ene gies
A icle
Hyb id Machine Lea ning Models o Classi ying
Powe Quali y Dis u bances: A Compa a i e S udy
Juan Ca los B a o-Rod íguez * , F ancisco J. To es and Ma ía D. Bo ás
Escuela Poli écnica Supe io , Uni e sidad de Se illa, c/Vi gen de Á ica 9, 41011 Se illa, Spain;
[email p o ec ed] (F.J.T.); [email p o ec ed] (M.D.B.)
*Co espondence: [email p o ec ed]; Tel.: +34-954-552-847
Recei ed: 10 Ma ch 2020; Accep ed: 25 May 2020; Published: 1 June 2020

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Abs ac :
The economic impac associa ed wi h powe quali y (PQ) p oblems in elec ical sys ems is
inc easing, so PQ imp o emen esea ch becomes a key ask. In his pape , a S ockwell ans o m
(ST)-based hyb id machine lea ning app oach was used o he ecogni ion and classi ica ion o
powe quali y dis u bances (PQDs). The ST o he PQDs was used o ex ac signi ican wa e o m
ea u es which cons i u e he inpu ec o s o di e en machine lea ning app oaches, including he
K-nea es neighbo s’ algo i hm (K-NN), decision ee (DT), and suppo ec o machine (SVM) used
o classi ying he PQDs. The p ocedu e was op imized by using he gene ic algo i hm (GA) and
he compe i i e swa m op imiza ion algo i hm (CSO). To es he p oposed me hodology, syn he ic
PQD wa e o ms we e gene a ed. Typical single dis u bances o he ol age signal, as well as
complex dis u bances esul ing om possible combina ions o hem, we e conside ed. Fu he mo e,
di e en le els o whi e Gaussian noise we e added o he PQD wa e o ms while main aining he
desi ed accu acy le el o he p oposed classi ica ion me hods. Finally, all he hyb id classi ica ion
p oposals we e e alua ed and he bes one was compa ed wi h some o he s p esen in he li e a u e.
The p oposed ST-based CSO-SVM me hod p o ides good esul s in e ms o classi ica ion accu acy
and noise immuni y.
Keywo ds:
powe quali y dis u bances; classi ica ion; ea u e selec ion; swa m op imiza ion; suppo
ec o machine; gene ic algo i hm; K-NN algo i hm; decision ee; S- ans o m
1. In oduc ion
Powe quali y (PQ) is essen ial o elec ical sys ems o ope a e p ope ly wi h he minimum
possible de e io a ion o pe o mance. Eme ging PQ challenges, such as he g owing in eg a ion o
la ge powe plan s based on enewable sou ces, imp o emen s in nonlinea loads, and he ecen
equi emen s o sma g ids, mus be conside ed o ob ain an op imal ope a ion o he exis ing powe
g id. These ac o s inc easingly equi e cons an e isions o he common powe quali y p oblems,
enhanced s anda ds, u he op imiza ion o con ol sys ems, and mo e powe ul capabili ies o
measu ing ins umen s.
The pu pose o his esea ch was o con ibu e o his ask, mee ing he pa icula PQ equi emen s
abou de ec ion and classi ica ion o powe quali y dis u bances (PQDs) h ough op imal hyb id
machine lea ning app oaches.
Usually, he PQDs’ iden i ica ion p ocedu e is ca ied ou in h ee s eps, i.e., signal analysis,
ea u e selec ion, and classi ica ion. In he s age o PQDs’ analysis, some ad anced ma hema ical
echniques we e used o ex ac he ea u e eigen ec o s ha enable dis u bance iden i ica ion.
The ime- equency analysis me hods include sho - ime Fou ie ans o m (STFT), S ockwell ans o m
(ST) [
1
], wa ele ans o m (WT) [
2
–
4
], Hilbe –Huang ans o m [
5
,
6
], Kalman il e [
7
,
8
], s ong ace
il e (STF) [
9
], spa se signal decomposi ion (SSD) [
10
], Gabo –Wigne ans o m [
11
], and empi ical
Ene gies 2020,13, 2761; doi:10.3390/en13112761 www.mdpi.com/jou nal/ene gies
Ene gies 2020,13, 2761 2 o 20
mode decomposi ion (EMD) [
12
,
13
]. In his s udy, ST was selec ed due mainly o i s noise immuni y,
simplici y o implemen a ion, and lexible and con ollable— o some ex en — ime- equency esolu ion.
These ad an ages a ou weigh he compu a ional cos ha may be equi ed.
The selec ion o sui able ea u e emains a key challenge ha equi es de eloping ools in a eas
such as s a is ical analysis, machine lea ning, o da a mining [
14
]. Valuable e o s ha e been made
in his sense and some echniques a e used o a p ecise selec ion o ea u es including he p incipal
componen analysis [
15
], K-means-based ap io i algo i hm [
16
], classi ica ion and eg ession ee
algo i hm [
17
], mul i-label ex eme lea ning machine [
18
], andom o es model [
19
], sequen ial
o wa d selec ion [
20
], and bionic algo i hms. This la e g oup has also been success ully used in
classi ica ion ule disco e y. Pa icula ly signi ican among bionic algo i hms a e gene ic algo i hms
(GA) [
20
–
22
] and swa m-based app oaches like an colonies [
23
,
24
] and, abo e all, pa icle swa m
op imize s (PSO) [
25
–
28
]. Fo example, ecen ly in [
25
], a combina ion o PSO and suppo ec o
machine (PSO-SVM) was used o op imize he e o o he classi ie by selec ing he bes ea u e
combina ion. Simila ly, in [
26
], PSO op imizes he noise cu -o h eshold o PQD signals wo king
oge he wi h a modi ied ST in he ea u e ex ac ion s age. Howe e , canonical PSO has some
limi a ions o ea u e selec ion. Imp o ed and implemen ed PSO a ian s include compe i i e swa m
op imize [
29
–
31
] (CSO), disc e e pa icle swa m op imize [
32
], and exponen ial ine ia weigh pa icle
swa m op imize [
33
]. I should be no ed ha , as PSO was i s ly designed o con inuous op imiza ion
p oblems, his may no always be he mos app op ia e me hod o sol e a combina o ial op imiza ion
p oblem such as ea u e selec ion. The CSO algo i hm, howe e , is speci ically adap ed o pe o m his
ype o p oblem wi h each pa icle lea ning om a pai o andomly selec ed compe i o s o ele a e
bo h global and local sea ch abili ies. In his pape , GA and CSO we e selec ed and compa ed o
minimize he numbe o selec ed ea u es.
