dpw(5)
The possibili y exis s ha he wind a m will gene a e mo e powe han is equi ed,
which is he opposi e o he o e es ima ion scena io. Unde es ima ed powe is he e m
used o desc ibe such a si ua ion. I he e is no p o ision o managing he ou pu powe
om coal-based uni s, he excess powe will be los . Rega ding he ex a powe , he ISO
needs o be penalized.
The penal y cha ge o he wind uni is gi en by:
CPw(Pwa −Pws)=KPw(Pwa −Pws)=KPw ZPw
Pws
(pw−Pws) w(pw)dpw(6)
2.5. Unce ain y Models o S ochas ic Wind Uni s
In he edesigned IEEE-30, he wind powe gene a ing uni s ins alled a buses 5 and
11, which we e o iginally he mal gene a o s, we e eplaced. I should be no ed ha , as
o a compa ison poin o iew wi h he published e e ence a icle [
23
], in his pape , he
he mal DGs we e also eplaced wi h he wind u bines. This will ensu e compa ibili y o
he esul s ob ained by he p oposed algo i hm o he al eady published esea ch a icle [
23
].
The scale (c) and shape (k) cons an s o he p oposed Weibull model a e de ailed in Table 2.

Elec onics 2022,11, 3825 6 o 34
Table 2. PDF cons an s o wind powe plan s [23].
Wind a m No. o
Tu bines
Ra ed
Powe
Weibull PDF
Pa ame e s
Cos Cons an s
Di ec Rese e Penal y
WG5 (bus 5) 25 75 c= 9, k= 2 1.60 3.0 1.50
WG11 (bus 11) 20 60 c= 10, k= 2 1.75 3.0 1.50
The Weibull cu e and wind equency dis ibu ions in Figu e 2( o he bus 5 wind
plan ) and Figu e 3( o he bus 11 wind plan ) we e p oduced using 8000 Mon e-Ca lo
se ings. The s anda d p o ided explains he need o wind u bine design and speci ies
he maximum u bulence class IA ha is con i med o ope a e a he highes yea ly a e age
wind eloci y o 10 m/s a hub heigh . The o med shape
(k)
and scale
(c
) pa ame e s o
wind a ms a e gi en pa icula a en ion, because hei highes Weibull mean alue is ixed
a a ound 10. I is commonly known ha he wind speed dis ibu ion ollows he Weibull
PDF cu e.
Elec onics 2022, 11, x FOR PEER REVIEW 7 o 37
Figu e 2. Weibull PDF (bus 5).
Figu e 3. Weibull PDF (bus 11).
2.6. A e age Powe Calcula ion o Wind Plan s
The combined ou pu s o he 25 u bines in he a m a e aken as he wind uni
connec ed a bus 5. E e y u bine has a 3 MW ou pu a ing. The wind eloci y a ec s
he wind u bine’s p ecise ou pu , which a ies. We used he ollowing equa ions o ex-
p ess u bine ou pu powe in e ms o wind eloci y ( ) [23]:
𝑝𝑤(𝑣)={0, o 𝑣〈𝑣𝑖𝑛 𝑎𝑛𝑑 𝑣〉𝑣𝑜𝑢𝑡
𝑝𝑤𝑟(𝑣−𝑣𝑖𝑛
𝑣𝑟−𝑣𝑖𝑛) o 𝑣𝑖𝑛⩽𝑣⩽𝑣𝑟
𝑝𝑤𝑟 o 𝑣𝑟<𝑣⩽𝑣𝑜𝑢𝑡
(10)
The Ene con E82-E4 design speci ica ion is e e ed o o he 3 MW wind u bine.
The a ious speeds a e 𝑣𝑖𝑛 = 3 m/s, 𝑣𝑟 = 16 m/s, and 𝑣𝑜𝑢𝑡 = 25 m/s.
Figu e 2. Weibull PDF (bus 5).
Elec onics 2022, 11, x FOR PEER REVIEW 7 o 37
Figu e 2. Weibull PDF (bus 5).
Figu e 3. Weibull PDF (bus 11).
2.6. A e age Powe Calcula ion o Wind Plan s
The combined ou pu s o he 25 u bines in he a m a e aken as he wind uni
connec ed a bus 5. E e y u bine has a 3 MW ou pu a ing. The wind eloci y a ec s
he wind u bine’s p ecise ou pu , which a ies. We used he ollowing equa ions o ex-
p ess u bine ou pu powe in e ms o wind eloci y ( ) [23]:
𝑝𝑤(𝑣)={0, o 𝑣〈𝑣𝑖𝑛 𝑎𝑛𝑑 𝑣〉𝑣𝑜𝑢𝑡
𝑝𝑤𝑟(𝑣−𝑣𝑖𝑛
𝑣𝑟−𝑣𝑖𝑛) o 𝑣𝑖𝑛⩽𝑣⩽𝑣𝑟
𝑝𝑤𝑟 o 𝑣𝑟<𝑣⩽𝑣𝑜𝑢𝑡
(10)
The Ene con E82-E4 design speci ica ion is e e ed o o he 3 MW wind u bine.
The a ious speeds a e 𝑣𝑖𝑛 = 3 m/s, 𝑣𝑟 = 16 m/s, and 𝑣𝑜𝑢𝑡 = 25 m/s.
Figu e 3. Weibull PDF (bus 11).
Elec onics 2022,11, 3825 7 o 34
The ollowing o mula can be used o calcula e he p obabili y o wind eloci y , in
m/s, pu suing he Weibull PDF wi h shape ac o (k) and scale ac o (c) [23]:
( ) = k
c
c(k−1)e−(
c)k o 0< <∞(7)
The Weibull dis ibu ion’s mean is gi en as ollows [23]:
Mwbl =c∗Γ1+k−1(8)
and he gamma unc ion Γ(x)is exp essed in Equa ion (9):
Γ(x) = Z∞
0e− x−1d (9)
2.6. A e age Powe Calcula ion o Wind Plan s
The combined ou pu s o he 25 u bines in he a m a e aken as he wind uni
connec ed a bus 5. E e y u bine has a 3 MW ou pu a ing. The wind eloci y a ec s he
wind u bine’s p ecise ou pu , which a ies. We used he ollowing equa ions o exp ess
u bine ou pu powe in e ms o wind eloci y ( ) [23]:
pw( ) = 




0, o h in and i ou
pw  − in
− in  o in ≤ ≤
pw o < ≤ ou
(10)
The Ene con E82-E4 design speci ica ion is e e ed o o he 3 MW wind u bine. The
a ious speeds a e in = 3 m/s, = 16 m/s, and ou = 25 m/s.
