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Ci a ion: Domenech, B.; Fe e -Ma í,
L.; Ga cía, F.; Hidalgo, G.; Pas o , R.;
Ponsich, A. Op imizing PV Mic og id
Isola ed Elec i ica ion P ojec s—A
Case S udy in Ecuado . Ma hema ics
2022,10, 1226. h ps://doi.o g/
10.3390/ ma h10081226
Academic Edi o s: Ripon Kuma
Chak abo y and A min Fügenschuh
Recei ed: 16 Decembe 2021
Accep ed: 5 Ap il 2022
Published: 8 Ap il 2022
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ma hema ics
A icle
Op imizing PV Mic og id Isola ed Elec i ica ion P ojec s—A Case
S udy in Ecuado
B uno Domenech 1,2 , Laia Fe e -Ma í1,3, Facundo Ga cía4, Geo gina Hidalgo 4, Ra ael Pas o 1,2
and An onin Ponsich 1,2,*
1
Supply Chain and Ope a ions Managemen Resea ch G oup, Escola Tècnica Supe io d’Enginye ia Indus ial,
Uni e si a Poli ècnica de Ca alunya-Ba celonaTech, 08028 Ba celona, Spain;
[email p o ec ed] (B.D.); laia. e [email p o ec ed] (L.F.-M.); [email p o ec ed] (R.P.)
2Depa men o Managemen , Uni e si a Poli ècnica de Ca alunya-Ba celonaTech, 08028 Ba celona, Spain
3Depa men o Mechanical Enginee ing, Uni e si a Poli ècnica de Ca alunya-Ba celonaTech,
08028 Ba celona, Spain
4Ca alan Associa ion o Enginee ing Wi hou Bo de s, 08027 Ba celona, Spain; [email p o ec ed] (F.G.);
[email p o ec ed] (G.H.)
*Co espondence: [email p o ec ed]
Abs ac :
Access o elec ici y o he u al and indigenous popula ion o Ecuado ’s Amazon Region
(RAE) is conside ed a c i ical issue by he na ional au ho i ies. The RAE is an isola ed zone wi h
communi ies sca e ed h oughou he ain o es , whe e he expansion o he na ional g id is no a
iable op ion. The e o e, au onomous elec i ica ion sys ems based on sola ene gy cons i u e an
impo an solu ion, allowing he de elopmen o indigenous popula ions. This wo k p oposes a
ool o he design o s and-alone u al elec i ica ion sys ems based on pho o ol aic echnologies,
including bo h mic og id o indi idual supply con igu a ions. This ool is o mula ed as a Mixed
In ege Linea P og amming model including economic, echnical and social aspec s. This app oach
is used o design elec i ica ion sys ems (equipmen loca ion and sizing, mic og id con igu a ions)
in h ee eal communi ies o he RAE. The esul s highligh he bene i s o he de eloped ool and
p o ide guidelines ega ding RAE’s elec i ica ion.
Keywo ds:
u al elec i ica ion; ma hema ical p og amming; Ecuado ’s Amazon egion; pho o ol aic
ene gy; mic og id
1. In oduc ion
Cu en ly, access o elec ici y emains a c i ical issue o a ound 1.1 billion people
wo ldwide, gene ally a ec ing hose li ing in u al con ex s, whe e po e y le els a e high.
Indeed, bo h aspec s (ene gy access and po e y) a e closely connec ed, as p o ed by he
Se en h Sus ainable De elopmen Goal s a ed by he Uni ed Na ions (“ensu e access o
a o dable, eliable sus ainable and mode n ene gy o all”) [
1
]. This goal explici ly iews
access o ene gy as key o imp o ing he li ing condi ions o he mos unde p i ileged
popula ions, “a backbone o any mode n economy” [
2
]. In his sense, se e al ecen s udies
highligh he ac ha elec ici y access is linked o inc eased incomes and p oduc i i y
(di e en o ag icul u al ac i i ies [
3
]), as well as bene i s in educa ion (inc eased s udy
ime) and heal h (dec ease in espi a o y diseases due o lowe ke osene usage) [
4
]. In
o de o imp o e elec ici y co e age, he ex ension o he na ional g id has been he main
s a egy o p o iding access o elec ici y. Howe e , in a eas wi h ough opog aphy o
emo e popula ion cen e s, he expansion o he na ional dis ibu ion g id may become
in easible [
5
]. In hose cases, he de elopmen o o -g id elec i ica ion sys ems and
mic og ids is among he mos economical solu ions and has been success ully adop ed
in many p ac ical amewo ks [
6
]. Besides his, o such s and-alone gene a ion sys ems,
he cu en conce n abou global wa ming and he esul ing in e es in al e na i e ene gy
Ma hema ics 2022,10, 1226. h ps://doi.o g/10.3390/ma h10081226 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 1226 2 o 24
sou ces ha e led o he in ensi e use o enewable esou ces. In pa icula , sola ene gy
emains one o he mos widely employed sou ces [7].
In his amewo k, di e en kinds o elec i ica ion sys ems ha e been ackled in he
de o ed li e a u e, in many dis inc wo ldwide con ex s. Fo ins ance, ega ding la ge-scale
elec i ica ion (na ional o egional), Ehsan and Yang [
8
] highligh he endency o o mula e
he in eg a ion and planning o gene a ion sys ems based on enewable ene gy as an
op imiza ion p oblem. The model p oposed in [
9
] o enewable alloca ion planning in la ge-
scale powe sys ems conside s a dynamic en i onmen leading o he expansion/ educ ion
o he cu en ne wo k. Hassan [
10
] uses simula ion o op imize sola pho o ol aic-based
gene a ion sys ems in I aq, conside ing wo con igu a ions, namely o -g id and on-g id
e sions (when possible). Taye e al. [
11
] use Geog aphical In o ma ion Sys ems (GIS) and
a mul i-c i e ia decision-making echnique o u al elec i ica ion planning, allowing o
an e alua ion o he adequacy o enewable ene gy in E hiopia. Th ough he applica ion o
he Analy ic Hie a chy P ocess, hey conclude ha wind and pa icula ly sola ene gies
should be p e e ed a he han g id ex ension. In addi ion, se e al compu a ional ools
we e ecen ly de eloped, in pa icula o u al elec i ica ion planning. Fo ins ance,
he REM (Re e ence Elec i ica ion Model [
12
]) ocuses on he use o o -g id gene a ion
sys ems in o de o plan elec ici y ne wo ks and has been used in se e al coun ies
(India, Colombia, Kenya, Rwanda, e c.). Fu he mo e, OnSSET [
13
] is an open-sou ce
ool using GIS ha is designed o complemen ene gy planning models no suppo ed
by geog aphical analysis and ha quan i ies he in es men , echnology ype and geo-
e e encing o na ional elec i ica ion p ojec s based ei he on con en ional o enewable
ene gy echnologies. Also obeying an open-sou ce philosophy, ecen ly de eloped Py hon-
based ools ep esen ee access al e na i es. Fo ins ance, PyPSA (Py hon o Powe
Sys em Analysis [
14
]) is a ee oolbox o simula ing and op imizing mode n powe
sys ems, accoun ing o al e na i e wo king modes (con en ional gene a o s, a iable wind
and sola gene a ion, s o age uni s, mixed al e na ing and di ec cu en
ne wo ks, e c.)
and
wi h an imp o ed scalabili y o dealing wi h la ge ne wo ks and long ime se ies. Ano he
ecen compu a ional ool o mic og id design, Sandia’s Mic og id Design Toolki [
15
],
allows o he gene a ion o design op imiza ions acco ding o se e al c i e ia (in es men
and ope a ion cos s, eliabili y, pe o mance le els) in o de o p oduce a Pa e o on ie o
e icien con igu a ions p omo ing he mic og id o e indi idual supply sys ems. Based on
ad anced op imiza ion and modeling app oaches, i has been used o p o ide elec ici y in
se e al mili a y and public in as uc u es wo ldwide.