Rega ding he dis u bance pa e n ecogni ion capabili y and acco ding o he op imal selec ion o
ea u es p o ided by he abo e-men ioned algo i hms, nume ous machine lea ning app oaches ha e
been widely u ilized o classi ying powe quali y dis u bances. Common classi ica ion echniques
include a i icial neu al ne wo k (ANN), K-nea es neighbo (K-NN) algo i hm, suppo ec o machine
(SVM), and decision ee (DT) me hods.
Suppo ec o machine (SVM) is a good op ion o classi ica ion pu poses, especially when
dealing wi h small samples, nonlinea i y, o high dimension in pa e n ecogni ion [
2
,
16
,
22
,
34
,
35
].
Among he ad an ages o SVM a e he lack o local ex emum, ea u e mapping o nonlinea sepa able
da a, low space complexi y, and he capabili y o adjus only a educed numbe o ea u es as compa ed
o, o example, he ANNs [
36
]. On he con a y, i s disad an ages include limi a ions esul ing om
speed and size, in bo h aining and es ing, as well as hose esul ing om an imp ope choice o he
ke nel. These handicaps in ol e, in p ac ical e ms, high algo i hmic complexi y and ex ensi e memo y
equi emen s. Imp o ed applica ions o SVMs algo i hms include mul iclass SVM [M-SVM] [
37
],
di ec ed acyclic g aph SVMs [DAG-SVMs] [
38
] and adial basis unc ion ke nel SVM [RBF-SVM] [
39
].
Fo i s pa , he ule-based DT classi ie is a good choice when he ea u es a e clea ly dis inguishable
om each o he [
40
,
41
]. PQDs’ classi ie s based on DT include uzzy decision ee [
42
–
44
] and he
a o emen ioned classi ica ion and eg ession ee algo i hm (CART) [
17
,
21
]. On he one hand,
DT ad an ages include emo ing unnecessa y compu a ions, a singula se o pa ame e s which
allows di e en ia ing be ween classes and a smalle numbe o ea u es a each non e minal node
while main aining pe o mance a an accep able le el. On he o he hand, i s p incipal disad an ages
include being s ongly dependen on he selec ed ea u es, accumula ion o e o s om le el o le el
in a la ge ee, and o e lap—which inc eases he sea ch ime and memo y space equi emen s when
he numbe o classes is la ge. Compa ed o o he app oaches, DT lowcha symbols con igu e a
simple and s aigh o wa d model in which he con ol pa ame e s a e easy o unde s and and apply.
Thus, DT is easie o se up and in e p e and, despi e he men ioned dependence o he classi ica ion
p ocess on he selec ed ea u es, i s execu ion o da a is be e han o he me hods. Fo example, in [
22
],
a compa a i e using DT/SVM, wa ele ans o m (WT) and ST is shown.
Ene gies 2020,13, 2761 3 o 20
Mos o he p e ious wo ks in he li e a u e a e mainly ocused on pa e n ecogni ion issues,
so PQDs a e gene ally ea ed as single-e en signals. Howe e , in elec ical sys ems, i is common o
ind se e al dis u bances consecu i ely in he same obse a ion window. These combined dis u bances
a e much mo e di icul o iden i y and ea han single ones. In his wo k, complex PQDs we e
designed h ough a consecu i e o simul aneous combina ion o wo simple ones in he same in e al.
F om ST, ime- equency ea u es we e ex ac ed, while ea u e selec ion was op imized by using
K-NN, GA, and CSO. In he classi ica ion s age, K-NN (again) and dis inc ypes o SVM and DT we e
conside ed. The e we e di e en p oposals o classi ie s depending on he op imiza ion-classi ica ion
sequence chosen. All hese p oposals ope a ed o e he same da ase ob ained a e op imiza ion.
A compa a i e in e ms o classi ica ion accu acy and noise immuni y o he p oposed models
was planned. In Figu e 1, a gene al block scheme o he p oposed classi ica ion plan is p esen ed.
The main s eps included PQD signal p ocessing ia ST, ea u e ex ac ion, op imal ea u e selec ion,
and classi ica ion. A de ailed o e iew o he p oposed compa a i e s udy including di e en hyb id
me hods can be ound in Sec ion 5. The MATLAB (Classi ica ion Lea ne Toolbox) so wa e was used o
implemen he whole machine lea ning me hods equi ed a bo h op imiza ion and classi ica ion s ages.
Ene gies 2019, 12, x FOR PEER REVIEW 3 o 20
Mos o he p e ious wo ks in he li e a u e a e mainly ocused on pa e n ecogni ion issues, so
PQDs a e gene ally ea ed as single-e en signals. Howe e , in elec ical sys ems, i is common o
ind se e al dis u bances consecu i ely in he same obse a ion window. These combined
dis u bances a e much mo e di icul o iden i y and ea han single ones. In his wo k, complex
PQDs we e designed h ough a consecu i e o simul aneous combina ion o wo simple ones in he
same in e al. F om ST, ime- equency ea u es we e ex ac ed, while ea u e selec ion was
op imized by using K-NN, GA, and CSO. In he classi ica ion s age, K-NN (again) and dis inc ypes
o SVM and DT we e conside ed. The e we e di e en p oposals o classi ie s depending on he
op imiza ion-classi ica ion sequence chosen. All hese p oposals ope a ed o e he same da ase
ob ained a e op imiza ion. A compa a i e in e ms o classi ica ion accu acy and noise immuni y o
he p oposed models was planned. In Figu e 1, a gene al block scheme o he p oposed classi ica ion
plan is p esen ed. The main s eps included PQD signal p ocessing ia ST, ea u e ex ac ion, op imal
ea u e selec ion, and classi ica ion. A de ailed o e iew o he p oposed compa a i e s udy
including di e en hyb id me hods can be ound in Sec ion 5. The MATLAB (Classi ica ion Lea ne
Toolbox) so wa e was used o implemen he whole machine lea ning me hods equi ed a bo h
op imiza ion and classi ica ion s ages.
Figu e 1. Gene al block scheme o he p oposed classi ica ion plan.
