applied
sciences
A icle
Fuzzy Logic App oach o Maximum Powe Poin T acking
Implemen ed in a Real Time Pho o ol aic Sys em
C is ian Napole * and Mohamed De beli and Osca Ba ambones
Ci a ion: Napole, C.; De beli, M.;
Ba ambones, O. Fuzzy Logic
App oach o Maximum Powe Poin
T acking Implemen ed in a Real Time
Pho o ol aic Sys em. Appl. Sci. 2021,
11, 5927. h ps://doi.o g/10.3390/
app11135927
Academic Edi o : Ludmila Dymo a
Recei ed: 10 June 2021
Accep ed: 23 June 2021
Published: 25 June 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
Sys em Enginee ing and Au oma ion Depa men , Facul y o Enginee ing o Vi o ia-Gas eiz,
Basque Coun y Uni e si y (UPV/EHU), 01006 Vi o ia-Gas eiz, Spain;
[email p o ec ed] (M.D.); osca [email p o ec ed] (O.B.)
*Co espondence: [email p o ec ed]
Abs ac :
Pho o ol aic (PV) panels a e de ices capable o con e ing sola ene gy o elec ical wi hou
emissions gene a ion, and can las o se e al yea s as he e a e no mo ing pa s in ol ed. The bes
pe o mance can be achie ed h ough maximum powe poin acking (MPPT), which is challenging
because i equi es a sophis ica ed design, since he sola ene gy luc ua es h oughou he day. The
PV used in his esea ch p o ided a low ou pu ol age and, he e o e, a boos -con e e wi h a
non-linea con ol law was implemen ed o each a sui able end-used ol age. The main con ibu ion
o his esea ch is a no el MPPT me hod based on a ol age e e ence es ima o (VRE) combined
wi h a uzzy logic con olle (FLC) in o de o ob ain he maximum powe om he PV panel. This
s uc u e was implemen ed in a dSpace 1104 boa d o a comme cial PV panel, PEIMAR SG340P.
The scheme was compa ed wi h a con en ional pe u ba ion and obse a ion (P&O) and wi h a
sliding mode con olle (SMC), whe e he ou comes demons a ed he supe io i y o he p oposed
ad anced me hod.
Keywo ds:
FLC; PV sys em; MPPT; P&O; nonlinea con ol; ol age e e ence es ima o ; boos con e e
1. In oduc ion
The inc easing deg ee o pollu ion h oughou he las decades made a change o
di ec ion in he policies o ene gy p oduc ion, whe e enewables a e p e e ed o e he
con en ional ones. Ac ually, i an inc emen o 25% in he wo ld ene gy consump ion
is expec ed o 2040, and he a ge is ha clean ones can p o ide 40% o he whole
p oduc ion [
1
]. Resea che s also o ecas ha sola PV panels could gene a e 15% o he
global demand in 2050 [2].
Despi e PV panels being highly dependen on sun i adia ion (which luc ua es su i-
cien ly depending on he geog aphical loca ion), majo ad an ages include he low main e-
nance and a la ge li e-span se ice [
3
]. In se e al applica ions whe e a PV ou pu ol age is
unsui able, DC–DC boos con e e s a e used o s ep-up his a iable, hus inc easing he
e iciency o he whole sys em. The ol age g ow h is ca ied ou h ough a swi ching o
elec ic ci cui s which indica es ha he end-use ol age is dependen on a du y cycle [
4
].
The swi ching signal can be managed by a pulse-wid h-modula ion (PWM) gene a o ;
his implies ha a sui able con ol law can be designed wi h he a ge o maximizing he
pe o mance o he MPPT.
Linea s a egies can be a sui able i s op ion o MPPT con ol. Despi e he con en-
ionali y o p opo ional–in eg al–de i a i e (PID) con olle s, hey ha e been widely used
in PV powe sys ems. Fo ins ance, he au ho o [
5
] p oduced a linea simula ion o a
DC–DC con e e wi h a PV panel; despi e he con olle being achie ed h ough classic
Bode analysis, he esul s epo ed a educ ion o o e shoo and se ling ime. Likewise,
an ad anced linea quad a ic egula o (LQR) has been de eloped o boos -con e e s [
6
];
al hough he measu emen o he pe o mance o his algo i hm needs expe imen al alida-
ion, he simula ion ou comes e ealed inc emen s in obus ness in compa ison wi h simila
Appl. Sci. 2021,11, 5927. h ps://doi.o g/10.3390/app11135927 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2021,11, 5927 2 o 18
s a egies. Howe e , linea con olle s p oduce ce ain issues o eal boos con e e s
due o inconsis encies such as he in insic non-minimum phase [
7
]. Addi ionally, he e
a e nonlinea i ies, such as sola ene gy, ha depend on e e yday wea he ; panels ha e
asso ed ol age-cu en oscilla ions and boos con e e s ha e a discon inuous ac ion due
o he swi ching [8,9].