A a local le el, Bah ama a e al. [
16
] e iew wo ks using he well-es ablished com-
me cial ool HOMER [
17
], pa icula ly ocusing on he design o hyb id enewable ene gy
sys ems. Fo ins ance, in [
18
], such a ool is used o design and analyze he obus ness o
en i onmen al- iendly sys ems o be ins alled in Malaysian islands. In addi ion, in e es is
inc easingly de o ed o he design o mic og ids, which a e educed-size g ids connec ing
se e al use s isola ed om he na ional g id. Se e al applica ions ha e demons a ed
ha he design o sel -su icien mic og ids p oduces highe bene i s han hose ob ained
by indi idual sys ems [
19
] and also educes he li e cycle en i onmen al impac o he
elec i ica ion sys ems [
20
]. Despi e he addi ional complexi y associa ed wi h he design o
mic og id opologies, his s a egy allows he ene gy supply o be independen om he
esou ces a ailable a demand poin s, cos sa ings hanks o economies o scale o sha ed
equipmen and supply lexibili y in case o an inc ease in demand [
21
]. Acco dingly, many
s udies ha e ackled he design o mic og id dis ibu ion s uc u es a ound he wo ld. The
eade is e e ed o he ecen su eys o
Pe e s e al. [22]
as well as
Cas illa e al. [23]
o
a pe spec i e on he use o mic og ids in La in Ame ica,
Mahomed e al. [24]
in Uganda,
Tenenbaum e al. [
25
] in sub-Saha an A ica o
Lukuyu e al. [26]
in Eas A ica. These
s udies insis on he need o adequa e design and planning ools o mic og id-based elec i-
ica ion p ojec s, suppo ed by objec i e demand p ojec ion me hodologies and de eloped
in collabo a ion wi h local ac o s o accoun o he popula ion’s speci ic equi emen s.
Ma hema ics 2022,10, 1226 3 o 24
Rega ding he sol ing p ocedu es, ma hema ical models ha e been de eloped wi h
he aim o p o iding e ec i e con igu a ions (in e ms o cos o any o he pe o mance
c i e ion). Indeed, hese models a e adap ed o he pa icula applica ion add essed and
may in ol e dis inc con igu a ions (indi idual/mic og id), ene gy sou ces o o he ea-
u es speci ic o each case s udy and o he communi ies o be elec i ied [
27
]. Fo ins ance,
Lei hon e al. [
28
] de elop a model o ene gy alloca ion policy ha minimizes ene gy
usage. In [
29
], an op imiza ion app oach is de eloped o he elec i ica ion o highland
communi ies in Pe u h ough a model in eg a ing social cons ain s associa ed wi h sys em
managemen and communi y bene i s. Ranaboldo e al. [
30
] de elop op imiza ion models
including he pa icula conside a ions o elec i ica ion sys ems in Cabo Ve de. Heu is ic
p ocedu es ha e also been used. Fo ins ance, se e al ma heu is ics a e in oduced in [
31
],
ep esen ing compu a ionally e icien ools o op imize u al elec i ica ion sys ems in ol -
ing bo h mic og ids and indi idual supply. Mo eo e , di e en mul i-c i e ia app oaches
(VIKOR and AHP) a e compa ed in [
32
] o mic og id design in Venezuela, including
economic (in es men , main enance and ope a ion cos s), social and en i onmen al c i e ia.
Finally, Py hon-based ee access lib a ies ha e also been de eloped in his con ex , such as
Mic og idsPy [
33
]. This ool ackles he p oblem o gene a ion equipmen sizing (Li-ion
ba e ies, diesel gene a o s and PV panels) and ene gy dispa ching in emo e and isola ed
con ex s by minimizing he Le elized Cos o Ene gy (LCOE).
The abo e-men ioned s udies emphasize he possibili y o designing economically
e icien and en i onmen ally esilien o -g id gene a ion sys ems based on enewable
ene gies. This is pa icula ly ele an o he so-called “las -mile” elec i ica ion (as opposed
o mass elec i ica ion, o which di e en s a egies such as g id expansion may p o ide
be e esul s). “Las -mile” elec i ica ion p ojec s a e no only use ul in he con ex o , o
example, small islands [
34
], whe e diesel gene a o s migh be eplaced by pho o ol aic and
wind ene gies, as hey ha e a g ea impac on g eenhouse gas emissions while p oducing
signi ican economic bene i s. In se e al La in Ame ican coun ies, bo h he Amazon
( ain o es ) and Andean (highlands) egions ypically show ough e ains and ep esen
massi e challenges, such as epo ed o B azil [
35
], Colombia [
36
] o Boli ia, Pe u and
A gen ina [
37
]. In his amewo k, Ecuado is a coun y wi h a wide-sp ead na ional
g id and high global access o elec ici y, bu he indigenous popula ions o he Amazon
basin a e sca e ed o e la ge a eas co e ed by ain o es , leading o p ohibi i e cos s
o expanding he na ional g id. Fu he mo e, he di icul y in eaching hese isola ed
communi ies and hei agile economic esou ces make elec i ica ion p ojec s nei he
p o i able o p i a e dis ibu o s no sus ainable o go e nmen s. So, he a o emen ioned
s and-alone, enewable-based sys ems appea as a iable elec i ica ion s a egy. To he
bes o ou knowledge, only one s udy has epo ed on he design o such sys ems in
Ecuado [
38
]. Ca ied ou in he San a Elena p o ince, his wo k epo s he design o
hyb id wind–pho o ol aic sys ems h ough HOMER, concluding ha mos o he ene gy
is supplied by PV cells. Howe e , he pape does no accoun o mic og id o ma ion,
al hough such con igu a ions a e p omo ed among he guidelines s a ed by Ecuado ’s
Minis y o Elec ici y and Renewable Ene gies (MEER) [39].
Thus, he p esen wo k add esses he de elopmen o au onomous elec i ica ion
sys ems o isola ed communi ies in he Amazon Region o Ecuado (RAE) by op imizing
he design o PV-based sys ems in ol ing mic og ids. Thanks o a de ailed analysis o
he ele an local ac o s o be accoun ed o , a Mixed In ege Linea P og am (MILP) is
in oduced as a compu a ional ool o he au oma ic design o such elec i ica ion sys ems.
This model ex ends se e al s a e-o - he-a app oaches. In pa icula , Fe e -Ma íe al. [
40
]
se ou he basis o u al elec i ica ion sys em modeling in ol ing hyb id gene a ion and
mic og ids. Domenech e al. [
29
], in addi ion, include managemen and social cons ain s
in he design phase. Despi e conside ing some elemen s o hese p e ious s udies, ou
model accoun s o cha ac e is ics ha a e speci ic o he RAE and he e o e ep esen s
a no el design ool, which could ye be ex apola ed o o he con ex s. In pa icula , he
con ibu ion in e ms o new modeling ea u es is h ee old:
Ma hema ics 2022,10, 1226 4 o 24
i.
A se o po en ial connec ions is es ablished, indica ing impossible wi ing be ween
di e en geog aphical poin s. This new cons ain , mo i a ed by local ac o s
(explained in he ollowing sec ions), is add essed h ough a decomposi ion s a egy
o he o iginal p oblems ha allows a mo e e icien solu ion p ocess.
ii.
A new objec i e unc ion now inco po a es he pa ame ized ponde a ion o he
cos s o mic og ids e sus indi idual sys ems. This no el ea u e is mo i a ed by
he elec i ica ion policies dic a ed by Ecuado ’s na ional go e nmen , p omo ing
mic og id con igu a ions. Howe e , he e sa ili y o he o mula ion p oposed
he e allows ei he one o he o he con igu a ion o be a o ed acco ding o any
policy make s’ decisions.
iii.
Fo cul u al easons in he RAE, i ems sha ed by he membe s o a communi y
canno be s o ed on p i a e g ound. This equi emen is e lec ed in he model by
new cons ain s, which p e en mic og id gene a ion uni s om being loca ed a
demand poin s.
Thus, his model cons i u es a ool adap ed o RAE condi ions o assis ing p ojec
p omo e s in he design o elec i ica ion sys ems. Th ee case s udies add essing he
elec i ica ion o indigenous communi ies in he RAE a e subsequen ly sol ed in o de o
alida e he p oposed ool.
The emainde o his wo k is o ganized as ollows. Sec ion 2p esen s a con ex
analysis ega ding elec i ica ion p ocesses in Ecuado , a desc ip ion o he elec i ica ion
sys ems accoun ed o and hei speci ic condi ioning ea u es in he RAE. The p oposed
ma hema ical model is desc ibed in Sec ion 3, while Sec ion 4p esen s he case s udies
add essed and he nume ical esul s ob ained. Finally, some conclusions and p ospec s o
u u e wo ks a e p o ided in Sec ion 5.