The es o his pape is o ganized as ollows: In Sec ion 2, a simpli ied ou line o he ex ac ion
o he ini ial ea u e se is p esen ed. Sec ion 3 is de o ed o he op imal selec ion o ea u es,
desc ibing he op imize s used o his ask. Sec ion 4 b ie ly desc ibes he machine lea ning me hods
used o classi y. In Sec ion 5, a de ailed o e iew o he p oposed classi ica ion plan is shown. In
Sec ion 6, PQ dis u bances’ syn hesis and he esul ing aining da ase s a e de ailed. In Sec ion 7,
esul s a e discussed. The las sec ion d aws conclusions om he esul s.
2. Ini ial Fea u e Se Ex ac ion Based on S-T ans o m and S a is ical Pa ame e s
Each p oposed dis u bance signal was gene a ed in a disc e e o m o compu e i s S- ans o m,
he de ailed desc ip ion o which is gi en in he Appendix. ST was chosen o i s inhe en noise
immuni y and accep able ime- equency esolu ion.
The esul ing complex S-ma ix p o ided aluable ime- equency da a on which PQD ea u es
we e ex ac ed by compu ing se e al s a is ics and igu es o me i . In his wo-dimensional S-ma ix,
he signal was spli in o di e en equencies (M = 1280 ows) and dis inc samples (N = 2560
columns). This ex ac ion o ea u es had a ele an e ec on he accu acy o classi ica ion because o
i s g ea in luence on he o e all pe o mance o machine lea ning app oaches.
In a i s app oxima ion, he chosen ini ial ea u e se should ha e been enough o gua an ee a
co ec iden i ica ion o e e y one o he conside ed dis u bed signals. In his wo k, he ex ac ed se
was o med by nine ea u es (k1-k9) and included he in oduced dis u bance ene gy a io (DER)
Figu e 1. Gene al block scheme o he p oposed classi ica ion plan.
The es o his pape is o ganized as ollows: In Sec ion 2, a simpli ied ou line o he ex ac ion o
he ini ial ea u e se is p esen ed. Sec ion 3is de o ed o he op imal selec ion o ea u es, desc ibing
he op imize s used o his ask. Sec ion 4b ie ly desc ibes he machine lea ning me hods used o
classi y. In Sec ion 5, a de ailed o e iew o he p oposed classi ica ion plan is shown. In Sec ion 6,
PQ dis u bances’ syn hesis and he esul ing aining da ase s a e de ailed. In Sec ion 7, esul s a e
discussed. The las sec ion d aws conclusions om he esul s.
2. Ini ial Fea u e Se Ex ac ion Based on S-T ans o m and S a is ical Pa ame e s
Each p oposed dis u bance signal was gene a ed in a disc e e o m o compu e i s S- ans o m,
he de ailed desc ip ion o which is gi en in he Appendix A. ST was chosen o i s inhe en noise
immuni y and accep able ime- equency esolu ion.
The esul ing complex S-ma ix p o ided aluable ime- equency da a on which PQD ea u es
we e ex ac ed by compu ing se e al s a is ics and igu es o me i . In his wo-dimensional S-ma ix,
he signal was spli in o di e en equencies (M =1280 ows) and dis inc samples (N =2560 columns).
This ex ac ion o ea u es had a ele an e ec on he accu acy o classi ica ion because o i s g ea
in luence on he o e all pe o mance o machine lea ning app oaches.
In a i s app oxima ion, he chosen ini ial ea u e se should ha e been enough o gua an ee a
co ec iden i ica ion o e e y one o he conside ed dis u bed signals. In his wo k, he ex ac ed se
was o med by nine ea u es (k1–k9) and included he in oduced dis u bance ene gy a io (DER) index
Ene gies 2020,13, 2761 4 o 20
as well as some o he well-known s a is ical pa ame e s, such as maximum, minimum, oo mean
squa e and mean alues, s anda d de ia ion, a iance, skewness, and ku osis. These ea u es we e
calcula ed ollowing he equa ions shown in Table 1.
Table 1. Ma hema ical equa ions o he ini ial ea u e se .
Ex ac ed Fea u es
K1 Maximum M=maxnAjno1S anda d
de ia ion σ= PM
j=1PN
n=1(Ajn−µj)2
(M−1)(N−1)K6
K2 Minimum m=minnAjnoVa iance σ2=PM
j=1PN
n=1(Ajn−µj)2
(M−1)(N−1)K7
K3 Mean
alue µ=PM
j=1PN
n=1Ajn
M·N
Skewness
(phase) 2SK(φ)=PM
j=1PN
n=1(φjn−µ(φ)j)3
M·N·σ3
(φ)
K8
K4 RMS RMS = PM
j=1PN
n=1A2
jn
M·NKu osis KT =PM
j=1PN
n=1(Ajn−µj)4
M·N·σ4K9
K5 DER DER =RMS>50
RMS50Hz - -
1Ajn,2φjn a e he absolu e alue and phase alue o he jn- h elemen in he S-ma ix.
All s a is ical pa ame e s we e calcula ed om bo h ime samples (N =2560) and equency
(
M=1280
) in e als. The skewness pa ame e was compu ed based on he phase alues o he complex
elemen s in he S-ma ix. Fo he es o he pa ame e s, calcula ions we e done om he absolu e
alues o such elemen s.
Dis u bance Ene gy Ra io (DER) Index
The in oduced DER index ep esen s he a io be ween he ene gy o he signal wi h equency
componen s g ea e han 50 Hz and ha one whose componen s a e equal o o less han 50 Hz. Thus,
he de ini ion o DER pa ame e includes he e ms
RMS>50 =
eq=6400 Hz
X
eq=50.1 Hz
RMSj(1)
and
RMS50Hz =X eq=50Hz
eq=0Hz RMSj. (2)
This index is e y use ul o he cha ac e iza ion o PQ dis u bances wi h high- equency con en
as, o example, oscilla o y ansien s.
Sample da ase s o aining/ es ing consis o single obse a ions, each o which is compu ed
om ea u es, as shown in Table 1.
3. Op imal Fea u e Selec ion: GA and CSO
The main pu pose o using an op imize is o educe as much as possible he dimension o inpu
ea u e da ase o he p edic ion models. Once da a ha e been ob ained by S- ans o m, u he
analysis is necessa y o achie e he op imal ea u e ec o . As seen abo e, a ec o wi h nine di e en
ea u es was p oposed. Howe e , he gi en ea u e ec o con ained a ibu es whose in o ma ion was
edundan o dis inguish he mos disc imina ing ea u es o PQDs. The in aclass compac ion could
be minimized and he in e class di ision could be maximized by educing he numbe o ea u es.