2.7. Wind Powe P obabili ies Calcula ion
In ce ain anges o wind speeds, unce ain wind gene a ion is no iceable. The gene -
a ed powe would be 0 i he wind speed was g ea e han o less han he cu -ou speed o
cu -in speed. The u bine he eby p oduces he speci ied amoun o powe wi hin he ange
o he a ed and cu -ou wind speeds. These a e possible ways o desc ibe he likelihood o
hese a eas [23]:
w(pw){pw=0}=1−exp− in
ck+exp− ou
ck(11)
w(pw){pw=pw }=exp−
ck−exp− ou
ck(12)
Be ween he cu -in eloci y and he a ed eloci y o he wind, he wind p oduc ion
emains cons an . The ollowing can be used o exp ess he likelihood o he con inuous
zone [23]:
w(pw)=β( − in)
αβ∗pw  in +pw
pw
( − in)β−1
exp
− in +pw
pw ( − in)
α!β
(13)
2.8. Thy is o -Con olled Se ies Compensa o (TCSC) Modeling
The basic ci cui y o he TCSC is depic ed in Figu e 4. I consis s o a ixed se ies
capaci o (XC) and a eac o (XL) ope a ed by a hy is o . Fo he TCSC o unc ion as a
a iable capaci i e eac ance, eac ance
XC<XL
is aken in o conside a ion. By a ying he
i ing angle (
α
) o he hy is o s, he induc i e eac ance is changed, and o high alues o
induc i e eac ance, he leas co esponding capaci i e eac ance is p oduced (Open ci cui
Elec onics 2022,11, 3825 8 o 34
Induc i e b anch). As a esul , he TCSC’s e ec i e eac ance wi h cons an capaci i e
eac ance XCand a iable induc i e eac ance XL(α)can be w i en as [23]:
XTCSC(α) = XCXL(α)
XL(α)−XC
=−jXC(14)
Elec onics 2022, 11, x FOR PEER REVIEW 8 o 37
2.7. Wind Powe P obabili ies Calcula ion
In ce ain anges o wind speeds, unce ain wind gene a ion is no iceable. The gen-
e a ed powe would be 0 i he wind speed was g ea e han o less han he cu -ou
speed o cu -in speed. The u bine he eby p oduces he speci ied amoun o powe
wi hin he ange o he a ed and cu -ou wind speeds. These a e possible ways o de-
sc ibe he likelihood o hese a eas [23]:
𝑓𝑤(𝑝𝑤){𝑝𝑤=0}=1−exp[−(𝑣𝑖𝑛
𝑐)𝑘]+exp[−(𝑣𝑜𝑢𝑡
𝑐)𝑘]
(11)
𝑓𝑤(𝑝𝑤){𝑝𝑤=𝑝𝑤𝑟}=exp[−(𝑣𝑟
𝑐)𝑘]−exp[−(𝑣𝑜𝑢𝑡
𝑐)𝑘]
(12)
Be ween he cu -in eloci y and he a ed eloci y o he wind, he wind p oduc ion
emains cons an . The ollowing can be used o exp ess he likelihood o he con inuous
zone [23]:
𝑓𝑤(𝑝𝑤)=𝛽(𝑣𝑟−𝑣𝑖𝑛)
𝛼𝛽∗𝑝𝑤𝑟 [𝑣𝑖𝑛+𝑝𝑤
𝑝𝑤𝑟(𝑣𝑟−𝑣𝑖𝑛)]𝛽−1exp[−(𝑣𝑖𝑛+𝑝𝑤
𝑝𝑤𝑟(𝑣𝑟−𝑣𝑖𝑛)
𝛼)𝛽]
(13)
2.8. Thy is o -Con olled Se ies Compensa o (TCSC) Modeling
The basic ci cui y o he TCSC is depic ed in Figu e 4. I consis s o a ixed se ies
capaci o (XC) and a eac o (XL) ope a ed by a hy is o . Fo he TCSC o unc ion as a
a iable capaci i e eac ance, eac ance 𝑋𝐶<𝑋𝐿 is aken in o conside a ion. By a ying
he i ing angle (𝛼) o he hy is o s, he induc i e eac ance is changed, and o high
alues o induc i e eac ance, he leas co esponding capaci i e eac ance is p oduced
(Open ci cui Induc i e b anch). As a esul , he TCSC’s e ec i e eac ance wi h cons an
capaci i e eac ance 𝑋𝐶 and a iable induc i e eac ance 𝑋𝐿(𝛼) can be w i en as [23]:
𝑋𝑇𝐶𝑆𝐶(𝛼)=𝑋𝐶𝑋𝐿(𝛼)
𝑋𝐿(𝛼)−𝑋𝐶=−𝑗𝑋𝐶
(14)
Figu e 4. Basic S uc u e and model o TCSC [23].
The TCSC s a ic model, which is si ua ed in he pa h be ween buses m and n, is
shown in Figu e 4. Following he TCSC’s in eg a ion (desc ibed as a a iable capaci i e
eac ance mode), he ansmission line’s adjus ed eac ance (𝑋𝑒𝑞) is gi en by [23]:
𝑋𝑒𝑞=𝑋𝑚𝑛−𝑋𝑇𝐶𝑆𝐶=(1−𝜏)𝑋𝑚𝑛
(15)
whe e
𝜏=𝑋𝑇𝐶𝑆𝐶
𝑋𝑚𝑛
(16)
Figu e 4. Basic S uc u e and model o TCSC [23].
The TCSC s a ic model, which is si ua ed in he pa h be ween buses mand n, is shown
in Figu e 4. Following he TCSC’s in eg a ion (desc ibed as a a iable capaci i e eac ance
mode), he ansmission line’s adjus ed eac ance (Xeq) is gi en by [23]:
Xeq =Xmn −XTCSC =(1−τ)Xmn (15)
whe e
τ=XTCSC
Xmn (16)
The powe low equa ions o he line inco po a ing he TCSC a e w i en as [23]:
Pmn =V2
mgmn −VmVngmn cos(δm−δn).
−VmVnbmn sin(δm−δn)(17)
Qmn =−V2
mbmn −VmVngmn sin(δm−δn).
+VmVnbmn cos(δm−δn)(18)
Pnm =V2
ngmn −VmVngmn cos(δm−δn).
+VmVnbmn sin(δm−δn)(19)
Qnm =−V2
nbmn +VmVngmn sin(δm−δn).
+VmVnbmn cos(δm−δn)(20)
whe e
gmn = mn
2
mn + (xmn −xc)2(21)
bmn =−xmn −xc
2
mn + (xmn −xc)2(22)
Elec onics 2022,11, 3825 9 o 34
2.9. Model o Thy is o -Con olled Phase Shi e (TCPS)
Figu e 5displays he model o he TCPS placed be ween he line ha connec s buses
m
and
n
. The powe low equa ions o he line can be exp essed as below, assuming ha is
he phase shi angle φis in oduced by he TCPS:
Pmn =V2
mgmn
cos2φ−VmVn
cos φ[gmn cos(δm−δn+φ).
+bmn sin(δm−δn+φ)] (23)
Qmn =−V2
mbmn
cos2φ−VmVn
cos φ[gmn sin(δm−δn+φ).
−bmn cos(δm−δn+φ)](24)
Pnm =V2
ngmn −VmVn
cos φ[gmn cos(δm−δn+φ).
−bmn sin(δm−δn+ϕ)](25)
Qnm =−V2
nbmn +VmVn
cos φ[gmn sin(δm−δn+φ).
+bmn cos(δm−δn+ϕ)](26)
Elec onics 2022, 11, x FOR PEER REVIEW 9 o 37
The powe low equa ions o he line inco po a ing he TCSC a e w i en as [23]:
P𝑚𝑛=𝑉𝑚2𝑔𝑚𝑛−𝑉𝑚𝑉𝑛𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛)
−𝑉𝑚𝑉𝑛𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛)
(17)
𝑄𝑚𝑛=−𝑉𝑚2𝑏𝑚𝑛−𝑉𝑚𝑉𝑛𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛)
+𝑉𝑚𝑉𝑛𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛)
(18)
P𝑛𝑚=𝑉𝑛2𝑔𝑚𝑛−𝑉𝑚𝑉𝑛𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛)
+𝑉𝑚𝑉𝑛𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛)
(19)
𝑄𝑛𝑚=−𝑉𝑛2𝑏𝑚𝑛+𝑉𝑚𝑉𝑛𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛)
+𝑉𝑚𝑉𝑛𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛)
(20)
whe e
𝑔𝑚𝑛=𝑟𝑚𝑛
𝑟𝑚𝑛
2+(𝑥𝑚𝑛−𝑥𝑐)2
(21)
𝑏𝑚𝑛=− 𝑥𝑚𝑛−𝑥𝑐
𝑟𝑚𝑛
2+(𝑥𝑚𝑛−𝑥𝑐)2
(22)
2.9. Model o Thy is o -Con olled Phase Shi e (TCPS)
Figu e 5 displays he model o he TCPS placed be ween he line ha connec s bus-
es 𝑚 and 𝑛. The powe low equa ions o he line can be exp essed as below, assuming
ha is he phase shi angle 𝜙 is in oduced by he TCPS:
Figu e 5. Model o TCP [23].
P𝑚𝑛=𝑉𝑚
2𝑔𝑚𝑛
cos2𝜙−𝑉𝑚𝑉𝑛
cos𝜙[𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜙)
+𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜙)]
(23)
𝑄𝑚𝑛=−𝑉𝑚
2𝑏𝑚𝑛
cos2𝜙−𝑉𝑚𝑉𝑛
cos𝜙[𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜙)
−𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜙)]
(24)
P𝑛𝑚=𝑉𝑛2𝑔𝑚𝑛−𝑉𝑚𝑉𝑛
cos𝜙[𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜙)
−𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜑)]
(25)
𝑄𝑛𝑚=−𝑉𝑛2𝑏𝑚𝑛+𝑉𝑚𝑉𝑛
cos𝜙[𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜙)
+𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜑)]
(26)
The inse ed ac ual and eac i e powe o he TCPS a bus 𝑚 and 𝑛 is [23]:
P𝑚𝑠=−𝑔𝑚𝑛𝑉𝑚2 an2𝜙−𝑉𝑚𝑉𝑛 an𝜙[𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛)
−𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛)]
(27)
Figu e 5. Model o TCP [23].
The inse ed ac ual and eac i e powe o he TCPS a bus mand nis [23]:
Pms =−gmnV2
m an2φ−VmVn an φ[gmn sin(δm−δn).