2. Con ex Analysis o he Design o S and-Alone Elec i ica ion Sys ems in he RAE
2.1. O e iew o he Elec i ica ion P ocess in Ecuado
The Republic o Ecuado is a La in Ame ican coun y wi h an a ea o 283,561 km
2
and
abou 17 million inhabi an s. Despi e he g ow h o i s G oss Domes ic P oduc , s ong
inequali ies mean ha mo e han 20% o Ecuado ians su e om po e y, pa icula ly in
u al and indigenous communi ies. Besides, Ecuado has some o he g ea es biodi e si y
in he wo ld, meaning ha en i onmen al p o ec ion is a p ominen ea u e o he coun-
y’s landscape. In his con ex , go e nmen policies o he de elopmen o he na ional
elec ici y sys em mus ind a ade-o be ween socio-economic de elopmen and en i-
onmen al conse a ion [
41
]. Acco dingly, e o s ha e been made by na ional au ho i ies
o gua an ee a eliable and compe i i e elec ici y supply, supe ised by he Regula o y
Agency and Elec ici y Con ol. These measu es ha e led o a signi ican inc ease in he
o al elec ici y gene a ed [
42
] and he expansion o he na ional g id, eaching one o he
bes co e ages in he sub-con inen : 97.33% na ional access and 94% o he u al popula ion
in 2017 [
43
]. Howe e , he de elopmen o Ecuado ’s elec i ica ion p ocess s ill aces wo
impo an challenges.
Fi s , some di e si ica ion is needed ega ding he gene a ion ma ix. As s essed
in [
44
], he only consis en end in Ecuado ian ene gy policies has been he de elopmen
o hyd oelec ici y, which ep esen ed mo e han 73% o he gene a ion ma ix in 2017 [
42
].
Howe e , his s a egy is c i icized by indigenous communi ies and en i onmen alis asso-
cia ions due o i s en i onmen al impac [
45
]. Se e al s udies ha e also emphasized he
g ea po en ial o sola and wind ene gies and hei ad an ages ega ding socio-economic
de elopmen and en i onmen al conse a ion [
41
]. Adop ing pho o ol aic echnologies
may yield long- e m bene i s in e ms o pollu ion aba emen and clima e change mi -
iga ion [
39
]. Acco dingly, he Fund o Elec i ica ion o Ru al and o Ma ginal U ban
A eas (c ea ed in 2004, in o de o imp o e elec ici y co e age in disad an aged a eas)
ini ia ed u al elec i ica ion p og ams elying on PV gene a ion in he Amazonian and
highland egions o hyb id sys ems in he Galapagos islands [
46
]. Mo e ecen ly, he cu en
adminis a ion is p omo ing PV-powe ed mic og id designs o emo e a eas a he han
Ma hema ics 2022,10, 1226 5 o 24
indi idual sys ems [
39
]. Despi e hese e o s, sola and wind ene gies did no ep esen
mo e han 0.5% o he p ima y sou ces used o elec ici y gene a ion in 2017.
On he o he hand, he e a e s ill some gla ing inequali ies in elec ici y access in
he RAE. In spi e o he e o s made o inc ease co e age in he co esponding p o inces
(Pas aza, Sucumbíos, O ellana, Napo, Mo ona and Zamo a), hei impac was unde mined
by he limi ed amoun o economic esou ces. Indeed, he RAE (40% o Ecuado ’s o al a ea)
is cha ac e ized by he highes po e y le els in Ecuado and has he lowes elec i ica ion
a es, pa icula ly in u al a eas, since he g id expansion s a egy has le a g ea pa
o his e i o y unco e ed [
43
]. Renewable ene gy echnologies ha e been p omo ed,
bu se e al elec i ica ion p og ams we e no conside ed p o i able by he dis ibu ion
ope a o s and we e hus in e up ed. Indeed, he main enance ope a ions o s and-alone
sys ems in ough e ain ( ain o es ) would ei he be oo cos ly o equi e aining p og ams
o communi y membe s. In addi ion, he sca ce economic esou ces o hese mainly
indigenous communi ies mean ha use s o en canno a o d he elec ici y se ice cos s,
making hese p ojec s unsus ainable o go e nmen s.
Thus, he six RAE p o inces p oduce elec ici y almos en i ely by he mal gene a ion.
In he Pas aza p o ince, whe e he p esen s udy was ca ied ou , he elec ici y co e age
cu en ly equals 89.3% (compa ed o 97.33% a na ional le el), bu he u al elec i ica ion
a e is much lowe (65.9% in 2010 [42]). This si ua ion is pa icula ly unsus ainable in one
o he mos p oli ic zones o sola ene gy, whe e he anspo a ion, i ewood consump ion
and elec ici y needs (mo e han 1000 GWh/yea ) could be en i ely co e ed by PV echnolo-
gies [
47
]. Accoun ing o hese gene al conside a ions, he sys emic app oach in oduced in
his wo k aims o p o ide an au oma ic ool o he design o elec i ica ion sys ems in u al
and disad an aged a eas o he RAE, aking ad an age o he high PV po en ial a ailable
and p omo ing he o ma ion o mic og id-based sys ems.
2.2. Technical Desc ip ion o S and-Alone Sys ems
In [
47
], he au ho s demons a e ha while he RAE bene i s om high sola esou ces,
on he con a y, wind does no cons i u e a p omising ene gy sou ce. Fu he mo e, due
o he p e ailing dense ain o es ege a ion, he cons uc ion and ins alla ion o wind
u bines aises p ac ical issues. The e o e, he elec i ica ion sys ems ep esen ed he e a e
only based on pho o ol aic echnology, as illus a ed in Figu e 1(wi h mic og id-based
dis ibu ion). The elec ici y is p oduced by he PV panels, while he con olle s p o ec
ba e ies om o e loads and deep discha ges. The elec ici y is hen s o ed in ba e ies
o b idge he gap be ween gene a ion and consump ion. Nex , he in e e s ans o m
he di ec cu en om ba e ies in o al e na ing cu en , which is mo e sui able o mos
elec ical appliances. Finally, he elec ici y is dis ibu ed o demand poin s (households,
schools, heal h cen e s, e c.) ia mic og ids o indi idual sys ems (indi idual sys ems a e
de o ed o supplying single-use consump ion).
Ma hema ics 2022, 10, x FOR PEER REVIEW 6 o 25
Figu e 1. Scheme o a PV sys em wi h mic og id dis ibu ion.
2.3. Condi ioning Fac o s o S and-Alone Elec i ica ion Sys ems in he RAE
This sec ion analyzes he main ea u es condi ioning he design o u al elec i ica ion
p ojec s wi h PV echnology o indigenous communi ies in he RAE. This analysis in-
ol ed s udying and unde s anding he conc e e eali y o he a ge ed communi ies
h ough h ee main ac ion lines. Fi s , he in o ma ion ob ained om he li e a u e was
enhanced wi h echnical documen s [49,50] p oduced by Ecuado ’s MEER in coo dina ion
wi h he NGO “Enginee s Wi hou Bo de s” (EWB). Second, in o ma ion on he local con-
ex and needs was comp ehensi ely collec ed h ough a 5-mon h ield su ey pe o med
by some o he au ho s o his wo k. This s ay allowed he indigenous popula ion’s li e-
s yle o be obse ed as well as he economic, social, en i onmen al o cul u al ea u es
ele an o he ins alla ion o au onomous sys ems o hese communi ies o be iden i ied.
I also made possible an e alua ion o he logis ics needed o equipmen anspo a ion
wi hin ain o es condi ions. Thi d, se e al key ac o s o u al elec i ica ion in Ecuado
we e in e iewed o iden i y and alida e he condi ioning ea u es adap ed o he RAE’s
eali y. This phase in ol ed h ee echnicians om he MEER, he leade o he enewable
ene gies a ea in he powe company Emp esa Eléc ica Amba o S.A. (Pas aza p o ince)
and h ee coo dina o s, echnicians and olun ee s om he ene gy line o EWB.
The esul ing lis o condi ioning ac o s encompasses all he ea u es o be accoun ed
o wi hin he design p ocess o elec i ica ion sys ems in he RAE. These conside a ions
a e cap u ed in he ma hema ical model de eloped o he solu ion o he add essed p ob-
lem (see nex sec ion).