Fo his pu pose, a e ob aining he da ase ea u es, i was necessa y o selec he bes op imize .
W appe -based echniques a e a signi ican g oup wi hin ea u e selec ion me hods ha a e e y
accu a e and popula and elimina e edundan ea u es by using a lea ning algo i hm wi h classi ie
Ene gies 2020,13, 2761 5 o 20
pe o mance eedback. The wo main op imiza ion me hods used in his wo k, namely GA and CSO,
belong o his g oup.
3.1. Gene ic Algo i hm
Da win’s heo y o e olu ion, “Su i al o he Fi es ”, inspi ed he design o gene ic algo i hms
in he 1960s [
45
]. GA is an adap ed heu is ic sea ch algo i hm [
45
] ha uses op imiza ion me hods
based on gene ics and ules o na u al selec ion. The lowcha ha desc ibes he ope a ion o GA is
shown in Figu e 2.
Ene gies 2019, 12, x FOR PEER REVIEW 5 o 20
classi ie pe o mance eedback. The wo main op imiza ion me hods used in his wo k, namely GA
and CSO, belong o his g oup.
3.1. Gene ic Algo i hm
Da win's heo y o e olu ion, "Su i al o he Fi es ", inspi ed he design o gene ic algo i hms
in he 1960s [45]. GA is an adap ed heu is ic sea ch algo i hm [45] ha uses op imiza ion me hods
based on gene ics and ules o na u al selec ion. The lowcha ha desc ibes he ope a ion o GA is
shown in Figu e 2.
Figu e 2. P ep ocessing s age using K-nea es neighbo (K-NN) algo i hm o he e alua ion o
gene ic algo i hms (GA) membe s.
In GA [46], an op imal ea u e ec o can be ep esen ed by a ch omosome, which includes he
mos disc imina i e ea u es. In u n, ch omosomes comp ise mul iple genes, each one
co esponding o a ea u e. The popula ion is a ini e se o ch omosomes manipula ed by he
algo i hm in a simila way o he p ocess o na u al e olu ion. In his p ocess, ch omosomes a e
enabled o c osso e and o mu a e. The c ossing o wo ch omosomes c ea es wo o sp ing and
hese wo each p oduce wo mo e, and so on. A gene ic mu a ion in he o sp ing gene a es an almos
iden ical copy o he combina ion o hei pa en s bu wi h some pa o he ch omosome mo ed.
Gene a ions a e he cycles whe e he op imiza ion p ocess is ca ied ou . C osso e , mu a ion, and
e alua ion make i possible o c ea e a se o new ch omosomes du ing each gene a ion. A p ede ined
numbe o he (bes ) ch omosomes su i es o he nex cycle o he eplica due o he ini e size o he
popula ion. The popula ion can achie e a as adap a ion despi e i s limi ed size, which esul s in
quick op imiza ion o he c i e ion unc ion (sco e). The mos impo an s ep o GA is he c osso e ,
in which exchanges o in o ma ion among ch omosomes a e implemen ed. Once he bes indi iduals
a e selec ed, i is necessa y o c osso e hese solu ions be ween hemsel es. The main pu pose o
his s ep is o ge a g ea e di e en ia ion be ween popula ions based on new solu ions ha could be
be e han he p e ious ones. A second impo an s ep is a mu a ion, which inc eases he a iableness
o he popula ion.
Figu e 2.
P ep ocessing s age using K-nea es neighbo (K-NN) algo i hm o he e alua ion o gene ic
algo i hms (GA) membe s.
In GA [
46
], an op imal ea u e ec o can be ep esen ed by a ch omosome, which includes he
mos disc imina i e ea u es. In u n, ch omosomes comp ise mul iple genes, each one co esponding
o a ea u e. The popula ion is a ini e se o ch omosomes manipula ed by he algo i hm in a simila
way o he p ocess o na u al e olu ion. In his p ocess, ch omosomes a e enabled o c osso e and o
mu a e. The c ossing o wo ch omosomes c ea es wo o sp ing and hese wo each p oduce wo mo e,
and so on. A gene ic mu a ion in he o sp ing gene a es an almos iden ical copy o he combina ion
o hei pa en s bu wi h some pa o he ch omosome mo ed. Gene a ions a e he cycles whe e he
op imiza ion p ocess is ca ied ou . C osso e , mu a ion, and e alua ion make i possible o c ea e a se
o new ch omosomes du ing each gene a ion. A p ede ined numbe o he (bes ) ch omosomes su i es
o he nex cycle o he eplica due o he ini e size o he popula ion. The popula ion can achie e a
as adap a ion despi e i s limi ed size, which esul s in quick op imiza ion o he c i e ion unc ion
(sco e). The mos impo an s ep o GA is he c osso e , in which exchanges o in o ma ion among
ch omosomes a e implemen ed. Once he bes indi iduals a e selec ed, i is necessa y o c osso e
hese solu ions be ween hemsel es. The main pu pose o his s ep is o ge a g ea e di e en ia ion
be ween popula ions based on new solu ions ha could be be e han he p e ious ones. A second
impo an s ep is a mu a ion, which inc eases he a iableness o he popula ion.

Ene gies 2020,13, 2761 6 o 20
Ano he key piece is he i ness unc ion. I is necessa y o ob ain an e ec i e en o cemen -o ien ed
e sion o GA. The i ness unc ion is he p ocedu e o de ice ha is esponsible o assessing he
quali y o each ch omosome, speci ying which one is he bes om he popula ion. Once he i ness
unc ion is calcula ed wi h each indi idual o he ini ial popula ion, he nex s age is he so-called
selec ion, in which ch omosomes wi h he bes quali ies a e selec ed o gene a e he new e olu ion o
he popula ion using disc imina ion c i e ia.
Di e en GA implemen a ions use speci ic impo an pa ame e s o de e mine he execu ion
and pe o mance o he gene ic sea ch. Howe e , some o he pa ame e s, including c osso e a e,
popula ion size, and mu a ion a e, a e usual o all implemen a ions. The p obabili y o aking an
eligible pai o ch omosomes o c osso e is called a e c osso e . Con e sely, he p obabili y o
changing a bi o andomly selec ed ch omosomes is called mu a ion a e. The c osso e a e usually
p esen s high alues, close o o equal o 1, while he mu a ion a e is usually small (1% o 15%).