Nonlinea con olle s a e known no only o he pe o mance imp o emen o boos
con e e s bu also o hei p ac ical implemen a ion [
10
]. The P&O algo i hm is a eliable
nonlinea algo i hm o MPPT ha compa es p e ious powe measu emen s o cu en
ones o ind he MPP, using he du y cycle as a a iable [
11
]. In he ield o PV panels, his
algo i hm has e en been s udied in ad anced and complex sys ems, such as PVs in space
applica ions, as he au ho s o [
12
] explained in hei esea ch. The combina ion o his
algo i hm wi h a sola acke p o ided sui able esul s. Due o i s p ope ies, i has also
been used in asso ed enewable sys ems such as wa e ene gy gene a o s [
13
], PV based
mic og ids [
14
], wind gene a o s [
15
], and so o h. Despi e i s obus ness and p ac icali y,
he downsides o P&O a e ela ed o he gene a ed pe u ba ions o pa ial shadow cases,
which can p oduce, espec i ely, oscilla ions a ound he MPP o all o a local maximum,
which p oduces a educ ion in accu acy [16,17].
Ano he classic nonlinea s uc u e is SMC, which no only enhances he dynamics o
a sys em bu also educes he e ec s o dis u bances and unce ain ies [
18
]. Fo ins ance,
De beli e al. [
19
] p oposed SMC o a boos -con e e used in a uel cell. The ou comes
showed ha one o he p ima y disad an ages o hese s uc u es is he cha e ing, which
causes ex a ene gy consump ion. E en hough he cha e ing can be educed by using
high o de sliding modes—which ha e been implemen ed in he pas [
20
]—i is impossible
o elimina e his phenomenon since he e a e always unmodeled dynamics in ol ed wi h
con ol digi aliza ion [
21
]. Backs epping depic s ano he nonlinea s uc u e, known o
i s obus ness in e ms o unmodeled dynamics and unce ain ies, based on a Lyapuno
app oach by spli ing he main sys em in o subdi isions o achie e a sui able design [
22
].
In p ac ical e ms, backs epping has been implemen ed in a boos con e e by he au ho s
o [
23
], whe e he obus ness and se ling ime ha e been imp o ed; ne e heless, one o
he main d awbacks o he backs epping is he complex design o he con ol law.
Ano he non-linea echnique is FLC, a p ac ical, uled-based app oach in which
he con igu a ion is dependen on he ope a o knowledge wi h ega ds o he main
sys em [
24
,
25
]. FLC has been used in adap i e expe imen al DC–DC boos con e e s,
whe e he induc o ea u es changed o achie e high e iciencies and he ou comes showed
signi ican enhancemen s [
26
]. In e ms o PV, he esea ch conduc ed by he au ho s o [
27
]
was based on an analysis o he powe o ecas o e a yea , whe e hey employed an
adap i e neu al uzzy in e ence sys em and accu a e esul s we e achie ed. Conce ning
MPPT o PV sys ems linked wi h boos con e e s, FLC was used by he au ho s o [
28
],
whe e hey showed ha he ou comes de eloped a as dynamic esponse and ensu ed he
obus ness. FLC has also been embedded o a PV ba e y sys em in a s udy based on an
op imisa ion [
29
]; in his case, he s a e o cha ge and consump ion we e s udied and he
esul s e ealed signi ican imp o emen s.
This esea ch is ocused on a ype-1 FLC, which is known o i s highe pe o mance
compa ed o con en ional con olle s due o i s capabili y o manage unce ain ies h ough
ules [
30
,
31
]. A VRE was designed wi h eal da a om he PV panel and was assembled
wi h a ype-1 FLC, an e o -based con olle used o acking. The i s compa ison
was pe o med agains a P&O, as i is commonly used o he compa ison o new MPPT
con olle s [
32
,
33
]. The second con as was execu ed wi h an SMC combined wi h he
VRE o highligh he bene i s o he p oposed FLC. The d awbacks a e ela ed o sudden
luc ua ions a ound he maximum powe poin (MPP) due o sudden luc ua ions in he
i adia ion o unde pa ial shadowing condi ions [33,34].