(a) Na ional and egional policies con empla e social aspec s, such as op ing o elec i-
ica ion designs including mic og ids a he han indi idual sys ems. Indeed, he
communi y-based managemen o join ins alla ions p o ides social bene i s, such as
he coo dina ion and coope a ion o amilies sha ing he same objec i es. In o de o
encou age mic og id o ma ion, p io i y is gi en o designs including such con igu-
a ions, e en hough hey en ail a highe cos han indi idual supply sys ems (up o
20% highe , as p oposed by MEER).
(b) The ins i u ional amewo k o elec i ica ion p ojec s may ensu e economies o scale
( o ins ance, when equipmen is pu chased o dis ic o egional p ojec s) bu may
also es ic he echnical cha ac e is ics o powe gene a ion and dis ibu ion i ems.
In he case s udy p esen ed he e, he limi e boxes allow only wo ou pu cables,
which may ha e an impac on he mic og id s uc u e.
(c) The communica ion pa hs a ailable ( i e s, ai ways) and he anspo a ion means
o ge o he communi y in ques ion ha e an impac on he echnical equipmen em-
ployed. Mo eo e , he cu en s a e o hese pa hs may also in ol e space and weigh
limi a ions o he equipmen uni s o be shipped. Fo ins ance, he a ying wa e
dep h in a i e (o landing s ip dimensions) can limi he size o canoes (o ai c a )
ha can be used.
Figu e 1. Scheme o a PV sys em wi h mic og id dis ibu ion.

Ma hema ics 2022,10, 1226 6 o 24
Rega ding he mic og id opology, a s uc u e ensu ing minimal cos s is chosen,
espec ing he ollowing condi ions [
48
]: (i) powe gene a ion is cen alized a a single
poin and (ii) he mic og id has a adial s uc u e (each use can ecei e elec ici y om
only one poin ). This s uc u e is illus a ed in Figu e 1, whe e he gene a ion uni s deli e
powe o ou use s connec ed acco ding o a ee-like con igu a ion.
2.3. Condi ioning Fac o s o S and-Alone Elec i ica ion Sys ems in he RAE
This sec ion analyzes he main ea u es condi ioning he design o u al elec i ica ion
p ojec s wi h PV echnology o indigenous communi ies in he RAE. This analysis in ol ed
s udying and unde s anding he conc e e eali y o he a ge ed communi ies h ough h ee
main ac ion lines. Fi s , he in o ma ion ob ained om he li e a u e was enhanced wi h
echnical documen s [
49
,
50
] p oduced by Ecuado ’s MEER in coo dina ion wi h he NGO
“Enginee s Wi hou Bo de s” (EWB). Second, in o ma ion on he local con ex and needs
was comp ehensi ely collec ed h ough a 5-mon h ield su ey pe o med by some o
he au ho s o his wo k. This s ay allowed he indigenous popula ion’s li es yle o be
obse ed as well as he economic, social, en i onmen al o cul u al ea u es ele an o
he ins alla ion o au onomous sys ems o hese communi ies o be iden i ied. I also
made possible an e alua ion o he logis ics needed o equipmen anspo a ion wi hin
ain o es condi ions. Thi d, se e al key ac o s o u al elec i ica ion in Ecuado we e
in e iewed o iden i y and alida e he condi ioning ea u es adap ed o he RAE’s eali y.
This phase in ol ed h ee echnicians om he MEER, he leade o he enewable ene gies
a ea in he powe company Emp esa Eléc ica Amba o S.A. (Pas aza p o ince) and h ee
coo dina o s, echnicians and olun ee s om he ene gy line o EWB.
The esul ing lis o condi ioning ac o s encompasses all he ea u es o be accoun ed
o wi hin he design p ocess o elec i ica ion sys ems in he RAE. These conside a ions a e
cap u ed in he ma hema ical model de eloped o he solu ion o he add essed p oblem
(see nex sec ion).
(a)
Na ional and egional policies con empla e social aspec s, such as op ing o elec i-
ica ion designs including mic og ids a he han indi idual sys ems. Indeed, he
communi y-based managemen o join ins alla ions p o ides social bene i s, such as
he coo dina ion and coope a ion o amilies sha ing he same objec i es. In o de o
encou age mic og id o ma ion, p io i y is gi en o designs including such con igu a-
ions, e en hough hey en ail a highe cos han indi idual supply sys ems (up o
20% highe , as p oposed by MEER).
(b)
The ins i u ional amewo k o elec i ica ion p ojec s may ensu e economies o scale
( o ins ance, when equipmen is pu chased o dis ic o egional p ojec s) bu may
also es ic he echnical cha ac e is ics o powe gene a ion and dis ibu ion i ems. In
he case s udy p esen ed he e, he limi e boxes allow only wo ou pu cables, which
may ha e an impac on he mic og id s uc u e.
(c)
The communica ion pa hs a ailable ( i e s, ai ways) and he anspo a ion means
o ge o he communi y in ques ion ha e an impac on he echnical equipmen
employed. Mo eo e , he cu en s a e o hese pa hs may also in ol e space and
weigh limi a ions o he equipmen uni s o be shipped. Fo ins ance, he a ying
wa e dep h in a i e (o landing s ip dimensions) can limi he size o canoes (o
ai c a ) ha can be used.
(d)
The p ope y concep in some indigenous communi ies means ha he equipmen
sha ed by he communi y canno be physically ins alled a a demand poin , which is
p i a e g ound. So, non-demand poin s should be iden i ied o po en ial mic og id
gene a ion. Fu he mo e, his means he cons uc ion o sheds o elec ic equipmen
s o age wi hin he a ea whe e PV panels a e o be loca ed in o de o p o ec ba e -
ies, in e e s and egula o s om wea he o animals. This incu s addi ional cos s
associa ed wi h he pu chase and ins alla ion o hese buildings.
(e)
In he Low Amazon egion, he inc eased conce n o en i onmen al aspec s leads o
he de elopmen o unde g ound connec ions a he han ai connec ions. Indeed,
Ma hema ics 2022,10, 1226 7 o 24
despi e he ad an ages o ai mic og ids in bo h p ac ical (a oiding obs acles such
as i e s, small buildings, e c.) and economic (cheape ins alla ion and main enance)
e ms, hey ha e a nega i e en i onmen al impac due o ee clea ing a ound he
mic og id ins alla ions. Unde g ound wi ing is also be e p o ec ed om ex e nal
agen s ( ain, animals) and p esen s echnical ad an ages [
51
]. Howe e , his policy
in ol es cons ain s due o physical obs acles ( i e , a ine, loodable a ea, landing
s ip, e c.) ha may p e en cable ins alla ion.
I should be no ed ha , in he abo e-men ioned ac o s, he es ima ion o end-use s’
demand is no men ioned. E alua ing he demand cons i u es a c i ical phase in he
design o elec i ica ion sys ems, since his pa ame e s ongly in luences he inal sys em
con igu a ion [
52
]. An o e es ima ed demand would lead o o e sized sys ems, bo h
was ing gene a ion capaci y and inc easing elec ici y p ices [
53
]. Se e al s udies ha e
highligh ed he complexi y o his es ima ion ask, in ol ing quali a i e and quan i a i e
assessmen s o he a ge communi y and i s su oundings, an analysis o he ene gy sou ces
used p io o elec i ica ion, he planned elec ici y usage and he p ojec ion o he demand
g ow h [
22
,
54
,
55
]. In he RAE, he MEER pe o med such a s udy, as e i ied by he ield
su ey ca ied ou in Ecuado by some o he au ho s o his wo k. In o de o gua an ee
equal oppo uni ies o e e y amily, he MEER decided o se s anda dized consump ion
le els o ene gy and peak powe o all use s. In his con ex , he demand alues accoun ed
o in he p esen s udy a e hose de e mined and imposed by he MEER (see Sec ion 4.2).
No e ha hese demands we e concei ed as a cons an alue, so p ojec s may ini ially be
sligh ly o e sized bu will be adequa e in he medium e m, when demand g ow h occu s
du ing he i s yea s o implemen a ion.
Hence, his analysis iden i ied he key elemen s ensu ing he sus ainabili y o elec i-
ica ion p ojec s in he RAE. This phase se s he concep ual amewo k necessa y o he
de elopmen o a solu ion p ocedu e o he design o s and-alone elec i ica ion sys ems
in his egion.