In he p esen wo k, he ch omosome consis ed o nine genes, each o which ep esen ed a ea u e.
As shown in Figu e 2, he ch omosome is ep esen ed as a ec o o bi s since all he genes could be
assigned wi h ei he 0 o 1 (0 when he co esponding ea u e was no selec ed and 1 when i was).
A popula ion o 560 indi iduals (ch omosomes) and 100 i e a ions (gene a ions) was selec ed o his
p oblem. The sea ch began ini ializing he pa ame e s o:
•Ini ial (pa en ) popula ion size: 10 (ch omosomes).
•C osso e a e: 0.8.
•Mu a ion a e: 0.01.
The pe o mance o he classi ie mus be kep abo e a ce ain speci ied le el. Fo his, he leas
expensi e subse o ea u es mus be ound. Fo his pu pose, he pe o mance is measu ed using he
e o o a classi ie . The iabili y o a subse is ensu ed when he e o a e o he classi ie is lowe
han he so-called easibili y h eshold. The goal is o ind he smalles subse o ea u es among all
easible ones.
In his case, he iden i ica ion accu acy o he K-NN algo i hm was se as he i ness alue o he
ch omosome. In o de o assess he quali y o he ch omosome h ough he i ness unc ion (accu acy),
he k pa ame e o he K-NN me hod was adjus ed o 10 and he alue o c oss- alida ion was se o
8 olds. The K-NN inpu da ase was exclusi ely designed o his alida ion p ocedu e (see Sec ion 6
o de ails).
3.2. Compe i i e Swa m Op imiza ion
Compe i i e swa m op imize [
29
] is a pa icula case o pa icle swa m op imize (PSO), hus
belonging o e olu iona y algo i hms inspi ed by locking and swa ming beha io . Swa m me hods
y o emula e he adap i e s a egy, which conside s collec i e in elligence as beha io wi hou any
s uc u e o cen alized con ol o e indi iduals. Usually, he o e all s uc u e o swa m op imize s
includes di e en algo i hms wi h each handle a speci ic ask. The c i ical one is he classi ica ion ule
disco e y algo i hm, which is, in essence, a s anda d GA. Thus, a g oup o indi iduals (pa icles) ac s
and e ol es ollowing he p inciples o na u al selec ion—su i al o he i es . In PSO, he op imal
solu ion o a p oblem is ob ained om he global in e ac ions among pa icles. In con as , he CSO
me hod in oduces pai wise in e ac ions andomly selec ed om he swa m (popula ion). Gene a ions
succeed one ano he a e each pai wise compe i ion, in which he i ness alue o he lose is upda ed
by lea ning om he winne ha goes di ec ly o he swa m o he nex gene a ion. CSO has p o en o
be be e han GA in op imiza ion asks ela ed o ea u e selec ion due o i s easy- o-use s uc u e,
ewe pa ame e s, and simple concep , e en hough i s compu a ional cos is sligh ly highe . Howe e ,
as will be shown below in he conclusions, he supe io i y o CSO o e GA is clea om he solu ion
quali y, bu in e ms o success a e, i is no so.
In his wo k, pa icles we e de ined by he ea u e se (K1...K9) in he same way as ch omosomes
(indi iduals) in GA. They also de i ed om he same da ase (560 indi iduals) om which pa icles
Ene gies 2020,13, 2761 7 o 20
we e andomly selec ed. Then, he swa m size was se o 100 and he maximal numbe o gene a ions
(i e a ions) was se as 200.
Following a pa allel p ocess o ha ca ied ou in he GA op imize , a K-NN simple iden i ica ion
model was used o check he e iciency o he CSO-based ea u e selec ion, in his case wi h k =5. Once
again, he accu acy o he K-NN iden i ie was es ablished as he i ness unc ion o he CSO op imize .
Bo h ypes o op imiza ion me hods, GA and CSO, educed he numbe o ea u es om nine o
i e, bu hey we e no he same.
As was men ioned abo e, K-NN was chosen o ac as a as alida ion ool in he ea u e op imal
selec ion s age. A his s age, he aim was o educe ea u es and high accu acy was no as necessa y as
simplici y, speed, and e iciency. In hese aspec s, he K-NN was highly compe i i e. As shown below,
his me hod is going o be used again in he nex s age o compa e i s classi ica ion pe o mance wi h
ha o o he app oaches. In he nex sec ion, unlike his one, he aim is o achie e he highes possible
accu acy in he classi ica ion.
4. Classi ie s: K-NN, SVMs, and DTs
Once he op imized se o ea u e was de e mined, he nex p ocess was he classi ica ion o da a
wi h hese ea u es. In his wo k, a ious classi ica ion me hods we e used o ind be e e iciency
and he bes beha io wi h noise signals. These me hods included he K-nea es neighbo s’ algo i hm,
he suppo ec o machine, and he decision ees.
4.1. K-Nea es Neighbo s’ Algo i hm
One o he p oposed classi ica ion app oaches used he K-NN classi ie o iden i y bo h single
and complex dis u bances. K-NN [
47
], as a supe ised lea ning algo i hm, de e mines he dis ance
o he nea es neighbo ing aining samples in he ea u e space in o de o classi y a new objec .
This Euclidean dis ance is s a ed as ollows:
Djxi,yj= Xp
k=1xi,k−yj,k2(3)
whe e
Djxi,yj
is he Euclidean dis ance-based ela ionship be ween he i h p-dimensional inpu
ea u e ec o
xi
and he j h p-dimensional ea u e ec o
yj
in he aining se . A new inpu ec o
xi
is classi ied by K-NN in o he class ha allows a minimum o ksimila i ies be ween i s membe s.
The pa ame e ko he K-NN me hod is a use -speci ic pa ame e . O en kis se o a na u al numbe
close o
pN samples
[
47
], in which
N samples
is he numbe o samples in he aining da ase . In his
wo k, di e en K-NN classi ie s we e i on he aining da ase esul ing om alues o kbe ween 5
and 12. The lowes classi ica ion e o a e on he alida ion se pe mi ed selec ing he sough-a e
alue o k.