This esea ch pape is s uc u ed as ollows: Sec ion 2desc ibes he ha dwa e used
and i s in e connec ion, and p esen s a heo e ical explana ion o he models applied o he
PV and he boos con e e wi h u he de ails abou he con olle s handled o he MPPT
Appl. Sci. 2021,11, 5927 3 o 18
acking. Sec ion 3p esen s an expe imen al analysis o he PV ea u es such as ol age–
cu en and ol age–powe cu es, which a e signi ican o gene a ing a VRE. This sec ion
also concludes wi h he ou comes ga he ed wi h he implemen ed con olle s. Finally,
Sec ion 4comp ises a summa y o he highligh s accomplished h oughou his esea ch.
2. Ma e ials and Me hods
2.1. Ha dwa e Desc ip ion
Figu e 1shows he wo k low o he ha dwa e in ol ed in he expe imen s. The PV
panel used was a PEIMAR SG340P, he modules o which we e p oduced om poly-
c ys alline silicon, which ep esen s sui able e sa ili y and e iciency. These panels a e
equen ly employed in comme cial, esiden ial and indus ial ins alla ions. Since he
s uc u al design is ocused on he ligh ness and obus ness, i has high s i ness and a
easible ins alla ion. Supplemen a y echnical in o ma ion abou his de ice is de ailed in
Table 1.
Figu e 1. So wa e–ha dwa e wo k low diag am.
Table 1. Peima SG340P Speci ica ions.
P ope ies Values Uni s
Dimensions 156 ×156 mm
Maximum powe 340 W
Open ci cui ol age 45.2 V
Max powe ol age 36.7 V
Max powe cu en 9.28 A
Numbe o cells in se ies 6 uni
Numbe o cells in pa allel 12 uni s
Isc 9.9 A
The empe a u e and i adia ion we e measu ed wi h a silicon ex e nal senso om
he manu ac u e Ingenieu bü o Si-V-10TC-T, which is eliable o PV a iables moni o ing.
This module is o med o a monoc ys alline silicon sola cell ha is connec ed o a shun . I
is also capable o co ec ing he measu emen s due o an ac i e empe a u e compensa o ,
Appl. Sci. 2021,11, 5927 4 o 18
which wo ks h ough a senso ha is lamina ed o he back su ace o he module. As a
consequence o his ac ion, i p o ides addi ional in o ma ion abou he empe a u e. Bo h
measu ed signals a e ansmi ed as ol age a ia ions in he ange o 0–10 V. Fu he
de ails a e p o ided in Table 2.
Table 2. Ingenieu bü o Si-V-10TC-T Speci ica ions.
P ope ies Values Uni s
Vol age supply 12 o 28 VDC
I adiance measu emen ange Up o 1500 W/m2
Tempe a u e measu emen ange −40 o 90 ◦C
Rega ding he boos con e e , a TEP-192 p oduced by he esea ch g oup o Huel a
Uni e si y (Spain) was implemen ed in his case. This de ice has a PWM swi ching inpu o
20 kHz o he con ol p ocess and wo ks wi h inpu /ou pu signals o 0–10 V. The module
has wo Scho ky diodes MURF1560 GT, 2 TK capaci ances (1500
µ
F and 3000
µ
F), six
PVC2-564-08 induc ances and an HGT40N60B3 IGBT. Vol age and cu en lec u es a e
p o ided in signals om 0 o 10 V. Addi ional echnical da a on his con e e a e desc ibed
in Table 3.
Table 3. TEP-192 De ails.
P ope ies Values Uni s
Swi ching equency 20 kHz
Maximum inpu ol age 60 V
Maximum inpu cu en 30 A
Maximum ou pu ol age 250 V
Maximum ou pu cu en 30 A
To close he elec ic ci cui , an 8500B by BK P ecision was used since i is a DC
p og ammable load wi h wide lexibili y and i is sui able o es ing and e alua ing DC
powe sou ces such as DC–DC boos con e e s, ba e ies, cha ge s and especially o PV
panels. This p og ammable esis ance includes e e se pola i y p o ec ion o p o ec he
sys em om empe a u e, powe , ol age and cu en o e uns. The echnical speci ica ions
a e p esen ed in Table 4.
Table 4. BK 8500B Speci ica ions.