−bmn cos(δm−δn)](27)
Qms =bmnV2
m an2φ+VmVn an φ[gmn cos(δm−δn).
+bmn sin(δm−δn)](28)
Pns =−VmVn an φ[gmn sin(δm−δn)+bmn cos(δm−δn)] (29)
Qns =−VmVn an φ[gmn cos(δm−δn)−bmn sin(δm−δn)] (30)
2.10. Model o S a ic VAR Compensa o (SVC)
The basic ci cui a chi ec u e and he SVC model a e depic ed in Figu e 6. I is made
up o a hy is o -con olled eac o (
XL=ωL)
and a ixed capaci o (
XC=
1
/ωC)
. By
changing he hy is o i ing angle
(α)
, he eac ance can be changed. The equi alen
suscep ibili y is compu ed as:
Beq =BL(α) + BC(31)
whe e
BL(α) = −1
ωL1−2α
π,Bc=ω×C(32)
The eac i e powe o e ed by he SVC can be exp essed in e ms wi hin he con ex o
powe low:
QSVC =−V2
m·BSVC (33)
Elec onics 2022,11, 3825 16 o 34
Elec onics 2022, 11, x FOR PEER REVIEW 16 o 37
De ini ion 3: Pa e o Op imal Se
All Pa e o op imal solu ion se s a e called he Pa e o se , and a e exp essed as ol-
lows:
𝑃𝑠={𝑥,𝑦∈𝑋 | ∃𝐹(𝑦)≻𝐹(𝑥)}
(67)
De ini ion 4: Pa e o Op imal F on
The Pa e o op imal on is a collec i e o Pa e o op imal solu ions in he Pa e o op-
imal se , as shown in (68):
𝑃𝑓={𝐹(𝑥)|𝑥∈𝑃𝑠}
(68)
Any mul i-objec i e op imiza ion issue mus be sol ed using he Pa e o op imum
se du ing he mul i-objec i e op imiza ion p ocess. The sea ch space (se o domina ed
solu ions) and objec i e space (se o non-domina ed solu ions) a e depic ed in Figu e 9.
The Pa e o op imum on desc ibes he in e ac ion be ween he objec i e space and
sea ch space.
Figu e 9. Objec i e space and sea ch space in he mul i-objec i e op imiza ion p oblem.
4.6. Mul i-Objec i e Gene alized No mal Dis ibu ion Op imiza ion (MOGNDO)
The p oposed MOGNDO algo i hm op imize uses bo h he c owding dis ance
(CD) mechanism and he eli is non-domina ed so ing (NDS) me hod. The NDS consis s
o he ollowing s ages:
• Loca ing he non-domina ed solu ion is he i s s ep.
• The second s ep is he use o he NDS s a egy.
• Pe o ming non-domina ed anking (NDR) calcula ions on all non-domina ed solu-
ions.
Be ween wo on s, he NDR p ocess akes place. The i s on ’s solu ions o e a
“0” index because no solu ions a e domina ed by hem, bu a leas one solu ion om
he i s on domina es he second on ’s solu ions. A solu ion’s NDR is equal o he
numbe o solu ions ha p edomina e i . The CD p ocess is used o keep he c ea ed so-
lu ions di e se. The ollowing is a de ini ion o he CD mechanism:
𝐶𝐷𝑗𝑖 =𝑓𝑜𝑏𝑗𝑗𝑖+1−𝑓𝑜𝑏𝑗𝑗𝑖−1
𝑓𝑜𝑏𝑗𝑗𝑚𝑎𝑥−𝑓𝑜𝑏𝑗𝑗𝑚𝑖𝑛
(69)
whe e 𝑓𝑜𝑏𝑗𝑗𝑚𝑎𝑥 and 𝑓𝑜𝑏𝑗𝑗𝑚𝑖𝑛a e he maximum and minimum alues o 𝑗 h objec i e unc-
ion. The diag amma ic illus a ion o an NDS-based app oach is illus a ed in Figu e 10.
Figu e 9. Objec i e space and sea ch space in he mul i-objec i e op imiza ion p oblem.
4.6. Mul i-Objec i e Gene alized No mal Dis ibu ion Op imiza ion (MOGNDO)
The p oposed MOGNDO algo i hm op imize uses bo h he c owding dis ance (CD)
mechanism and he eli is non-domina ed so ing (NDS) me hod. The NDS consis s o he
ollowing s ages:
•Loca ing he non-domina ed solu ion is he i s s ep.
•The second s ep is he use o he NDS s a egy.
•
Pe o ming non-domina ed anking (NDR) calcula ions on all non-domina ed solu ions.
Be ween wo on s, he NDR p ocess akes place. The i s on ’s solu ions o e a “0”
index because no solu ions a e domina ed by hem, bu a leas one solu ion om he i s
on domina es he second on ’s solu ions. A solu ion’s NDR is equal o he numbe o
solu ions ha p edomina e i . The CD p ocess is used o keep he c ea ed solu ions di e se.
The ollowing is a de ini ion o he CD mechanism:
CDi
j= obji+1
j− obji−1
j
objmax
j− objmin
j
(69)
whe e
objmax
j
and
objmin
j
a e he maximum and minimum alues o
j h
objec i e unc ion.
The diag amma ic illus a ion o an NDS-based app oach is illus a ed in Figu e 10.
The MOGNDO algo i hm’s pseudocode is displayed in Algo i hm 1. The MOGNDO
me hod begins by speci ying he necessa y inpu s, such as popula ion size (N
p
), e mina ion
c i e ia, he maximum numbe o gene a ions, and he maximum numbe o i e a ions
(Maxi ). Then, each objec i e unc ion in he objec i e space ec o F o P
o
is e alua ed
using a andomly gene a ed pa en popula ion P
o
in he easible sea ch space egion S.
Thi dly, P
o
is subjec ed o he eli is -based CD and NDS. Fou hly, P
o
is me ged wi h a esh
popula ion o P
j
o c ea e a popula ion, P
i
. This P
i
is so ed using he CD and NDR da a, as
well as eli is non-dominance. To es ablish a new pa en popula ion, he bes N
p
op ions a e
e alua ed. The p ocess is hen epea ed un il he e mina ion c i e ia a e me . MOGNDO’s
lowcha is displayed in Figu e 11.
Algo i hm 1: Pseudocode o Mul i-objec i e Gene alized No mal Dis ibu ion Op imiza ion
(MOGNDO).
S ep 1: Ini ially Gene a e popula ion (Po) andomly in solu ion space (S)
S ep 2: E alua e objec i e space (F) o he gene a ed popula ion (Po)
S ep 3: So he based on he eli is non-domina ed so me hod and ind he non-domina ed ank
(NDR) and on s
S ep 4: Compu e c owding dis ance (CD) o each on
S ep 5: Upda e solu ions (Pj)
S ep 6: Me ge Poand Pj o c ea e Pi=PoUPj
S ep 7: Fo Pipe o m S ep 2
S ep 8: Based on NDR and CD so Pi
S ep 9: Replace Powi h Pi o Np i s membe s o Pi

Elec onics 2022,11, 3825 17 o 34
Elec onics 2022, 11, x FOR PEER REVIEW 17 o 37
Figu e 10. The p ocedu e o he non-domina ed so ing app oach.
The MOGNDO algo i hm’s pseudocode is displayed in Algo i hm 1. The
MOGNDO me hod begins by speci ying he necessa y inpu s, such as popula ion size
(Np), e mina ion c i e ia, he maximum numbe o gene a ions, and he maximum
numbe o i e a ions (Maxi ). Then, each objec i e unc ion in he objec i e space ec o
F o Po is e alua ed using a andomly gene a ed pa en popula ion Po in he easible
sea ch space egion S. Thi dly, Po is subjec ed o he eli is -based CD and NDS. Fou hly,
Po is me ged wi h a esh popula ion o Pj o c ea e a popula ion, Pi. This Pi is so ed us-
ing he CD and NDR da a, as well as eli is non-dominance. To es ablish a new pa en
popula ion, he bes Np op ions a e e alua ed. The p ocess is hen epea ed un il he e -
mina ion c i e ia a e me . MOGNDO’s lowcha is displayed in Figu e 11.
Algo i hm 1: Pseudocode o Mul i-objec i e Gene alized No mal Dis ibu ion Op i-
miza ion (MOGNDO).