T adi ional K-NN app oach based on Euclidean dis ance becomes less disc imina ing as he
numbe o a ibu es inc eases. To imp o e he accu acy o he K-NN me hod o PQDs classi ica ion,
a weigh ed K-NN classi ica ion me hod can be used [
48
]. The weigh ac o is o en aken o be he
ecip ocal o he squa ed dis ance,
ωi=
1
/D2
jxi,yj
. Se e al schemes can be de eloped o a emp o
calcula e he weigh s o each a ibu e based on some disc iminabili y c i e ia in he aining se .
4.2. Suppo Vec o Machine
SVM is a s a is ical me hod o machine lea ning ha uses supe ised lea ning [
49
]. Al hough his
me hod was o iginally in ended o sol e bina y p oblems, i s use was easily ex ended o mul iclass
classi ica ion p oblems. The majo objec i e o SVM is he minimiza ion o he so-called s uc u al isk
by p oposing hypo heses o minimize he isk o making mis akes in u u e classi ica ions. This me hod
inds op imal hype planes sepa a ing he dis inc classes o aining da ase in a high-dimensional
ea u e space and, based on his, es da a can be classi ied. The hype plane is equidis an om he
Ene gies 2020,13, 2761 8 o 20
closes samples o each class o achie e a maximum ma gin on each side o i . Only he aining samples
o each class ha all igh a he bo de o hese ma gins a e conside ed o de ine he hype plane.
These samples a e called suppo ec o s [50,51].
Nex , a ough ske ch o SVM is ou lined below in an o e sigh -speci ic manne . Conside a da ase
con aining a da a pai de ined as
xi,yj(i=1,. . . ,M)
, whe e Mis he numbe o samples,
yi∈{−1,1}
.
Based on an n-dimensional ec o
w
no mal o he hype plane and a scala b, he issue is o ind he
minimum alue o
kwk
in he objec i e equa ion
(x)=DwT·x+bE
. The posi ion o he sepa a ing
hype plane can be de e mined based on he alues o
w
and b ha ul il he cons ain
yi·wT·xi+b≥
1.
The key pa ame e
b/kwk
gi es he dis ance om he o igin
(x0,y0)
o he closes da a poin along
w
.
Fu he mo e, o deal wi h he case o he linea insepa able p oblem, whe e empi ical isk is no ze o,
a penal y ac o
C
and slack a iables
ξi
a e in oduced. The op imal sepa a ing hype plane can be
de e mined by sol ing he ollowing cons ained op imiza ion p oblem [24,52]:
Minimize
1
2·kwk2+C·
M
X
i=1
ξi(4)
subjec o
yi·wT·xi+b≥1−ξi o i =1,2, . . . ,M
ξi≥0 o all i (5)
whe e ξiis he dis ance be ween he ma gin and w ongly loca ed samples xi.
Despi e SVM being a linea unc ion se , i is possible o sol e nonlinea classi ica ion p oblems by
using a ke nel unc ion. As shown in Figu e 3, he mapping ansla es he classi ied ea u es on o a
high-dimensional space whe e he linea classi ica ion is easible.
Ene gies 2019, 12, x FOR PEER REVIEW 8 o 20
Nex , a ough ske ch o SVM is ou lined below in an o e sigh -speci ic manne . Conside a
da ase con aining a da a pai de ined as (𝑥𝑖,𝑦𝑗)(𝑖=1,…,𝑀), whe e M is he numbe o samples,
𝑦𝑖∈{−1,1}. Based on an n-dimensional ec o 𝑤 no mal o he hype plane and a scala b, he issue
is o ind he minimum alue o ‖𝑤‖ in he objec i e equa ion 𝑓(𝑥)=〈𝑤𝑇∙𝑥+𝑏〉. The posi ion o
he sepa a ing hype plane can be de e mined based on he alues o 𝑤 and b ha ul il he cons ain
𝑦𝑖∙(𝑤𝑇∙𝑥𝑖+𝑏)≥1. The key pa ame e 𝑏‖𝑤‖⁄ gi es he dis ance om he o igin (𝑥0,𝑦0) o he
closes da a poin along 𝑤. Fu he mo e, o deal wi h he case o he linea insepa able p oblem,
whe e empi ical isk is no ze o, a penal y ac o 𝐶 and slack a iables 𝜉𝑖 a e in oduced. The
op imal sepa a ing hype plane can be de e mined by sol ing he ollowing cons ained op imiza ion
p oblem [24,52]:
Minimize
1
2∙‖𝑤‖2+𝐶∙∑𝜉𝑖
𝑀
𝑖=1
(4)
subjec o
𝑦𝑖∙(𝑤𝑇∙𝑥𝑖+𝑏)≥1−𝜉𝑖 𝑓𝑜𝑟 𝑖=1,2,…,𝑀
𝜉𝑖 ≥0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖
(5)
whe e 𝜉𝑖 is he dis ance be ween he ma gin and w ongly loca ed samples 𝑥𝑖.
Despi e SVM being a linea unc ion se , i is possible o sol e nonlinea classi ica ion p oblems
by using a ke nel unc ion. As shown in Figu e 3, he mapping ansla es he classi ied ea u es on o
a high-dimensional space whe e he linea classi ica ion is easible.
Figu e 3. Mapping ke nel unc ions: E ec on he sepa a ion hype plane o wo-class da ase s.
In SVM me hod, he e a e di e en ypes o speci ic ke nel unc ions o imp o e he classi ie ,
including he linea ke nel ( he easies o in e p e ), Gaussian, o adial basis unc ion ke nel (RBF),
quad a ic, cubic, e c. These ke nels di e in he complexi y o de ini ion and p ecision in he
classi ica ion o di e en classes. In his wo k, bo h quad a ic and cubic ke nel unc ions we e used.
Two app oaches ha combine mul iple bina y SVMs we e used o add ess mul iclass
classi ica ion p oblems: One e sus one (OVO) and one e sus all (OVA). The OVO app oach needs
𝑚∙(𝑚−1) 2
⁄ SVM classi ie s o dis inguish be ween m classes [2]. The classi ie s a e ained o
di e en ia e he samples o one class om hose o ano he class. Based upon a o e o each SVM, an
unknown pa e n is classi ied. Thus, he s a egy o accomplish a single class decision ollows a
majo i y o ing scheme based on 𝑠𝑖𝑔𝑛 (𝑦𝑖∙(𝑤𝑇∙𝑥𝑖+𝑏)) [52]. The class ha wins he mos o es is
he one p edic ed o x. This winning class is di ec ly assigned o he es pa e n.