3. A Ma hema ical Model o he Design o Au onomous Ru al Elec i ica ion Sys ems
in he RAE
Since op imiza ion app oaches ha e p o en hei e ec i eness in he amewo k o
elec i ica ion sys em design [
56
], a ma hema ical model o s and-alone elec i ica ion
sys ems, including he conside a ions analyzed abo e, is in oduced in his sec ion. The
ma hema ical o mula ion p oposed in his wo k includes pa s o he p ocedu e p esen ed
in [
57
] and u he accoun s o all he condi ioning ac o s desc ibed in he p e ious sec ion.
The p oposed me hod o he de elopmen o he ma hema ical ool in oduced he e is
shown in Figu e 2.
This new MILP aims o de e mine he de ails o he elec i ica ion design (indi id-
ual and/o mic og id con igu a ions, equipmen loca ion and selec ion, e c.) in o de o
minimize he p ojec cos . To u he highligh he o iginal ea u es o he ackled p oblem
and he no el y o he model de eloped acco dingly, hei main cha ac e is ics a e lis ed in
Table 1, and he compa ison wi h simila wo ks emphasizes ha no o he app oach consid-
e s all o hem. In pa icula , he e sa ili y in p omo ing ei he mic og id o indi idual
supply con igu a ions and he es ic ion on mic og id gene a ion equipmen being loca ed
a demand poin s a e in oduced o he i s ime in his wo k. The e o e, he elec i ica ion
solu ions ob ained wi h he model p oposed he e, which esponds o RAE’s social, cul u al
and en i onmen al equi emen s, could no be ob ained wi h o he ools, hus con i ming
he con ibu ion o he p esen wo k. A he same ime, since hese equi emen s may
be ound in plen y o con ex s di e en om RAE’s, such as many isola ed egions in
La in Ame ica [
58
] and A ica [
24
,
26
], and since he model p oposed he e also includes he
classical ea u es needed o he design o elec i ica ion sys ems based on enewables, i
can be e-used o o he applica ions in dis inc con ex s.
Ma hema ics 2022,10, 1226 8 o 24
Ma hema ics 2022, 10, x FOR PEER REVIEW 8 o 25
Figu e 2. Flowcha o he p oposed me hod.
This new MILP aims o de e mine he de ails o he elec i ica ion design (indi idual
and/o mic og id con igu a ions, equipmen loca ion and selec ion, e c.) in o de o mini-
mize he p ojec cos . To u he highligh he o iginal ea u es o he ackled p oblem and
he no el y o he model de eloped acco dingly, hei main cha ac e is ics a e lis ed in
Table 1, and he compa ison wi h simila wo ks emphasizes ha no o he app oach con-
side s all o hem. In pa icula , he e sa ili y in p omo ing ei he mic og id o indi idual
supply con igu a ions and he es ic ion on mic og id gene a ion equipmen being lo-
ca ed a demand poin s a e in oduced o he i s ime in his wo k. The e o e, he elec-
i ica ion solu ions ob ained wi h he model p oposed he e, which esponds o RAE’s
social, cul u al and en i onmen al equi emen s, could no be ob ained wi h o he ools,
hus con i ming he con ibu ion o he p esen wo k. A he same ime, since hese e-
qui emen s may be ound in plen y o con ex s di e en om RAE’s, such as many iso-
la ed egions in La in Ame ica [58] and A ica [24,26], and since he model p oposed he e
also includes he classical ea u es needed o he design o elec i ica ion sys ems based
on enewables, i can be e-used o o he applica ions in dis inc con ex s.
Table 1. Compa ison o main p oblem ea u es add essed in his s udy and in p e ious ela ed
wo ks.
Fea u e a.* b.* c.* d.* e.* .* g.* h.*
PV gene a ion X X X X X X X
Ba e y s o age X X X X X X X
Mic og id dis ibu ion X X X X X X X
Se e al mic og ids X X X X X X
Indi idual supply X X X X X X
Ve sa ile (mic og ids s. indi idual) X
Fo bidden connec ions X X
Res ic ion on mic og id gene a ion poin s X
Demand/non-demand poin s X X X X X
Economic assessmen X X X X X X X X
Addi ional ins alla ion cos s X X X X
* whe e a. HOMER [17]; b. ViPOR [59]; c. Fe e -Ma í e al. [40]; d. REM [12]; e. Domenech e al.
[57]; . Mic og idsPy [33]; g. Sandia’s Mic og id Design Toolki [15]; h. The app oach p oposed in
his wo k.
Field su ey:
- Iden i ica ion o condi ioning ac o s
- Da a collec ion
Li e a u e su ey: s a e-
o - he-a solu ion ools
Fo mula ion o he
ma hema ical model
Valida ion wi h es
ins ances
Compu a ional
expe imen s
Figu e 2. Flowcha o he p oposed me hod.
Table 1.
Compa ison o main p oblem ea u es add essed in his s udy and in p e ious ela ed wo ks.
Fea u e a.* b.* c.* d.* e.* .* g.* h.*
PV gene a ion X X X X X X X
Ba e y s o age X X X X X X X
Mic og id dis ibu ion X X X X X X X
Se e al mic og ids X X X X X X
Indi idual supply X X X X X X
Ve sa ile (mic og ids s. indi idual) X
Fo bidden connec ions X X
Res ic ion on mic og id gene a ion poin s X
Demand/non-demand poin s X X X X X
Economic assessmen X X X X X X X X
Addi ional ins alla ion cos s X X X X
* whe e a. HOMER [
17
]; b. ViPOR [
59
]; c. Fe e -Ma íe al. [
40
]; d. REM [
12
]; e. Domenech e al. [
57
];
. Mic og idsPy [33]; g. Sandia’s Mic og id Design Toolki [15]; h. The app oach p oposed in his wo k.
The nume ical pa ame e s o he model include in o ma ion ega ding demand poin
loca ions and equi emen s, he cha ac e is ics o he powe gene a ion and dis ibu ion
equipmen , as well as he da a associa ed wi h he speci ic RAE ea u es. These pa ame e s
a e comp ehensi ely de ined in Table 2, bu u he explana ions a e p o ided he e on he
Ma ix o Po en ial Connec ions.
As s a ed p e iously, only unde g ound-wi ed mic og ids a e conside ed. Howe e ,
he ain o es en i onmen in which he indigenous communi ies li e is no mally cha -
ac e ized by loodable e ain. Besides his, obs acles such as i e s o landing s ips can
p e en unde g ound wi ing, hus impeding some di ec connec ions be ween wo poin s.
Thus, all he allowed connec ions a e conca ena ed in a bina y ma ix called he Ma ix o
Po en ial Connec ions (MPC), de ined o each case s udy. This ma ix o size |P|
×
|D|
has elemen s equal o 1 i a connec ion is possible be ween he wo poin s and 0 o he wise.
I is wo h men ioning ha his ma ix may highligh clus e s o poin s comple ely isola ed
om he o he s (i.e., nei he di ec no indi ec connec ions can be es ablished be ween any
poin s o wo di e en clus e s).
This in o ma ion o e s wo op ions: ea ing each communi y as a whole o b eaking
i down in o se e al independen ly sol ed sub-p oblems. Indeed, he e is no eason o
simul aneously sol e se e al physically sepa a ed zones, since hey canno be included
in he same mic og id anyway. Addi ionally, due o he combina o ial na u e o he
MILP model p esen ed he ea e , ea ing each sub-p oblem independen ly migh allow
signi ican CPU ime sa ings when compa ed wi h sol ing he global p oblem. Thus, be o e
using he op imiza ion ool p oposed he e, a p e-p ocessing s ep mus be pe o med o
Ma hema ics 2022,10, 1226 9 o 24
each pa icula case in o de o decide i he ackled communi y should be ea ed en i ely
o di ided in o isola ed clus e s o be sol ed independen ly one om ano he .
In addi ion, he decision a iables ha e o accoun o he sys em con igu a ion wi h
i s associa ed ene gy/powe lows, as well as he numbe and ype o equipmen uni s o be
used o powe gene a ion and dis ibu ion (see Table 3). The objec i e unc ion consis s in
minimizing he o al in es men cos o wind u bines, PV panels, PV con olle s, ba e ies,
in e e s, me e s ( o be ins alled a all he demand poin s in a mic og id, o g an equali y
o he p o ided se ices o all use s) and wi es. In addi ion, as s a ed in poin d) o he
condi ioning ac o s (Sec ion 2.3), a cul u al equi emen is ha he gene a ion equipmen
o (communi y-sha ed) mic og id sys ems canno be se a demand poin s (p i a e g ound),
meaning ha addi ional cos s co esponding o he ins alla ion o a shed o equipmen
s o age ha e o be included.