S ep 1: Ini ially Gene a e popula ion (Po) andomly in solu ion space (S)
S ep 2: E alua e objec i e space (F) o he gene a ed popula ion (Po)
S ep 3: So he based on he eli is non-domina ed so me hod and ind he non-
domina ed ank (NDR) and on s
S ep 4: Compu e c owding dis ance (CD) o each on
S ep 5: Upda e solu ions (Pj)
S ep 6: Me ge Po and Pj o c ea e Pi=Po U Pj
S ep 7: Fo Pi pe o m S ep 2
S ep 8: Based on NDR and CD so Pi
S ep 9: Replace Po wi h Pi o Np i s membe s o Pi
Figu e 10. The p ocedu e o he non-domina ed so ing app oach.
Elec onics 2022, 11, x FOR PEER REVIEW 18 o 37
Figu e 11. Flowcha o MOGNDO algo i hm.
4.7. Cons ain Handling App oach
The majo i y o enginee ing design issues in he ac ual wo ld a e mul i-objec i e
and highly nonlinea ly cons ained. To sol e cons ained MOPs, managing all con-
s ain s wi hin hei bounds is c ucial. A s a ic penal y echnique is used in he
MOGNDO algo i hm because i ans o ms a cons ained p oblem in o an uncons ained
p oblem, despi e he li e a u e su ey gi ing a ious cons ained handling app oaches.
This app oach adds a signi ican penal y o he ele an goal unc ion i a cons ain is
b oken. The ollowing is a p esen a ion o he s a ic penal y sys em:
𝑓𝑗(𝑋)=𝑓𝑗(𝑋)+∑𝑝𝑖=1 𝑃𝑖max{𝑔𝑖(𝑋),0}+∑𝑁𝐶
𝑖=𝑝 𝑃𝑖max{|ℎ𝑖(𝑋)|−𝛿,0}
(70)
whe e 𝑓𝑗(𝑋),𝑗=1,2…𝑛 is he objec i e unc ion o be op imized (he e minimized), 𝑋=
{𝑥1,𝑥2,…𝑥𝑚} a e design a iables, 𝑔𝑖(𝑋)⩽0,𝑖=1,2…𝑝 a e inequali y cons ain s,
ℎ𝑖(𝑋)=0,𝑖=𝑝+1…𝑁𝐶 a e equali y cons ain s, and 𝛿 is he ole ance in equali y con-
s ain s.
4.8. Fuzzy App oach o he Mul i-Objec i e P oblem
The uzzy membe ship app oach can be used in mul i-objec i e unc ions o iden i-
y he bes comp omising ou come ou o all he non-in e io esul s. The uzzy membe -
ship unc ion 𝜇𝑓𝑖uses a uzzy membe ship unc ion o keep ack o he minimum 𝑓𝑖𝑚𝑖𝑛
and maximum 𝑓𝑖𝑚𝑎𝑥 alues o each objec i e aim. Now, he membe ship unc ion o he
𝑖𝑡ℎ𝑡ℎ𝑒 objec i e is gi en as:
𝜇𝑓𝑖=
{
1 𝑓𝑖≤ 𝑓𝑖𝑚𝑖𝑛
𝑓𝑖𝑚𝑎𝑥−𝑓𝑖
𝑓𝑖𝑚𝑎𝑥−𝑓𝑖𝑚𝑖𝑛𝑓𝑖𝑚𝑖𝑛<𝑓𝑖<
0 𝑓𝑖≥ 𝑓𝑖𝑚𝑎𝑥𝑓𝑖𝑚𝑎𝑥
(71)
The s anda ds o membe ship unc ions lie on he measu e o (0–1) and display in
he way ha sa is ies he unc ion 𝑓𝑖. La e , he decision-making unc ion 𝜇𝑘 should be
calcula ed as ollows:
Figu e 11. Flowcha o MOGNDO algo i hm.
4.7. Cons ain Handling App oach
The majo i y o enginee ing design issues in he ac ual wo ld a e mul i-objec i e
and highly nonlinea ly cons ained. To sol e cons ained MOPs, managing all cons ain s
wi hin hei bounds is c ucial. A s a ic penal y echnique is used in he MOGNDO algo i hm
Elec onics 2022,11, 3825 18 o 34
because i ans o ms a cons ained p oblem in o an uncons ained p oblem, despi e he
li e a u e su ey gi ing a ious cons ained handling app oaches. This app oach adds a
signi ican penal y o he ele an goal unc ion i a cons ain is b oken. The ollowing is a
p esen a ion o he s a ic penal y sys em:
j(X) = j(X) + ∑p
i=1Pimax{gi(X), 0}+∑NC
i=pPimax{|hi(X)|−δ, 0}(70)
whe e
j(X)
,
j=
1, 2
. . . n
is he objec i e unc ion o be op imized (he e minimized),
X={x1,x2, . . . xm}
a e design a iables,
gi(X)≤
0,
i=
1, 2
. . . p
a e inequali y cons ain s,
hi(X) =
0,
i=p+
1
. . . NC
a e equali y cons ain s, and
δ
is he ole ance in equali y
cons ain s.
4.8. Fuzzy App oach o he Mul i-Objec i e P oblem
The uzzy membe ship app oach can be used in mul i-objec i e unc ions o iden i y
he bes comp omising ou come ou o all he non-in e io esul s. The uzzy membe ship
unc ion
µ i
uses a uzzy membe ship unc ion o keep ack o he minimum
min
i
and
maximum
max
i
alues o each objec i e aim. Now, he membe ship unc ion o he i h he
objec i e is gi en as:
µ i=




1 i≤ min
i
max
i− i
max
i− min
i
min
i< i< max
i
0 i≥ max
i
(71)
The s anda ds o membe ship unc ions lie on he measu e o (0–1) and display in
he way ha sa is ies he unc ion
i
. La e , he decision-making unc ion
µk
should be
calcula ed as ollows:
µk=∑N
i=1µk
i
∑M
k=1∑N
i=1µk
i
(72)
Fo non-in e io indings, he decision-making unc ion can also be hough o as he
no malized membe ship unc ion, which displays he o de ing o he undomina ed esul s.
The end ou come is ega ded as he bes a ainable comp omise among all PFs, wi h a
maximum alue o maximum nµk:k=1, 2, 3 . . . . . . Mo.
5. Simula ion Resul s, Analysis, and Compa a i e S udy
This sec ion discusses he ou comes o he MOGNDO algo i hm, which op imized
he op imal powe low wi h non-con en ional and FACTS de ice p oblems wi h con ol
a iables. The ini ializa ion o he algo i hm’s popula ion size, a chi e size, he maximum
numbe o i e a ions, and bounda y condi ion o op imal powe low p oblems all came
i s . To iden i y he bes op imal adeo poin s be ween mul iple objec i e unc ions, he
MOGNDO algo i hm was hen used o ob ain he ini ial posi ion and objec i e unc ion
alues. Op imal powe low wi h non-con en ional sou ces and FACTS de ices we e used
o apply he MOGNDO algo i hm’s pe o mance, which was ini ially e i ied on eigh
uncons ained mul i-objec i e p oblems. On a compu e wi h 4 GB o RAM and a 3.20
GHz clock speed, he simula ion was un using he MATLAB p og am. The benchma k
unc ions o each uncons ained es we e sol ed using 10 sepa a e uns. The popula ion
size was se o 30, he maximum numbe o i e a ions was se o 100, and he a chi e size
was se o 30 when he con ol pa ame e s o he p oposed MOGWO algo i hm we e
i s se . The pe o mance measu es o he MOGNDO algo i hm, including Gene a ional
Dis ance (GD), In e sion Gene a ional Dis ance (IGD), Spacing Me ics (SP), Di e si y
Me ics (DM), and Sp ead Me ics (SD), a e co e ed in his sec ion.