4.3. Decision T ee
X2
X1
φ2(x)
φ1(x)
X
F
φ (x)=[φ1(x), φ2(x)]
x=[x1, x2]
Φ
Figu e 3. Mapping ke nel unc ions: E ec on he sepa a ion hype plane o wo-class da ase s.
In SVM me hod, he e a e di e en ypes o speci ic ke nel unc ions o imp o e he classi ie ,
including he linea ke nel ( he easies o in e p e ), Gaussian, o adial basis unc ion ke nel (RBF),
quad a ic, cubic, e c. These ke nels di e in he complexi y o de ini ion and p ecision in he
classi ica ion o di e en classes. In his wo k, bo h quad a ic and cubic ke nel unc ions we e used.
Two app oaches ha combine mul iple bina y SVMs we e used o add ess mul iclass classi ica ion
p oblems: One e sus one (OVO) and one e sus all (OVA). The OVO app oach needs
m·(m−1)/
2
SVM classi ie s o dis inguish be ween mclasses [
2
]. The classi ie s a e ained o di e en ia e he
samples o one class om hose o ano he class. Based upon a o e o each SVM, an unknown
pa e n is classi ied. Thus, he s a egy o accomplish a single class decision ollows a majo i y o ing
scheme based on
sign yi·wT·xi+b
[
52
]. The class ha wins he mos o es is he one p edic ed o x.
This winning class is di ec ly assigned o he es pa e n.
Ene gies 2020,13, 2761 9 o 20
4.3. Decision T ee
The decision ee is a classi ica ion ool, based on decision ules, which uses a bina y ee g aph
o ind an unknown ela ionship be ween inpu and ou pu pa ame e s. A ypical ee s uc u e is
cha ac e ized by in e nal nodes ep esen ing es on a ibu es, b anches symbolizing ou comes o he
es , and lea nodes (o e minal nodes) de ining class labels. Decisions a a node a e aken wi h he
help o ules ob ained om da a [43,53].
The DT should ha e as many le els as necessa y o classi y he inpu ea u e da a. Depending
on he numbe o le els o his DT, he classi ica ion can be mo e o less accu a e, and mo e o less
calcula ion complex. A key poin , in his sense, is he sui able choice o he maximum numbe o spli s.
I is well known ha high classi ica ion accu acy on he aining da ase can be achie ed h ough a ine
ee wi h many lea es. Howe e , such a lea y ee usually o e i s he model and o en educes i s
alida ion accu acy in espec o he p ope aining accu acy. On he con a y, coa se ees do no
each such a high aining accu acy, bu hey a e easie o in e p e and can also be mo e obus in he
sense o app oaching he accu acy be ween bo h aining and ep esen a i e es da ase .
Based upon he o egoing and in o de o achie e he equi ed deg ee o accu acy, in his wo k,
he maximum numbe o spli s was se o 91 and he so-called Gini’s di e si y index was chosen as he
spli c i e ion. A a node, his index was de ined as ollows
GINI index =1−X
j
p2
j(6)
and i is he p obabili y o class jcomplying wi h he c i e ia o he selec ed node. Gini’s di e si y
index gi es an es ima ion o node impu i y since he op imiza ion p ocedu e in ee classi ie s ends o
nodes wi h jus one class (pu e nodes). Thus, a Gini index o 0 is de i ed om nodes ha con ain only
one class; o he wise, he Gini index is posi i e. The e o e, he op imal si ua ion o a gi en da ase is o
achie e a Gini index wi h a alue as small as possible.
4.4. Bagged Decision T ee Ensemble
Ensemble classi ie compiles he esul s o many weak lea ne s and combines hem in o a single
high-quali y ensemble model. The quali y o ha app oach depends on he ype o algo i hm chosen.
In his s udy, he selec ed bagged ee classi ie s we e based on B eiman’s andom o es algo i hm [
54
].
In he bagged me hod, he o iginal g oup o da a di ides in o di e en da ase s by andom selec ion
wi h eplacemen , and hen a classi ica ion o each one o hem is ob ained by a decision ee me hod.
The esul o each lea ne is submi ed o a o ing p ocess and he winne inally se s he bes
classi ica ion model o he bagged DT Ensemble me hod.
This me hod pe mi s ob aining lowe da a a iance han a single DT and also ge s a educed
o e -adjus men . The model can be imp o ed by p ope ly selec ing he numbe o lea ne s. I should
be no ed ha a la ge numbe o hem can p oduce high accu acy bu also slow down he classi ica ion
p ocess. In his wo k, a comp omise solu ion was ound by se ing he numbe o lea ne s o 30.
5. Full Compa a i e Classi ica ion o PQDs: De ailed O e iew
A de ailed o e iew o he p oposed hyb id classi ica ion plan is shown in Figu e 4, whe e he
main s eps desc ibed in p e ious sec ions ha e been highligh ed. The i s one is he analysis s age,
whe e signal p ocessing o he PQDs was ob ained ia S- ans o m. Then, an ini ial ea u e se , which
was de ined by s a is ical pa ame e s, was ex ac ed. Nex , ea u e ec o s we e op imized employing
bo h GA and CSO algo i hms, including an ex a alida ion p o ided by K-NN algo i hm. The las
s age consis ed o classi ica ion in ol ing he de e mina ion o PQ mul i-e en by using DT ( ine ee),
bagged decision ee ensemble, weigh ed K-NN, and bo h quad a ic and cubic SVMs.
Ene gies 2020,13, 2761 16 o 20
noise immuni y as indica ed by i s accu acy a es. This compa a i e s udy shows ha he p oposed
CSO-QSVM model, a las , equaled he be e esul s o classi ica ion accu acy ob ained in he li e a u e,
bu using only i e ea u es pe sample and dealing wi h 13 PQDs classes. These esul s, oge he wi h
he compa ison be ween al e na i e p oposals (Table 3) and he de ailed analysis o noise immuni y
(Table 4), cons i u e he main con ibu ions o his pape .
Al hough he p esen wo k deal wi h simula ed signals, he esul s we e so good ha hey could
be ex apola ed when applied o expe imen al da a. In such a case, a compa a i e wi h hose s udies
based on eal signals could be applied p ope ly.