Mo eo e , as explained in poin (a) o Sec ion 2.3, MEER’s policy p io i izes mic og id-
based designs o e indi idual sys ems. The e o e, con igu a ions including mic og ids
will be p e e ed e en i hei cos is up o
α
% highe han indi idual supply sys ems.
P ac ically, he cos associa ed wi h all he i ems belonging o a mic og id will be mul iplied
by 1/(1 +
α
/100), which is equi alen o educing he co esponding cos by a ac o
α/(α+ 100)%.
The gene a ion cos s o indi idual sys ems (a demand poin s) can easily
be dis inguished om hose o mic og id con igu a ions by de ining mic og id gene a ion
poin s as no-demand poin s. Wi h ega d o α, he MEER p oposes 20%, bu he in luence
o his alue is also s udied h ough a sensi i i y analysis (see nex sec ion).
Table 2. Pa ame e s o he ma hema ical model.
Pa ame e Desc ip ion Uni
Demand poin s
PSe o po en ial gene a ion poin s, including he demand poin s. -
DSe o demand poin s, D∈P. -
Lpd Dis ance be ween wo poin s pand d(p∈P,d∈D). [m]
Lmax Maximum leng h o a wi e segmen o he mic og id. [m]
MPCpd (p,d)-elemen o he ma ix o po en ial connec ions (p∈P,d∈D). ∀p∈P,
∀d∈D,MPCpd ∈{0,1}. -
QpSubse o poin s o which poin pcan be di ec ly connec ed wi h a wi e segmen (p∈P,
d∈D:p6=d,MPCpd = 1, Lpd ≤Lmax). -
EDpEne gy demand a p(p∈D). [Wh/day]
PDpPowe demand a p, conside ing he simul anei y ac o (p∈D). [W]
PV gene a ion
S,NS Se o PV panel ypes and maximum numbe o PV panels ha can be placed a a poin ,
espec i ely. -
ESsEne gy gene a ed by a PV panel o ype s (s∈S). [Wh/day]
PSsMaximum powe o a PV panel o ype s (s∈S). [W]
CSsCos o a PV panel o ype s (s∈S). [US$]
ZSe o PV con olle ypes. -
PZzMaximum powe o a PV con olle o ype z(z∈Z). [W]
CZzCos o a PV con olle o ype z(z∈Z). [US$]
Elec ic equipmen
BSe o ba e y ypes. -
EBbCapaci y o a ba e y o ype b(b∈B). [Wh]
CBbCos o a ba e y o ype b(b∈B). [US$]
ηbBa e y e iciency. [%]
DB Maximum discha ge p opo ion admi ed o he ba e ies. [%]
DA Requi ed au onomy o he ba e ies. [days]
ISe o in e e ypes. -
PIiMaximum powe o an in e e o ype i(i∈I). [W]
CIiCos o an in e e o ype i(i∈I). [US$]
ηiIn e e e iciency. [%]
CL Cos o an elec ic me e de ice. [US$]
Ma hema ics 2022,10, 1226 16 o 24
Ma hema ics 2022, 10, x FOR PEER REVIEW 17 o 25
Figu e 6. Demand and po en ial gene a ion poin s in Conambo. Numbe s e e o he households
(blue) and po en ial gene a ion poin s ( ed).
4.3. Expe imen al Resul s
As indica ed p e iously, one con ibu ion o he ma hema ical ool de eloped in Sec-
ion 3 is he pa ame e α which, in he objec i e unc ion, deno es he pe cen age o cos
o e head accep ed pe mic og id wi h ega d o indi idual sys ems. The in oduc ion o
his pa ame e was due o MEER’s desi e o p omo e g id o ma ion, conc e ely p oposing
a alue α o 20% (a mic og id con igu a ion is p e e ed i i s cos is up o 20% highe han
he cos o indi idual sys ems). Howe e , he aim is o design a e sa ile ool ha may be
used in bo h ways—i.e., ei he o p omo e o penalize mic og id o ma ion o e indi id-
ual sys ems, depending on he policy make s’ decisions. In o de o pe o m a sensi i i y
analysis using his pa ame e and compa e he con igu a ions ob ained in di e en cases,
h ee execu ions we e pe o med o each communi y, wi h di e en α alues: 20% (mi-
c og id o ma ion is p omo ed as p oposed by he MEER), 0% (mic og ids a e nei he pe-
nalized no p omo ed) and −20% (mic og id o ma ion is penalized, wi h an in e se
amoun compa ed o ha p oposed by he MEER). No e ha o he alues could be chosen
o α in o de o u he p omo e/penalize g id o ma ion, bu he h ee selec ed alues (-
20%, 0%, 20%) seem o be enough o demons a e he abili y o he p oposed ool o p o-
duce di e en kinds o con igu a ions.
The compu a ional expe imen s p esen ed he e we e ca ied ou sol ing he p e i-
ously desc ibed MILP p oblem, using IBM ILOG CPLEX 12.2, un on an In elCo e i7-6700
3.40GHz p ocesso (16 Gb RAM). The MILP sol e employed in CPLEX is based on a
B anch and Cu algo i hm. The co esponding OPL code o ou model, as well as comple e
inpu da a (al eady a ailable in Sec ion 4.2) and he de ailed solu ions ob ained in e e y
op imiza ion p ocess (decision a iables and objec i e unc ion), a e a ailable om he
ollowing websi e ( ee public access): h ps://gi ioc.upc.edu/ioc/2022_ma hema -
ics_equado (accessed on 6 Ap il 2022). In his way, he alidi y o he solu ions ob ained
can be checked by he eade , and his s udy is comple ely ep oducible. All he execu ions
a e pe o med using a ela i e op imali y gap o 10–6 wi h a 1 h ime limi , which p o ed
o be enough o sol e all he ackled ins ances op imally excep he C-RC sub-p oblem
(independen ly om he alue o α). No e ha , o he San a Rosa and Conambo commu-
ni ies, op imali y could be ob ained o he co esponding sub-p oblems hanks o he “di-
ide-and-conque ” s a egy p oposed he e, which decomposes communi ies in o sub-
Figu e 6.
Demand and po en ial gene a ion poin s in Conambo. Numbe s e e o he households
(blue) and po en ial gene a ion poin s ( ed).
4.3. Expe imen al Resul s
As indica ed p e iously, one con ibu ion o he ma hema ical ool de eloped in
Sec ion 3is he pa ame e
α
which, in he objec i e unc ion, deno es he pe cen age o cos
o e head accep ed pe mic og id wi h ega d o indi idual sys ems. The in oduc ion o
his pa ame e was due o MEER’s desi e o p omo e g id o ma ion, conc e ely p oposing
a alue
α
o 20% (a mic og id con igu a ion is p e e ed i i s cos is up o 20% highe
han he cos o indi idual sys ems). Howe e , he aim is o design a e sa ile ool ha
may be used in bo h ways—i.e., ei he o p omo e o penalize mic og id o ma ion o e
indi idual sys ems, depending on he policy make s’ decisions. In o de o pe o m
a sensi i i y analysis using his pa ame e and compa e he con igu a ions ob ained in
di e en cases, h ee execu ions we e pe o med o each communi y, wi h di e en
α
alues: 20% (mic og id o ma ion is p omo ed as p oposed by he MEER), 0% (mic og ids
a e nei he penalized no p omo ed) and
−
20% (mic og id o ma ion is penalized, wi h an
in e se amoun compa ed o ha p oposed by he MEER). No e ha o he alues could be
chosen o
α
in o de o u he p omo e/penalize g id o ma ion, bu he h ee selec ed
alues (
−
20%, 0%, 20%) seem o be enough o demons a e he abili y o he p oposed ool
o p oduce di e en kinds o con igu a ions.