Elec onics 2022,11, 3825 19 o 34
5.1. MOGNDO Resul s o Tes Benchma k P oblems
Be o e ackling eal-wo ld issues, he MOGNDO was used o e alua e he pe o -
mance o he benchma k uncons ain es unc ion p o ided in [
26
]. Eigh benchma k
uncons ained es unc ions—ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, KURSAVE, SCHAFFER-1,
and SCHAFFER-2 (Figu e 12) we e aken in o accoun , and a ho ough simula ion was
pe o med using he MOGNDO echnique. Any algo i hm’s con ol pa ame e s a e c ucial
o he esolu ion o he op imiza ion p oblem. As a esul , he numbe o popula ions was
decided a e conduc ing a compa a i e analysis ha ook in o accoun a ious popula ion
sizes, while holding all o he a iables cons an . Following ca e ul conside a ion, he
popula ion size, maximum i e a ions, and a chi e size we e chosen as 30, 100, and 30,
espec i ely, o he uncons ained es benchma k unc ions. The MOGNDO algo i hm’s
pe o mance was e alua ed using pe o mance me ics, such as Gene a ional Dis ance
(GD), In e sion Gene a ional Dis ance (IGD), Spacing Me ics (SP), Di e si y Me ics (DM),
and Sp ead Me ics (SD), o con e gence measu emen . Tables 3–7demons a e ha
MOGNDO could achie e he bes ou comes o all pe o mance me ics, including Gen-
e a ional Dis ance (GD), In e sion Gene a ional Dis ance (IGD), Spacing Me ics (SP),
Di e si y Me ics (DM), and Sp ead Me ics (SD), which co e con e gence and solu ion
accu acy. I ollows ha he sugges ed MOGNDO can p o ide he bes con e gence on
all benchma k unc ions. The ou comes (a chi e solu ions) o all eigh es benchma k
issues a e displayed in Figu es 1–5. As can be shown, he MOGNDO me hod was capable
o app oxima ing he PF. By compa ing he PF es ima ions, i can also be seen ha he
sugges ed MOGNDO could p o ide accep able pe o mance. Thus, i was de e mined ha
he MOGNDO algo i hm is mo e sui able o he s ochas ic OPF p oblem wi h h ee FACTS
de ices and wind powe plan s.
Table 3. Resul s o GDMETRICS on es unc ions.
TEST
FUNCTIONS Minimum A e age Median Maximum S d De
ZDT-1 0.00014795 0.00025051 0.00025043 0.00033133 6.658 ×10−5
ZDT-2 0.00015293 0.00016665 0.00016915 0.00018271 8.9928 ×10−6
ZDT-3 0.00048314 0.00059601 0.00057943 0.00077354 9.9396 ×10−5
ZDT-4 0.00012104 0.00022808 0.00020834 0.0004758 9.4401 ×10−5
ZDT-6 9.084 ×10−50.046963 0.00011316 0.22777 0.084189
KURSAVE 0.00079741 0.0014452 0.0012905 0.0025211 0.00051722
SCHAFFER-1 0.00033607 0.00040992 0.00042203 0.00047897 4.714 ×10−5
SCHAFFER-2 4.3236 ×10−57.2824 ×10−55.2463 ×10−50.00016939 4.0743 ×10−5
Table 4. Resul s o IGD METRICS on es unc ions.
TEST
FUNCTIONS Minimum A e age Median Maximum S d De
ZDT-1 0.00086623 0.00098862 0.0010348 0.0010718 8.6005 ×10−5
ZDT-2 0.00086578 0.00098848 0.00094837 0.0012358 0.00012678
ZDT-3 0.001211 0.0020746 0.0013386 0.0077205 0.0020015
ZDT-4 0.00078066 0.00089518 0.00087008 0.0012276 0.00012621
ZDT-6 0.0004207 0.00045525 0.00044688 0.00049748 2.8268 ×10−5
KURSAVE 0.00054436 0.00062399 0.00057845 0.00085408 9.8135 ×10−5
SCHAFFER-1 0.0013154 0.0015246 0.0015027 0.0017064 0.0001302
SCHAFFER-2 0.00038808 0.00042594 0.00042586 0.00044933 1.8173 ×10−5
Elec onics 2022,11, 3825 20 o 34
Elec onics 2022, 11, x FOR PEER REVIEW 20 o 37
Figu e 12.
Bes Pa e o op imal on ob ained o uncons ained es unc ions by MOGNDO algo i hm.
Elec onics 2022,11, 3825 21 o 34
Table 5. Resul s o SPACINGMETRICS on es unc ions.
TEST
FUNCTIONS Minimum A e age Median Maximum S d De
ZDT-1 0.055513 0.068922 0.068773 0.078086 0.0087192
ZDT-2 0.055269 0.066982 0.06225 0.082706 0.010406
ZDT-3 0.23708 0.30199 0.30489 0.37195 0.038816
ZDT-4 0.055461 0.079673 0.082798 0.0984 0.012829
ZDT-6 0.060275 0.36467 0.082672 1.2984 0.50689
KURSAVE 1.7123 2.0817 2.0541 2.3807 0.20644
SCHAFFER-1 0.47734 0.6444 0.65722 0.78476 0.086121
SCHAFFER-2 4.3184 6.228 6.1458 8.1797 0.99173
Table 6. Resul s o DIVERSITYMETRICS on es unc ions.
TEST
FUNCTIONS Minimum A e age Median Maximum S d De
ZDT-1 0.39774 0.4726 0.46959 0.53772 0.060856
ZDT-2 0.34666 0.4369 0.44215 0.57139 0.069382
ZDT-3 0.42402 0.52122 0.50625 0.64813 0.07401
ZDT-4 0.3035 0.37573 0.36822 0.43095 0.043003
ZDT-6 0.36148 0.6907 0.45236 1.3455 0.42847
KURSAVE 0.28774 0.3839 0.39009 0.45763 0.050257
SCHAFFER-1 0.28774 0.3839 0.39009 0.45763 0.050257
SCHAFFER-2 0.9161 0.95607 0.95992 1.0033 0.030768
Table 7. Resul s o SPREAD METRICS on es unc ions.
TEST
FUNCTIONS Minimum A e age Median Maximum S d De
ZDT-1 0.38649 0.45794 0.45049 0.5251 0.059019
ZDT-2 0.33511 0.42459 0.42842 0.56709 0.070091
ZDT-3 0.5604 0.64503 0.64749 0.73814 0.066426
ZDT-4 0.30364 0.36741 0.36204 0.4213 0.040601
ZDT-6 0.35291 0.67963 0.4479 1.3293 0.42341
KURSAVE 0.37729 0.43491 0.43448 0.4872 0.030263
SCHAFFER-1 0.27113 0.37375 0.37832 0.44441 0.051198
SCHAFFER-2 0.56431 0.61293 0.61561 0.65022 0.02504
5.2. Mul i-Objec i es OPF P oblem wi h Wind Powe Plan s and Th ee FACTS De ices
The GNDO algo i hm was used o add ess he s ochas ic OPF p oblem wi h wind
powe plan s and h ee FACTS de ices in his s udy. The solu ion o he op imum powe
low p oblem was e alua ed in pa allel using newly c ea ed algo i hms, such as he Mul i-
Ve se Op imiza ion (MVO), he Sine-Cosine Algo i hm (SCA) [
27
], he G ey Wol Op imiza-
ion (GWO), he Mo h Fame Op imiza ion (MFO), he An Lion Op imiza ion (ALO) [
28
],
and Ion Mo ion Algo i hms (IMA) [
29
]. The p oposed app oach was demons a ed using a
modi ied IEEE-30 bus in as uc u e wi h wind powe plan s and FACTS de ices. Table 1
lis s he majo cha ac e is ics o he cus omized IEEE-30 bus amewo k. The ollowing a e
wo scena ios:
•Scena io-1 (Solo objec i e OPF wi h wind powe plan s and FACTS de ices)
•Scena io-2 (Mul i-objec i e OPF wi h wind powe plan s and FACTS de ices)
As shown in Table 8, he e we e a o al o hi een di e en es scena ios o e alua e. In
his sec ion, he esul s o case s udies using a ious me aheu is ics me hodologies a e ab-
ula ed and p esen ed. The i s six case s udies a e o single-objec i e op imiza ion, while
he la e se en a e mul i-objec i e op imiza ion p oblems ha include non-con en ional
sou ces o ene gy esou ces, as well as op imal FACTS de ice sizes and loca ions. The
sea ch agen alue was se o 40, and each algo i hm unde wen 500 i e a ions o analysis.
Please e e o he o iginal esea ch o a de ailed discussion o hose p ocedu es. Table 4
shows he pa ame e se ings o hese me hods.

Elec onics 2022,11, 3825 22 o 34
Table 8. Summa y o case s udies o adap ed IEEE-30 bus es sys em.
Tes Sys em Case # Single and Mul i-Objec i es Func ions
Adap ed
IEEE 30-bus es sys em
Case # 1 TFC includes FACTS de ices, wind a ms, and coal-based plan s
Case # 2 Reduc ion o o al oxic gas emana ions wi h he use o coal-based, wind, and FACTS
echnologies.
Case # 3 Minimiza ion o APL in FACTS de ices, wind a ms, and coal-based plan s.
Case # 4 Minimiza ion o he o al ol age a ia ion using coal-based, wind, and FACTS de ices.
Case # 5 Vol age s abili y imp o emen in coal-based, wind, and FACTS equipmen .
Case # 6 To al G oss Gene a ion Cos , includes FACTS de ices, wind a ms, and coal-based plan s.