P ope ies Values Uni s
Powe 300
Ra ed Vol age 500 V
Ra ed Cu en 15 A
Inpu Vol age 0–150 V
Inpu Cu en 0–15 A A
Resis ance ange 0.05–10 Ω
In he acquisi ion and con ol signal gene a ion, a dSpace DSP1104 was linked because
i is con enien ha dwa e o collec ing and gene a ing analog signals, which a y be ween
0 and 10 V. Addi ionally, ex e nal so wa e designed in Simulink can be embedded o
con ol pu poses. This pla o m also suppo s Real-Time-In e ace (RTI) and Con olDesk,
whe e he la e is he so wa e used o obse ing da a in eal ime. All he expe imen s
we e ca ied ou wi h a sampling equency o 10 kHz as his sa is ies he equi emen s o
he quali y o he da a acqui ed and he ha dwa e limi a ions.
Appl. Sci. 2021,11, 5927 5 o 18
2.2. PV Model
The single diode model was conside ed o his s udy since i p o ides accu acy and
simplici y [
35
]. This heo y sugges s ha PV should be conside ed as a DC sou ce o
an elec ic ci cui , as shown in Figu e 2, which p oduces a cu en
Iph
, gene a ed by he
i adia ion o he sun. In addi ion, wo esis ances a e in ol ed, whe e
Rsh
is ela ed o
he leakage cu en o he diode o he p-n in e ace [
36
] and
Rs
ep esen s he esis ance
a hwa he PV [
37
]; he la e causes a de imen al phenomenon in e ms o he maximum
powe o he sys em [38].
Figu e 2. PV Model.
The Ki cho cu en law p o ides Equa ion
(1)
, whe e he
Ish
and
Ic
a e de ined
acco ding o he esea ch o [39].
Ic=Iph −Id−Ish (1)
Ic=Iph −Ioeq(V+RsIc)
αKTc−1−V+RsIc
Rsh
, (2)
whe e
Io
,K,qand
Tc
a e he e e se sa u a ion cu en , Bol zmann cons an , elemen a y
cha ge and he ope a ing empe a u e, espec i ely [
40
]. The gene a ed cu en om he
PV panel is also exp essed as Equa ion
(3)
, whe e
G/GSRC
ep esen s he ela ion o he
sola adiance measu ed and he adiance a s anda d a ing condi ions (SRC); he
Isc_ e
is
he sho ci cui cu en o he PV and
T e
is he PV empe a u e, bo h a iables a SRC [
41
].
The e m kI_ e comp ises a he mal ac o o he sho ci cui cu en .
Iph =G
G e Isc_ e +kI_ e (T−T e ). (3)
A PV panel was buil wi h se e al modules in pa allel (
Np
) and in se ies (
Ns
), which
led o an ou pu cu en (
Im
) and ol age (
Vm
) o he whole module based on Equa ion
(4)
.
The e o e, wi h he p e ious equa ions, he ou pu cu en o he PVG can be exp essed as
Equa ion (5).
Im=NpIc
Vm=NsVc
(4)
Ic=IphNp−NpIoeq(V+RsIc)
αKTc−1−NpV+RsIc
Rsh
. (5)
2.3. Boos Con e e Model
In his in es iga ion, a con e e was used o ans e he ene gy om he PV panel o
he esis i e load. The elec ical con igu a ion is a boos ype ha a ge s a ol age s ep up
om he sou ce o deli e a sui able end-used ol age. The ela ionship be ween he inpu
and ou pu is gi en by he du y cycle (d) h ough Equa ion
(6)
. This a iable is dependen
Appl. Sci. 2021,11, 5927 6 o 18
on a PWM signal gene a o , which is con olled h ough a physical de ice, which, in his
case, is a me al–oxide–silicon ield-e ec ansis o (MOSFET). Ce ainly, he alue o d
was bounded be ween 0.1 and 0.9 o he con inuous conduc ion mode.
Vou =Vin
1−d. (6)
The s a e–space ep esen a ion o he boos con e e is exp essed as in Equa ion
(7)
,
such ha x1=Iin and x2=Vou [20].
˙
x="0dc−1
L
1−dc
C
−1
RC #"x1
x2#+"1
L
0#Vin
y=h0 1i"x1
x2#.
(7)
2.4. Type-1 Fuzzy Con olle
FLC con olle s a e s e ling, as hei con igu a ion is based on he designe ’s expe ience
a he han knowledge o he sys em’s ma hema ical model. The implemen ed ype-1 uzzy
con olle (shown in Figu e 3) uses no malized ules in i s s uc u e and, hus, scale
ac o s
Ke
and
Kd
a e used o he inpu s, which a e he e o and i s de i a i e. An
ex a ou pu scale ac o
Kb
was also used o egula e he con ol signal in e ms o he
equi ed pe o mance.