The compu a ional expe imen s p esen ed he e we e ca ied ou sol ing he p e i-
ously desc ibed MILP p oblem, using IBM ILOG CPLEX 12.2, un on an In elCo e i7-6700
3.40 GHz
p ocesso (16 Gb RAM). The MILP sol e employed in CPLEX is based on a
B anch and Cu algo i hm. The co esponding OPL code o ou model, as well as comple e
inpu da a (al eady a ailable in Sec ion 4.2) and he de ailed solu ions ob ained in e e y op-
imiza ion p ocess (decision a iables and objec i e unc ion), a e a ailable om he ollow-
ing websi e ( ee public access): h ps://gi ioc.upc.edu/ioc/2022_ma hema ics_equado
(accessed on 6 Ap il 2022). In his way, he alidi y o he solu ions ob ained can be checked
by he eade , and his s udy is comple ely ep oducible. All he execu ions a e pe o med
using a ela i e op imali y gap o 10–6 wi h a 1 h ime limi , which p o ed o be enough
o sol e all he ackled ins ances op imally excep he C-RC sub-p oblem (independen ly
om he alue o
α
). No e ha , o he San a Rosa and Conambo communi ies, op imali y
could be ob ained o he co esponding sub-p oblems hanks o he “di ide-and-conque ”
s a egy p oposed he e, which decomposes communi ies in o sub-p oblems sol ed inde-

Ma hema ics 2022,10, 1226 17 o 24
penden ly and allows o a educ ion o ins ance sizes. This is why a classical solu ion
ool such as CPLEX could be applied he e, despi e he complexi y o he MILP model,
wi hou he need o de elop a new ad-hoc solu ion echnique. In he case o sub-p oblem
C-RC, he emo eness o ce ain demand poin s wi h ega d o he po en ial gene a ion
loca ion allowed he p oblem o be u he di ided. The pa ial solu ions ob ained we e
eused wi hin he comple e model o ob ain good quali y, easible solu ions whose op i-
mali y could be subsequen ly demons a ed. Speci ic commen s ega ding he mic og id
s uc u es ob ained o his ins ance a e p o ided nex .
The esul s a e p esen ed in Table 5, which shows, in addi ion o he size o each
sub-p oblem, he objec i e unc ion alue and he eal cos o he solu ion ob ained (which
may be di e en om he objec i e unc ion acco ding o
α
alue) as well a desc ip ion o
he p oposed con igu a ion. In addi ion, he mic og id opologies p oduced ( o a alue
α= 20%,
such as ha p oposed by he MEER) a e p esen ed in Figu es 7–9 o Su aka, San a
Rosa and Conambo, espec i ely. Please no e ha , in hese h ee igu es, yellow/whi e
do s s and o demand poin s wi h/wi hou indi idual gene a ion sys ems. In addi ion,
yellow squa es ep esen he loca ions used as gene a ion poin s o mic og ids while
g ay-ou lined whi e squa es a e o non-used po en ial gene a ion poin s.
Ma hema ics 2022, 10, x FOR PEER REVIEW 18 o 25
p oblems sol ed independen ly and allows o a educ ion o ins ance sizes. This is why
a classical solu ion ool such as CPLEX could be applied he e, despi e he complexi y o
he MILP model, wi hou he need o de elop a new ad-hoc solu ion echnique. In he case
o sub-p oblem C-RC, he emo eness o ce ain demand poin s wi h ega d o he po en-
ial gene a ion loca ion allowed he p oblem o be u he di ided. The pa ial solu ions
ob ained we e eused wi hin he comple e model o ob ain good quali y, easible solu ions
whose op imali y could be subsequen ly demons a ed. Speci ic commen s ega ding he
mic og id s uc u es ob ained o his ins ance a e p o ided nex .
The esul s a e p esen ed in Table 5, which shows, in addi ion o he size o each sub-
p oblem, he objec i e unc ion alue and he eal cos o he solu ion ob ained (which
may be di e en om he objec i e unc ion acco ding o α alue) as well a desc ip ion o
he p oposed con igu a ion. In addi ion, he mic og id opologies p oduced ( o a alue
α = 20%, such as ha p oposed by he MEER) a e p esen ed in Figu es 7–9 o Su aka,
San a Rosa and Conambo, espec i ely. Please no e ha , in hese h ee igu es, yel-
low/whi e do s s and o demand poin s wi h/wi hou indi idual gene a ion sys ems. In
addi ion, yellow squa es ep esen he loca ions used as gene a ion poin s o mic og ids
while g ay-ou lined whi e squa es a e o non-used po en ial gene a ion poin s.
Figu e 7. Mic og id ob ained o Su aka.
Fi s , o e e y sub-p oblem, di e en con igu a ions a e de e mined depending on
he alue o α (see Table 5), p o ing he e sa ili y o he o mula ion de eloped, which is
capable o adap ing o use ’s p e e ences ega ding mic og id o indi idual sys ems.
Howe e , apa om his global conclusion, di e en beha io s eme ge om a close
analysis. Fo he Su aka and C-RA sub-p oblems, a mic og id in ol ing all demand poin s
is de e mined when α = 0 o 20%, while all use s a e supplied by indi idual sys ems when
α = −20%. A compa able end is obse ed in SR-L and C L (one-mic og id con igu a ion
when α = 0 o 20%), bu , in hese cases, some demand poin s loca ed a away o isola ed
by obs acles a e supplied by indi idual sys ems. In con as , a di e en end is obse ed
in he emaining ins ances. Fo SR-R and C-RD, he minimum cos con igu a ions only
in ol e indi idual supplies when α = 0 o −20%, while mic og ids a e p oposed o α =
20%. These mic og ids may include all use s (SR-R) o exclude use s isola ed by i e me-
ande s (C-RD). Finally, C-RB and C-RC u he demons a e he e sa ili y o he compu-
a ional ool de eloped, which p oposes h ee di e en con igu a ions acco ding o he α
alue. When α = −20%, all use s a e supplied h ough indi idual sys ems; when α = 0, a
combina ion o indi idual supplies and one mic og id is designed. The p e ious mi-
c og id is expanded when α inc eases o 20%, including all use s o C-RB and pa o
hem o C-RC.
Figu e 7. Mic og id ob ained o Su aka.
Fi s , o e e y sub-p oblem, di e en con igu a ions a e de e mined depending on
he alue o
α
(see Table 5), p o ing he e sa ili y o he o mula ion de eloped, which
is capable o adap ing o use ’s p e e ences ega ding mic og id o indi idual sys ems.
Howe e , apa om his global conclusion, di e en beha io s eme ge om a close
analysis. Fo he Su aka and C-RA sub-p oblems, a mic og id in ol ing all demand poin s
is de e mined when
α
= 0 o 20%, while all use s a e supplied by indi idual sys ems when
α=−20%. A compa able end is obse ed in SR-L and C L (one-mic og id con igu a ion
when
α
= 0 o 20%), bu , in hese cases, some demand poin s loca ed a away o isola ed
by obs acles a e supplied by indi idual sys ems. In con as , a di e en end is obse ed
in he emaining ins ances. Fo SR-R and C-RD, he minimum cos con igu a ions only
in ol e indi idual supplies when
α
= 0 o
−
20%, while mic og ids a e p oposed o
α
= 20%. These mic og ids may include all use s (SR-R) o exclude use s isola ed by
i e meande s
(C-RD).
Finally, C-RB and C-RC u he demons a e he e sa ili y o he
compu a ional ool de eloped, which p oposes h ee di e en con igu a ions acco ding
o he
α
alue. When
α
=
−
20%, all use s a e supplied h ough indi idual sys ems; when
α
= 0, a combina ion o indi idual supplies and one mic og id is designed. The p e ious
mic og id is expanded when
α
inc eases o 20%, including all use s o C-RB and pa o
hem o C-RC.
Ma hema ics 2022,10, 1226 18 o 24
Table 5. Solu ions ob ained o he h ee communi ies.
Communi y Sub-P oblem Demand Poin s α(%) Obj. Func.