Case # 7 Minimizing TFCs and oxic gas emana ions while using non-con en ional sou ces o
ene gy and FACTS de ices
Case # 8 TFC and APL Minimiza ion Including non-con en ional sou ces and FACTS De ices
Case # 9 TFC and VSI minimiza ion including non-con en ional sou ces and FACTS de ices
Case # 10 To al G oss Gene a ion Cos and ol age de ia ion minimiza ion wi h non-con en ional
sou ces and FACTS de ices
Case # 11 TFC, Toxic gas emana ion, and APL minimiza ion oge he wi h non-con en ional
sou ces and FACTS de ices
Case # 12 TFC, APL, and VSI minimiza ion including non-con en ional sou ces and FACTS de ices
Case # 13 TFC, Toxic gas emana ion, APL, and ol age de ia ion minimiza ion including
non-con en ional sou ces and FACTS de ices
5.3. Scena io-1 (Single Objec i e OPF wi h Wind Powe Plan s and FACTS De ices)
Wi h he use o GNDO, MVO, ALO, SCA, and IMO me hods, all o he objec i e
goals indica ed in he ma hema ical o mula ion we e simul aneously handled as solo
objec i e op imiza ion issues. The limi a ions o all con ol a iables, as well as p ope
FACTS de ice loca ions and sizing, a e lis ed below. F om case 1 o case 6, he ou comes o
objec i e unc ions a e abula ed in Tables 9–11, wi h he bes minimum alues con aining
i e di e en ecen echniques.
Table 9. Single objec i es simula ion esul s o case 1 and case 2.
Con ol &
S a e Va iables Min Max Case-1 Case-2
GNDO MVO ALO SCA IMO GNDO MVO ALO SCA IMO
PTG2 20 80 41.427 40.311 40.960 35.458 30.881 46.634 46.639 46.634 48.357 46.726
PWG5 0 75 49.815 49.459 49.077 41.719 54.226 74.818 74.934 71.362 75.000 74.507
PTG8 10 35 10.000 10.352 13.038 15.172 14.361 35.000 35.000 35.000 35.000 35.000
PWG11 0 60 40.799 42.135 39.362 47.561 36.329 52.365 51.215 54.905 60.000 48.761
PTG13 12 40 12.002 12.000 12.000 14.321 16.648 40.000 40.000 40.000 40.000 40.000
V1 0.95 1.1 1.091 1.100 1.100 1.019 1.100 1.090 0.997 1.100 0.997 1.100
V2 0.95 1.1 1.075 1.090 1.091 0.992 1.100 1.080 1.013 1.078 0.950 1.100
V5 0.95 1.1 1.049 1.071 1.072 0.978 1.100 1.067 1.037 0.975 1.100 1.100
V8 0.95 1.1 1.046 1.076 1.079 0.950 1.100 0.962 1.100 1.100 1.100 1.100
V11 0.95 1.1 1.100 1.100 1.071 1.047 1.100 1.073 1.082 1.100 0.950 1.100
V13 0.95 1.1 1.069 1.062 1.038 0.950 1.100 1.087 0.970 1.047 1.100 1.100
T11 0.9 1.1 1.037 1.007 1.059 0.959 1.090 0.977 0.938 1.020 1.100 1.100
T12 0.9 1.1 0.997 1.087 1.084 0.900 1.090 0.943 1.055 1.073 0.965 1.100
T15 0.9 1.1 1.027 1.099 1.095 0.938 1.090 1.023 1.063 1.080 0.900 1.100
T36 0.9 1.1 0.957 1.024 1.087 0.900 1.090 1.017 1.024 1.054 0.900 1.100
SVC1 Loca ion - - 24 27 6 19 22 27 11 21 25 29
SVC2 Loca ion - - 7 27 10 9 15 6 20 30 25 30
SVC1 Ra ing −10 10 10.000 9.893 −6.541 3.882 1.740 −3.380 9.274 8.786 10.000 10.000
SVC2 Ra ing −10 10 5.764 −6.277 1.209 5.482 −3.094 −9.559 0.639 8.550 −10.000 4.480
TCSC1 Loca ion - - 15 39 10 3 14 37 38 32 34 33
TCSC2 Loca ion - - 12 29 20 4 32 36 24 37 41 39
TCSC1 Ra ing 0 0.5 0.491 0.219 0.218 0.000 0.452 0.110 0.196 0.497 0.000 0.500
TCSC2 Ra ing 0 0.5 0.496 0.239 0.440 0.000 0.492 0.454 0.060 0.485 0.193 0.500
TCPS1 Loca ion - - 14 16 34 15 14 38 14 26 40 40
TCPS2 Loca ion - - 16 22 19 1 30 35 5 32 1 41
TCPS1 Ra ing −5 5 2.714 4.503 −3.990 3.917 1.235 4.787 −0.175 0.926 5.000 5.000
TCPS2 Ra ing −5 5 1.705 3.761 −4.281 1.150 0.113 4.095 1.148 4.860 −3.963 4.566
TFC ($/h) 806.999 808.030 809.449 818.654 814.865 - - - - -
Emission
(Ton/h) - - - - - 0.138 0.138 0.138 0.138 0.138
Elec onics 2022,11, 3825 23 o 34
Table 10. Single objec i es simula ion esul s o case 3 and case 4.
Con ol &
S a e Va iables Min Max Case-3 Case-4
GNDO MVO ALO SCA IMO GNDO MVO ALO SCA IMO
PTG2 20 80 69.005 75.900 79.745 80.000 79.644 79.721 78.271 28.577 49.606 34.817
PWG5 0 75 75.000 74.966 75.000 73.825 74.667 47.504 11.236 13.419 0.000 47.506
PTG8 10 35 34.999 34.656 35.000 30.726 34.844 25.357 32.736 16.328 35.000 34.441
PWG11 0 60 59.999 58.422 60.000 50.635 59.748 26.234 20.292 4.602 0.000 55.391
PTG13 12 40 39.976 31.239 39.823 31.706 39.832 26.916 32.996 24.669 24.060 39.576
V1 0.95 1.1 1.036 1.100 1.098 1.098 1.089 1.008 0.958 0.967 0.950 0.955
V2 0.95 1.1 1.037 1.100 1.100 1.091 1.089 1.032 1.054 1.051 1.072 1.052
V5 0.95 1.1 1.027 1.090 1.091 1.100 1.089 1.010 1.016 1.017 0.960 0.955
V8 0.95 1.1 1.030 1.092 1.096 1.100 1.090 1.025 0.992 1.013 1.024 1.015
V11 0.95 1.1 1.100 1.099 1.100 1.100 1.089 0.950 1.069 1.009 1.100 1.004
V13 0.95 1.1 1.100 1.100 1.077 0.990 1.089 1.004 1.067 1.084 1.043 1.038
T11 0.9 1.1 1.012 1.047 1.046 1.030 1.090 0.938 1.063 0.949 1.017 0.905
T12 0.9 1.1 0.903 0.903 1.045 1.100 1.090 0.907 0.903 0.904 0.912 0.986
T15 0.9 1.1 0.998 1.042 1.100 1.018 1.090 0.945 1.048 1.065 1.044 1.069
T36 0.9 1.1 0.935 0.983 1.068 1.100 1.090 0.934 0.930 0.936 0.957 0.936
SVC1 Loca ion - - 18 12 27 11 27 19 24 28 19 19
SVC2 Loca ion - - 24 15 30 16 30 10 14 16 21 23
SVC1 Ra ing −10 10 4.901 6.871 3.951 6.535 9.956 8.082 8.769 −1.898 4.969 9.593
SVC2 Ra ing −10 10 10.000 5.448 5.238 −0.936 4.219 5.421 −2.342 −5.291 −1.465 9.547
TCSC1 Loca ion - - 34 5 40 3 40 14 13 2 2 40
TCSC2 Loca ion - - 11 40 41 2 41 18 25 9 5 29
TCSC1 Ra ing 0 0.5 0.494 0.107 0.500 0.000 0.471 0.353 0.300 0.075 0.027 0.500
TCSC2 Ra ing 0 0.5 0.500 0.198 0.497 0.000 0.498 0.499 0.391 0.087 0.000 0.486
TCPS1 Loca ion - - 16 12 40 4 33 19 38 3 1 37
TCPS2 Loca ion - - 19 15 41 1 34 15 41 5 5 36
TCPS1 Ra ing −5 5 1.744 −0.898 −3.378 0.433 4.979 −4.997 3.754 −1.638 5.000 2.660
TCPS2 Ra ing −5 5 0.257 4.748 −3.441 1.249 0.946 0.767 0.550 −3.874 0.128 −1.703
APL (MW) 1.647 1.735 1.686 2.482 1.880 - - - - -
Vol age
De ia ion (p.u) - - - - - 0.124 0.150 0.177 0.227 0.165
Table 11. Single objec i es simula ion esul s o case 5 and case 6.