The uzzy block includes h ee consequen modules, which a e he uzzi ica ion,
in e ence and de uzzi ica ion. The i s changes he inpu s in o uzzy inpu s o designa e he
deg ee o he membe ship; o his case, o e lapped iangula and uni o mly disc e ized
(be ween
−
1 and 1) unc ions a e based on nega i e big (NB), nega i e medium (NM),
nega i e small (NS), ze o (Z), posi i e small (PS), posi i e medium (PM) and posi i e big
(PB); his esul ed in 25 adap ed ules, which a e shown in Table 5. The in e ence is he
mechanism by which linguis ic ules a e e alua ed acco ding o he esul s ob ained in he
p e ious s ep o an i – hen ype [
42
]. Finally, he de uzzi ica ion comp ises he s age a
which he linguis ic exp essions a e ansla ed in o nume ical alues o he ou pu ; in his
case, equidis an and disc e ized cons an ou pu s we e con igu ed in a ange be ween
−
1
and 1.
Figu e 3. Type-1 FLC S uc u e.
Table 5. FLC linguis ic ules.
E ˙
ENB NS Z PS PB
NB NB NM NM NS Z
NS NM NM NS Z Z
Z NM NS Z PS PM
PS Z Z PS PM PM
PB Z PS PM PM PB
Appl. Sci. 2021,11, 5927 7 o 18
FLC a chi ec u e was implemen ed as in he schema ic diag am o Figu e 4. The ol -
age e e ence es ima o will be explained in u he sec ions as i is ela ed o he ea u es
ga he ed om he PV panel.
Figu e 4. Implemen a ion a chi ec u e o FLC con olle .
2.5. Pe u ba ion and Obse e Algo i hm
This algo i hm is based on an in en ional and pe iodical pe u ba ion on he con ol
command wi h a ollowing obse a ion and e alua ion o he sys em ou pu [
43
]. Applied
o he PV sys em, he pe u ba ion is gene a ed h ough a change in he ol age
Vin(k)
and cu en
Iin(k)
, such ha he powe o he PV is measu ed. This implies ha he slope
∆P/∆V
can be calcula ed, which helps wi h knowing whe he he MPP is achie ed, as
Figu e 5shows.
Figu e 5. Powe –Vol age cu e wi h he P&O mechanism o each he MPP.
Based on he p e ious desc ip ion and on he de ailed logic o Figu e 6, he knowledge
o
Pin(k)
and
Vin(k)
and i s delay in
k−
1 allows o he calcula ion o he men ioned slope.
The e o e, i he la e men ioned alue is posi i e, he du y cycle d(which modi ies he
ol age by Equa ion
(6)
) will inc ease such ha he algo i hm ou pu
u=d+∆d
and
aims o each he MPP; on he con a y, when he posi ion is a he igh side o he MPP,
he con ol signal dec eases h ough u=d+∆d.
Appl. Sci. 2021,11, 5927 8 o 18
Figu e 6. Flowcha o P&O algo i hm.
2.6. Sliding Mode Con olle
Conside ing ha he e o is de ined as Equa ion
(8)
, whe e
V e
is he e e ence
ol age, hen an in eg al sliding su ace is es ablished h ough Equa ion
(9)
such ha
λ>0.
e=V e −Vin, (8)
S=e−λZe·d . (9)
Acco ding o Slo ine e al. [
44
], he con ol signal is composed o an equi alen
ueq
and
a swi ching e m
usw
, which is de ined in Equa ion
(10)
. The i s men ioned is ga he ed
h ough he condi ion
˙
S=
0 [
45
]; he swi ching, which gua an ees obus ness, is exp essed
in Equa ion (11).
u=ueq +usw (10)
usw =−K·sign(S). (11)
As p e iously men ioned, ob aining he equi alen con ol signal implies ha he
su ace de i a i e should be equal o ze o. The e o e, as he e o was o me ly exp essed
in Equa ion
(8)
, and wi h he usage o he sys em in Equa ion
(7)
, he equi alen con ol
e m is ob ained as ollows.