As a u u e ex ension, an expe imen al se up would be used o es he e ec i eness o he
p oposed hyb id me hods unde common eal- ime wo king condi ions. Emula ed PQ incidence on
dis ibu ion ne wo ks could be modelled by low-cos ha dwa e p o o yping and so wa e componen s.
8. Conclusions
The mo i a ion o his wo k s emmed om challenges acing he elec ical sys ems and equipmen
in de e mining op imal, cos -e ec i e, and e icien powe quali y managemen . In his way, his pape
add essed he op imal hyb id classi ica ion me hods based on machine lea ning app oaches o
mee ing de ec ion, iden i ica ion, and classi ica ion o simula ed PQDs. Speci ically, ST was selec ed
o de ec ion and ea u e ex ac ion o PQDs, and, ollowing he end nowadays o u he op imize
he ecogni ion app oach, se e al op imiza ion algo i hms we e es ed o op imal ea u e selec ion.
A his s ep, his wo k unde lined he GA and CSO algo i hms since hey achie ed he bes esul s.
The esul ing op imal ea u e se s we e ed o se e al classi ie s, highligh ing among hem he QSVM,
CSVM, FT ee, ET ee, and WK-NN app oaches o showing imp o ed pe o mance.
The GA op imiza ion algo i hm associa ed wi h he FT ee, ET ee, and CSVM app oaches could
no classi y p ope ly all PQDs unde he condi ions es ablished in his analysis. Howe e , he esul s
ob ained h ough hese app oaches we e e y p omising and showed he g ea po en ial o hese kinds
o models when dealing wi h a ce ain g oup o PQDs.
Al e na i ely, CSO-based me hods including CSO-QSVM and CSO-WK-NN achie ed high
classi ica ion accu acy unde noisy condi ions. A ho ough compa a i e assessmen in e ms o noise
immuni y and classi ica ion accu acy led us o conclude ha he p o iciency o CSO-QSVM me hod is
sligh ly be e han CSO-WK-NN me hod.
I can also be no ed ha he esul s ound seemed o con i m he cu en end by which, despi e he
op imiza ion based on GA algo i hms being highligh ed by hei e iciency, GA-based me hodologies
a e p og essi ely being eplaced by he swa m op imiza ion algo i hms.
Finally, pe o mances o CSO-QSVM me hod we e compa ed o hose o o he classi ica ion
me hods al eady epo ed in he li e a u e, concluding ha he p oposed me hod achie ed a highe
deg ee o e iciency han mos o hem, and, based on he esul s, i may wo k well unde high noise
backg ound in p ac ical applica ions.
Au ho Con ibu ions:
J.C.B.-R. concei ed and de eloped he idea o his esea ch, designed he whole s uc u e o
he compa a i e s udy, and w o e he pape ; F.J.T. gene a ed he da a, pe o med he simula ions, and con ibu ed
o he me hodology; M.D.B. p o ided he heo e ical backg ound o he p oposed me hodology and con ibu ed
o i . All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This esea ch was unded by he Uni e sidad de Se illa (VI Plan P opio de In es igaci
ó
n y T ans e encia)
unde g an 2020/00000596.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Appendix A
The con inuous S ockwell ans o m (ST) [1] o a signal x( ) was o iginally desc ibed as
S(τ, )=ei2ππ τW(τ,d)(A1)

Ene gies 2020,13, 2761 17 o 20
i.e., a phase ac o ac ing on a con inuous wa ele ans o m (CWT)
W(τ,d)=Z+∞
−∞
x( )µ(τ− ,d)d (A2)
whe e
τ
is a ime displacemen ac o , dis a equency-scale dila ion ac o , and
µ( ,d)
is a speci ic
mo he wa ele ha includes he equency-modula ed Gaussian window
µ( ,d)= 
√2πe− 2 2
2e−i2ππ . (A3)
Ins ead, in his pape , he STFT pa hway o add ess he ST de ini ion was p e e ed,
STFT(τ, )=Z+∞
−∞
x( )w(τ− )e−i2ππ d (A4)
F om his app oach, ST can be w i en as
S(τ, )=Z+∞
−∞
x( )w(τ− , )e−i2ππ d (A5)
whe e
w( , )
is he Gaussian window unc ion simila o ha p oposed by Gabo (1946), bu now also
in oducing he a o emen ioned added equency dependence
w( , )= 
√2πe− 2 2
2(A6)
whe e he in e se o he equency 1/  ep esen s he window wid h.
F om ei he o he wo iewpoin s, he comple e ST de ini ion can be w i en as
S(τ, )=Z+∞
−∞
x( ) 
√2πe−(τ− )2 2
2e−i2π d (A7)
and also can be ep esen ed in e ms o i s ela ionship wi h FT and ela ed spec um X( ) o x( )
S(τ, )=Z+∞
−∞
X(α+ )e
2π2α2
2ei2πατdα ,0. (A8)
As is well known, he way o eco e he o iginal signal om con inuous ST is expensi e in e ms
o da a s o age due o o e sampling. This sampled e sion o he ST pe mi s calcula ing he widely
used ST complex ma ix ha is ob ained as (τ→jT, →n/NT)













Shjτ,n
NT i=N−1
P
m=0Xhm+n
NT iG(m,n)ei2πmj
N,n,0
S[jτ,0]=1
N
N−1
P
m=0Xhm
NT i,n=0
(A9)
whe e Tdeno es he sampling in e al, Nis he o al numbe o sample poin s, and bo h
Xhm+n
NT i
and
G(m,n)
esul a e disc e e as Fou ie ans o m (FFT), espec i ely, on he PQ dis u bance signal x( )
and he Gaussian window unc ion w( , ):
Xn
NT =1
N
N−1
X
k=0
x[kT]e−i2πnk
N(A10)
Ene gies 2020,13, 2761 18 o 20
G(m,n)=e−2π2m2
n2(A11)
whe e j,k,m, and na e in ege s in he ange o 0 o N−1.
The esul o disc e e ST is a 2D ime- equency ma ix ha is ep esen ed as
S(τ, )=A(τ, )e−iφ(τ, )(A12)
whe e
A(τ, )
is he ampli udeand
φ(τ, )
is he phase. Eachcolumn con ains he equencycomponen s
p esen in he signal a a pa icula ime. Each ow displays he magni ude o a pa icula equency
wi h ime a ying om 0 o N−1 samples.
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