Con ol &
S a e Va iables Min Max Case-5 Case-6
GNDO MVO ALO SCA IMO GNDO MVO ALO SCA IMO
PTG2 20 80 78.161 28.765 76.487 20.000 74.363 44.894 47.802 55.203 20.000 56.164
PWG5 0 75 75.000 16.020 74.240 0.000 67.677 74.998 74.564 71.256 75.000 68.555
PTG8 10 35 35.000 34.855 34.405 10.000 32.815 35.000 32.893 33.271 35.000 30.826
PWG11 0 60 54.355 0.000 55.983 49.379 14.173 59.006 58.030 50.102 60.000 58.732
PTG13 12 40 12.001 28.840 38.080 16.659 35.961 21.583 22.222 32.274 19.540 23.229
V1 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.046 1.100 1.100 1.100 1.100
V2 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.042 1.097 1.098 1.100 1.100
V5 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.032 1.087 1.087 1.100 1.100
V8 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.035 1.092 1.091 1.100 1.098
V11 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.098 1.100 1.100 1.100 1.100
V13 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.035 1.100 1.083 1.100 1.098
T11 0.9 1.1 0.905 0.910 0.990 1.100 1.089 1.077 1.054 1.007 1.100 1.084
T12 0.9 1.1 0.904 0.909 0.990 0.900 1.089 0.901 0.903 1.089 1.100 1.084
T15 0.9 1.1 0.901 0.900 0.930 0.900 1.089 1.065 1.053 1.081 1.100 1.100
T36 0.9 1.1 0.901 0.909 0.910 0.900 0.910 0.985 1.001 1.033 1.100 1.087
SVC1 Loca ion - - 10 29 29 29 26 24 28 26 15 24
SVC2 Loca ion - - 29 30 30 30 26 13 14 26 3 10
SVC1 Ra ing −10 10 9.999 4.260 7.322 10.000 9.312 9.999 −3.639 3.886 10.000 8.490
SVC2 Ra ing −10 10 9.999 2.982 9.608 9.747 9.622 3.844 −1.487 3.411 −0.065 7.591
TCSC1 Loca ion - - 38 38 38 24 36 16 7 39 1 18
TCSC2 Loca ion - - 15 36 40 1 38 19 29 34 3 31
TCSC1 Ra ing 0 0.5 0.500 0.490 0.500 0.002 0.499 0.500 0.468 0.470 0.002 0.500
TCSC2 Ra ing 0 0.5 0.500 0.325 0.475 0.013 0.499 0.013 0.490 0.355 0.000 0.147
TCPS1 Loca ion - - 36 4 33 3 38 14 4 35 31 30
TCPS2 Loca ion - - 41 17 38 1 39 4 2 25 11 33
TCPS1 Ra ing −5 5 −4.999 3.788 4.884 −5.000 4.540 3.173 0.486 4.591 2.761 1.494
TCPS2 Ra ing −5 5 −4.998 1.993 4.772 −1.126 4.697 −0.508 −1.527 −0.901 5.000 2.191
VSI 0.096 0.100 0.096 0.108 0.102 - - - - -
To al G oss Fuel
Cos ($/h) - - - - - 1120.996 1125.970 1138.357 1187.287 1148.359
The o e all uel cos wi h GNDO, which included he wo non-con en ional sou ces
o powe plan s and op imal placemen o FACTS de ices, was 806.999 $/h, which was
he bes in compa ison wi h he o he ci ed algo i hm shown in Table 9. The educ ions
in TFC in compa ison wi h MVO, ALO, SCA, IMO, SHADE-SF, DE-SF, ABC-SF, PSO-SF,
FPA-SF, and MSA-SF we e 1.031 $/h, 1.031 $/h, 2.45 $/h, 11.655 $/h 7.866 $/h, 0.0176 $/h,
0.4917 $/h, 0.4 $/h, 1.2553 $/h, 3.398 $/h, and 1.0403 $/h, espec i ely. This demons a ed
he GNDO algo i hm’s supe io i y o e o he ci ed me aheu is ics algo i hms.
Elec onics 2022,11, 3825 24 o 34
Figu e 13 illus a es he con e gence ai s o he TFC minimiza ion. Simila con e -
gence ai s o APL, ol age de ia ions, and VSI a e depic ed in Figu es 14–16. Figu e 17
also displays a compa ison o he uel cos dec ease wi h a ious algo i hms. In example 2,
he GNDO me hod esul ed in a pollu an gas emission o 0.138 ons pe hou . In ins ance
3, he APL o he a ious ansmission lines using he GNDO app oach was 1.647 MW. The
APL was 0.088 MW, 0.039 MW, 0.835 MW, 0.233 MW, 0.0997 MW, 0.0997 MW, 0.2598 MW,
0.2494 MW, 0.6127 MW, and 0.4972 MW less compa ed o MVO, ALO, SCA, IMO, SHADE-
SF, DE-SF, ABC-SF, PSO-SF, and MSA-SF, espec i ely. A c ucial ac o o he g id’s abili y
o ope a e eliably was he ol age di e gence o each bus om 1.0 pe uni . The e o e, in
scena io 4, he mo h lame algo i hm p oduced he lowes ol age a ia ion (0.124 p.u),
making i he bes o he i e op imiza ion me hods. The VSI, some imes e e ed o as he
L max index, a ied be ween ze o (no load) and one ( ol age collapse). The e o e, in case 5,
0.096 was he lowes alue o he L max index. In scena io 6, he o e all g oss uel cos
using he GNDO me hod was 1120.996 dolla s pe hou . I is in e es ing o no e he e ha in
Table 12, he o al g oss uel cos o he p oposed GNDO was mo e han he SHADE-SF,
which u he en o ces he na a i e o he “No ee lunch heo em,” which s a es ha no
algo i hm gi es he bes esul in e e y p oblem.
Elec onics 2022, 11, x FOR PEER REVIEW 26 o 37
index. In scena io 6, he o e all g oss uel cos using he GNDO me hod was 1120.996
dolla s pe hou . I is in e es ing o no e he e ha in Table 12, he o al g oss uel cos o
he p oposed GNDO was mo e han he SHADE-SF, which u he en o ces he na a-
i e o he “No ee lunch heo em,” which s a es ha no algo i hm gi es he bes esul
in e e y p oblem.
Figu e 13. Con e gence cha ac e is ics o TFC minimiza ion.
Figu e 14. Con e gence cha ac e is ics o APL minimiza ion.
Figu e 13. Con e gence cha ac e is ics o TFC minimiza ion.
Elec onics 2022, 11, x FOR PEER REVIEW 26 o 37
index. In scena io 6, he o e all g oss uel cos using he GNDO me hod was 1120.996
dolla s pe hou . I is in e es ing o no e he e ha in Table 12, he o al g oss uel cos o
he p oposed GNDO was mo e han he SHADE-SF, which u he en o ces he na a-
i e o he “No ee lunch heo em,” which s a es ha no algo i hm gi es he bes esul
in e e y p oblem.
Figu e 13. Con e gence cha ac e is ics o TFC minimiza ion.
Figu e 14. Con e gence cha ac e is ics o APL minimiza ion.
Figu e 14. Con e gence cha ac e is ics o APL minimiza ion.
Elec onics 2022,11, 3825 25 o 34
Elec onics 2022, 11, x FOR PEER REVIEW 27 o 37
Figu e 15. Con e gence cha ac e is ics o ol age de ia ion minimiza ion.
Figu e 16. Con e gence cha ac e is ics o VSI minimiza ion.
Table 12. Compa ison o he simula ion esul s o single objec i es.
Objec i es
Func ions
GNDO
MVO
ALO
SCA
IMO
SHADE-SF
DE-SF
ABC-SF
PSO-SF
FPA-SF
MSA-SF
To al F.C ($/h)
806.999
808.030
809.449
818.654
814.865
807.0166
807.4907
807.399
808.2543
810.397
808.0393
Emission (T/h)
0.138
0.138
0.138
0.138
0.138
-
-
-
-
-
-
Ploss (MW)
1.647
1.735
1.686
2.482
1.880
1.7467
1.7467
1.9068
1.8964
2.2597
2.1442
V.D (p.u)
0.124
0.150
0.177
0.227
0.165
-
-
-
-
-
-
Lmax
0.096
0.100
0.096
0.108
0.102
-
-
-
-
-
-
To al G oss
F.C ($/h)
1120.996
1125.970
1138.357
1187.287
1148.359
1104.077
1113.676
1116.365
1118.601
1164.719
1122.331
Figu e 15. Con e gence cha ac e is ics o ol age de ia ion minimiza ion.