ueq =−1
x2(Vin +λLe −x2). (12)
3. Resul s
3.1. Analysis o PV Fea u es and Vol age Re e ence Es ima o
The cha ac e is ics cu es shown in Figu e 7we e ob ained by a conside able alue
o esis ance linked o he PV sys em; his alue was dec eased g adually while he da a
we e being eco ded. Simul aneously, he en i onmen empe a u e and i adia ion we e
ga he ed, which a ied, espec i ely, be ween 14.6
◦
C and 36.6
◦
C and om 64 W/m
2
o
808 W/m2.
Figu e 7a indica es he ela ion be ween he ol age and he cu en whe e h ee
sec ions a e dis inguished and we e labeled in he esea ch conduc ed by he au ho s
o [
46
]. The i s is called he cu en sou ce pa , whe e his ea u e ends o s ay cons an ;
he second one, known as he knee o he I-V cu e, is he sec ion whe e he MPP is
achie ed; and inally, he ol age sou ce pa is whe e he cu en – ol age is linea ly ela ed.
Along he cu en sou ce sec ion, his ends o s ay almos cons an be ween 0 V and 35 V.
Appl. Sci. 2021,11, 5927 9 o 18
Howe e , his sec ion is highly dependen on he i adia ion, which, ideally, lean hese
cu es upwa ds, whe eas he empe a u e shi s in a ho izon al mo emen [
47
]. The knee
and he ol age sou ce pa a e mos ly condi ioned by he empe a u e, which mo es he
cu es o he igh hand side o he g aph [48].
Figu e 7b shows he ol age–powe cu e whe e, in his case, he i adia ion leans
he cu es upwa ds and he slope o he ini ial linea beha iou ascends. Ne e heless,
he empe a u e in luences a diagonal d i e o he cu es. The e o e, his means ha he
i adia ion mo es he MPP on a e ical axis, whe eas he empe a u e mo es he cu es
on an slan ed axis.
Figu e 7. PV panel cha ac e is ic cu es, whe e: (a) ol age-cu en ; (b) ol age–powe .
Fo me ly, he au ho s ha e been wo king wi h a cu en e e ence es ima o in powe
sou ces such as p o on exchange memb ane uel cells (PEMFC), whe e he ou pu e e -
ence was dependen on he empe a u e [
23
,
49
]. In his case, a VRE was de eloped by
highligh ing he MPP poin s om 790 cu es like hose in Figu e 7b, which a ied wi h
he adia ion, and hen a su ace
V e _MPP = (P
,
T)
was cons uc ed. This unc ion was
a ained wi h he Ma lab Cu e Fi ing Toolbox since i employs an op imized sol e o
achie e sui able pa ame e s ha could each he bes i quali y o cu es and su aces.
The amewo k p o ided a polynomial su ace ep esen ed by Equa ion
(13)
; hese pa am-
e e s we e ob ained h ough an op imiza ion o pe o mance me ics o acqui e a sui able
goodness o i ing (GOF); hese a e he sum o squa ed e o s (SSE), R-squa e, adjus ed
R-squa e and oo mean squa e o he e o s (RMSE).
Table 6p o ides he ob ained me ic alues, which a e app op ia e because he SSE
and he RMSE should be minimum whe eas he R-squa e and i s adjus ed e sion should
be nea he uni [
50
–
52
]. Fu he mo e, he coe icien s o Equa ion
(13)
a e also exhibi ed
wi h 95% con idence bounds. The ep esen a ion o he su ace is shown in Figu e 8based
on ol age, powe and empe a u e.
(x,y) = p00 +p10 ·x+p01 ·y+p20 ·x2+p11 ·x·y+p30 ·x3+p21 ·x2·y. (13)
Appl. Sci. 2021,11, 5927 16 o 18
Abb e ia ions
The ollowing abb e ia ions a e used in his manusc ip :
PV Pho o ol aic
MPPT Maximum powe poin acking
VRE Vol age e e ence es ima o
FLC Fuzzy logic con olle
P&O Pe u ba ion and obse a ion
PWM Pulse-wid h-modula ion
PID P opo ional-in eg al-de i a i e
LQR Linea quad a ic egula o
SMC Sliding mode con ol
RTI Real- ime-obse a ion
SRC S anda d a ing condi ions
MOSFET Me al-oxide-silicon ield-e ec ansis o
NB Nega i e Big
NM Nega i e Medium
NS Nega i e Small
Z Ze o
PS Posi i e Small
PM Posi i e Medium
PB Posi i e Big
MPP Maximum powe poin
PEMFC P o on exchange memb ane uel cell
GOF Goodness o i ing
SSE Sum o he squa ed e o s
RMSE Roo mean squa ed o he e o s
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