Elec onics 2022, 11, x FOR PEER REVIEW 27 o 37
Figu e 15. Con e gence cha ac e is ics o ol age de ia ion minimiza ion.
Figu e 16. Con e gence cha ac e is ics o VSI minimiza ion.
Table 12. Compa ison o he simula ion esul s o single objec i es.
Objec i es
Func ions
GNDO
MVO
ALO
SCA
IMO
SHADE-SF
DE-SF
ABC-SF
PSO-SF
FPA-SF
MSA-SF
To al F.C ($/h)
806.999
808.030
809.449
818.654
814.865
807.0166
807.4907
807.399
808.2543
810.397
808.0393
Emission (T/h)
0.138
0.138
0.138
0.138
0.138
-
-
-
-
-
-
Ploss (MW)
1.647
1.735
1.686
2.482
1.880
1.7467
1.7467
1.9068
1.8964
2.2597
2.1442
V.D (p.u)
0.124
0.150
0.177
0.227
0.165
-
-
-
-
-
-
Lmax
0.096
0.100
0.096
0.108
0.102
-
-
-
-
-
-
To al G oss
F.C ($/h)
1120.996
1125.970
1138.357
1187.287
1148.359
1104.077
1113.676
1116.365
1118.601
1164.719
1122.331
Figu e 16. Con e gence cha ac e is ics o VSI minimiza ion.
Elec onics 2022, 11, x FOR PEER REVIEW 28 o 37
Figu e 17. Compa ison o he TFC educ ion ($/h) wi h o he algo i hms.
Figu es 18 and 19 p o ide compa ison cha s wi h he minimizing o APL and
o e all uel cos . The esul s o he simula ions we e compa ed o hose o he mos e-
cen algo i hms, including MVO, ALO, SCA, IMO, and o he men ioned op imiza ion
app oaches. I was ound ha he p oposed me hod o Gene alized No mal Dis ibu ion
Op imiza ion me hodology p oduced supe io esul s.
Figu e 18. Compa ison cha o TFC minimiza ion wi h di e en algo i hms.
1.031
2.45
11.655
7.866
0.0176
0.4917
0.4
1.2553
3.398
1.0403
To al cos educ ion ($/h ) in compa ision wi h GNDO
algo i hm.
MVO ALO SCA IMO SHADE-SF
800
802
804
806
808
810
812
814
816
818
820
806.999 808.03
809.449
818.654
814.865
807.0166
807.4907807.399
808.2543
810.397
808.0393
TFC
Di e en Me a-Heu is ics Algo i hms
Compa ison o To al Fuel Cos wi h di e en MAs
GNDO MVO ALO SCA IMO SHADE-SF DE-SF ABC-SF PSO-SF FPA-SF MSA-SF
Figu e 17. Compa ison o he TFC educ ion ($/h) wi h o he algo i hms.
Elec onics 2022,11, 3825 32 o 34
solu ions o each si ua ion in ol ing op imal powe low. All o he esul s poin o
he sugges ed echnique’s signi ican ad an age in ob aining he bes solu ions o OPF
issues wi h one o mo e objec i es. Finally, i was shown ha by in eg a ing wind a ms
wi h FACTS de ices u ilizing a non-domina ed so ing echnique, MOGNDO could be
success ully employed o add ess small and la ge op imal powe low challenges. Based
on he ex ensi e analysis o he p oposed MOGNDO, he ollowing can be summa ized as
i s ad an ages—
•
Randomiza ion in MOGNDO includes he di e si y o he Pa e o on being en-
hanced, since all solu ions in he i s domina ed on will ha e an equal chance o
being selec ed, and mul i-objec i es a e made uni o mly signi ican while pe o ming
local explo a ion.
•MOGNDO can deal wi h la ge-scale sea ch spaces and is less dependen on p oblem
cha ac e is ics. Mo eo e , hese algo i hms a e capable o es ima ing mul iple poin s
in he sea ch domain simul aneously, due o hei popula ion-based na u e.
•
MOGNDO s ikes a good balance be ween exploi a ion and explo a ion, p o iding
powe ul sea chabili y o inding he op imum solu ion
•
MOGNDO is supe io in e ms o he balance o di e si y and con e gence, he
dis ibu ion o PF, and be e con e gence.
Au ho Con ibu ions:
Concep ualiza ion, S.B.P., J.V. and M.M.; Me hodology, S.B.P., J.V., M.M. and
T.K.M.; So wa e, T.K.M. and P.J.; Valida ion, S.B.P., M.M. and T.K.M.; Fo mal analysis, J.V., M.M.
and T.K.M.; In es iga ion, S.B.P.; Resou ces, P.J.; Da a cu a ion, J.V. and P.J.; W i ing—o iginal d a ,
S.B.P., J.V., M.M., T.K.M. and P.J.; W i ing— e iew & edi ing, S.B.P.; Visualiza ion, T.K.M. and P.J.;
Supe ision, S.B.P.; Funding acquisi ion, M.M. All au ho s ha e ead and ag eed o he published
e sion o he manusc ip .
Funding:
This wo k was suppo ed by he p ojec SP2022/60 Applied Resea ch in he A ea o
Machines and P ocess Con ol, which is suppo ed by he Minis y o Educa ion, You h and Spo s,
in he Czech Republic.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : The da a p esen ed in his s udy a e a ailable in he a icle.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Abb e ia ions
Ac onyms
OPF Op imal Powe Flow
Mas Me a Heu is ics Algo i hms
MOGNDO Mul i-Objec i e Gene alized No mal Dis ibu ion Op imiza ion
TG The mal Gene a ing uni
WG Wind Gene a ion
ISO Independen Sys em Ope a o
PDF P obabili y Densi y Func ion
BCS Bes Comp omise Solu ion
MOMFO Mul i-Objec i e Mo h Flame Op imiza ion
MOOPF Mul i-Objec i e Op imal Powe Flow
SHADE-SF Success His o y-based Adap i e Di e en ial E olu ion using Supe io i y o
Feasible solu ions me hod
DE-SF Di e en ial E olu ion using Supe io i y o Feasible solu ions me hod
ABC-SF A i icial Bee Colony using Supe io i y o Feasible solu ions me hod
PSO-SF Pa icle Swa m Op imiza ion using Supe io i y o Feasible solu ions me hod

Elec onics 2022,11, 3825 33 o 34
FPA-SF Flowe Pollina ion Algo i hm using Supe io i y o Feasible solu ions
me hod
MSA-SF Mo h Swa m Algo i hm using Supe io i y o Feasible solu ions me hod
TFC To al Fuel Cos
APL Ac i e powe Loss
VSI Vol age S abili y Index
Nomencla u e
ai,bi,ci,eiand diP ice cons an s o i h coal-based powe plan s.
αi,βi,γi,ωiand µiToxic gas emana ion cons an s conce ning he i h coal-based uni s.
gwDi ec cos cons an
Pws Scheduled powe o he wind uni .
KRw Rese e cos coe icien ega ding wind uni
KPw Penal y cos coe icien o wind uni
Pws Accessible powe om he wind uni
Pw Speci ied ou pu powe om he wind uni
w(pw)Wind ene gy p obabili y densi y unc ion o he wind uni .
in, and ou Cu -in, a ed, and cu -ou wind eloci y o he u bine espec i ely
pw Ra ed alue o he gene a ed ou pu o he wind u bine
τDeg ee o se ies compensa ion
Xmn Line induc i e eac ance linking buses mand n
Rmn Resis ance o he line linking buses mand n
Vmand VnBus ol age magni udes linking buses mand n.
δmand δnPhase angles o he linking buses mand n
gmn and bmn Conduc ance and suscep ance o he line linking buses mand n.
Npq Numbe o load (PQ) buses
ipu ol age le el o i h bus.
PGi and PDi Gene a ion and dispa ch a i h bus
Numbe o buses
Y1and Y2Sub-ma ices o
δij =δi−δjVa iance in phase angles o ol age among bus iand bus
PDi and QDi Real and VAR powe demand espec i ely a i h bus
PGi and QGi Real and VAR ou pu s espec i ely o i h bus by ei he uni
(coal-based o non-con en ional) as applicable
Gij and Bij Conduc ance and suscep ance be ween bus iand bus j
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