(USD)
Real Cos
(USD) Con igu a ion
Su aka 12
−20 34,800 34,800 12 indi idual sys ems
0 31,110 31,110 One mic og id (all 12 use s)
20 25,925
San a
Rosa
SR-R 4
−20 11,600 11,600 4 indi idual sys ems
0
20 10,231 12,277 One mic og id (all 4 use s)
SR-L 11
−20 31,900 31,900 11 indi idual sys ems
0 29,848 29,848
One mic og id (9 use s) and
2 indi idual sys ems
20 25,840
Conambo
C-L 9
−20 26,100 26,100 9 indi idual sys ems
0 25,925 25,925
One mic og id (5 use s) and
4 indi idual sys ems
20 23,538
C-RA 10
−20 29,000 29,000 10 indi idual sys ems
0 27,463 27,463 One mic og id (all 10 use s)
20 22,886
C-RB 15
−20 43,500 43,500 15 indi idual sys ems
0 39,353 39,353 One mic og id (13 use s)
and 2 indi idual sys ems
20 32,897 39,477 One mic og id (all 15 use s)
C-RC 20
−20 58,000 58,000 20 indi idual sys ems
0 53,649 53,649 One mic og id (15 use s)
and 5 indi idual sys ems
20 46,789 53,827 One mic og id (16 use s)
and 4 indi idual sys ems
C-RD 6
−20 17,400 17,400 6 indi idual sys ems
0
20 17,252 19,543
One mic og id (4 use s) and
2 indi idual sys ems
Wi h espec o he mic og id opologies shown in Figu es 7–9, hey illus a e beha io s
clea ly in luenced by he condi ions de ined in Sec ion 2.3. Fo ins ance, only one mic og id
is o med in he solu ion ob ained o Su aka, SR-L and C-RA, while ano he op ion would
ha e been o c ea e mo e independen mic og ids (se e al po en ial gene a ion poin s
a e a ailable). Such wo-g id con igu a ions a e disca ded due o he shed equi ed o
equipmen s o age, which is mo e expensi e han he addi ional wi ing necessa y o expand
he mic og id. Besides, ega ding C-RC, he con igu a ion o he mic og id ob ained wi h
α
= 20% (Figu e 9) is es ic ed by he maximum numbe o ou pu connec ions om
limi e boxes, C
max
= 2 (using C
max
= 3 allows cheape con igu a ions). In addi ion, some
connec ions a e limi ed by he ype o wi es a ailable (due o he space limi a ions in he
ai c a anspo ing he equipmen ), which p e en s u he expansion o he mic og id
while espec ing he allowed ol age d op.
In sho , he esul s highligh ha , in ou cases, using
α
> 0 allows mic og ids o be
c ea ed o expanded o eplace he indi idual sys ems ob ained when
α
= 0, which con i ms
he soundness o he incen i e p oposed by he MEER (in he ou o he cases, mic og ids
a e al eady ound o
α
= 0). Besides, he di e ence be ween he “ eal” cos o mic og id
con igu a ions ob ained wi h
α
= 20% and he cos o indi idual sys ems (designed when
α
= 0) is on a e age lowe han 5%. The e o e, he alue p oposed by he MEER (20%)
seems high enough o p omo e he o ma ion o mic og id-based sys ems. I migh e en be
dec eased in o de o appea mo e a ac i e om an economic iewpoin wi hou a ec ing
he inal esul s.
Ma hema ics 2022,10, 1226 19 o 24
Ma hema ics 2022, 10, x FOR PEER REVIEW 20 o 25
SR-R
SR-L
Figu e 8. Mic og id ob ained o San a Rosa.
In sho , he esul s highligh ha , in ou cases, using α > 0 allows mic og ids o be
c ea ed o expanded o eplace he indi idual sys ems ob ained when α = 0, which con-
i ms he soundness o he incen i e p oposed by he MEER (in he ou o he cases, mi-
c og ids a e al eady ound o α = 0). Besides, he di e ence be ween he “ eal” cos o
mic og id con igu a ions ob ained wi h α = 20% and he cos o indi idual sys ems (de-
signed when α = 0) is on a e age lowe han 5%. The e o e, he alue p oposed by he
MEER (20%) seems high enough o p omo e he o ma ion o mic og id-based sys ems. I
migh e en be dec eased in o de o appea mo e a ac i e om an economic iewpoin
wi hou a ec ing he inal esul s.
C-L
o poin s 54, 55, 59, 60
(indi . sys ems)
28
29
27
26
25
61
25m 50m
Mic og id gene a ion poin
Demand poin
Unde g ound connec ion
Figu e 8. Mic og id ob ained o San a Rosa.
The compa ison wi h classical p ocesses o elec i ica ion sys ems such as hose de-
sc ibed in Sec ion 1 u he highligh s he scien i ic con ibu ion o he app oach in oduced
he e and he bene i s ob ained acco dingly. Indeed, he pa icula ea u es included in
he ma hema ical model and de i ed om he condi ioning ac o s p ope o he RAE
allowed he design o speci ic con igu a ions. Fo example, in some ins ances o which
se e al mic og ids may ha e been ins alled, he equi emen o building a shed o gen-
e a ion equipmen led o a single mic og id. In addi ion, as men ioned be o e, hanks
o he e sa ili y o he design ool allowed by he
α
pa ame e , he sys ems p oposed
show mo e o la ge mic og ids han hose ha would ha e been ob ained by gene ic ools.
The e o e, he solu ion s a egy de eloped he e is jus i ied by i s con ibu ion and, in u n,
he new ea u es in oduced he e may be included in s anda d ools in o de o expand
hei applica ion scope.
Ma hema ics 2022,10, 1226 20 o 24
Ma hema ics 2022, 10, x FOR PEER REVIEW 21 o 25
C-L
C-RA
C-RB
C-RC
C-RD
Figu e 9. Mic og id/indi idual supply ob ained o Conambo.
o poin s 54, 55, 59, 60
(indi . sys ems)
28
29
27
26
25
61
25m 50m
Mic og id gene a ion poin
Demand poin
Unde g ound connec ion
15
17 16
18
62
63
19
21
20
22 23
24
50m 100m
Unused po en ial gene a ion poin
Mic og id gene a ion poin
Demand poin
Unde g ound connec ion
64
2
37
36
39
38
41
40
1
12
30
31
34 33 35
32
30m 60m
Mic og id gene a ion poin
Demand poin
Unde g ound connec ion
46
48
65
49
50
51
3
4
14
13
9
47
8
7
5
6
11
10 45
43 44
50m 100m
Mic og id gene a ion poin
Demand poin
Unde g ound connec ion
66
58
42
57
56
o poin s 52, 53
(indi . sys ems)
50m 100m
Mic og id gene a ion poin
Demand poin
Unde g ound connec ion
Figu e 9. Mic og id/indi idual supply ob ained o Conambo.
Ma hema ics 2022,10, 1226 21 o 24
5. Conclusions
Despi e global elec i ica ion a es oughly compa able o hose o de eloped coun-
ies, Ecuado shows eno mous dispa i ies ega ding access o his se ice, pa icula ly
in he u al a eas o i s Amazon basin. As a p oposal o o e come hese inequali ies, his
s udy p esen s a ool o he design o s and-alone elec i ica ion sys ems based on sola
ene gy specially adap ed o he case o he RAE. De eloping his ool equi ed a deep
unde s anding o he conside ed communi y’s way-o -li e, as well as close collabo a ion
wi h na ional ins i u ions and local ac o s in ol ed in he u al elec i ica ion p ocess. This
i s s ep led o he o mula ion o an MILP model o design sys ems adap ed o he speci ic
condi ioning ac o s in he RAE, accoun ing o ei he indi idual sys ems, mic og ids o a
combina ion o bo h, as well as he loca ion, ype and size o he equipmen employed.
This ma hema ical ool was used o de e mine minimal cos designs o h ee commu-
ni ies in he RAE, leading o se e al conclusions. Fi s , a ma ix o po en ial connec ions
was in oduced o allow he possible b eaking-down o la ge p oblems in o sub-p oblems,
whose size is ac able h ough ma hema ical p og amming. Besides, he in oduc ion o
RAE-speci ic ea u es had impo an consequences on he con igu a ions p oposed. In
he i s place, mic og id gene a ion equipmen canno be loca ed a demand poin s, and
he numbe o ou pu connec ions om limi e boxes is bounded, in luencing he inal
mic og id s uc u es. Mo e signi ican ly, he inclusion o p e e ences ega ding mic og id o
indi idual sys ems highligh ed he e sa ili y o he solu ion ool, which p oposes di e en
con igu a ions when a ying he co esponding pa ame e
α
. Nume ical expe imen s p o-
ided gene al guidelines o suppo elec i ica ion policy making, in pa icula ega ding
he adequacy o mic og id p omo ion and he se ing o
α
, which may be dec eased wi h
ega d o MEER’s p oposal (20%) o appea mo e a ac i e. In any case, he ool in oduced
he e can be used o p o ide e icien solu ions independen o he
α
alue es ablished by
policy make s, wi h ob ious bene i s o he RAE’s isola ed communi ies.
Mo e gene ally, his wo k shows ha elec i ica ion o he RAE is echnically easible,
p o ides highe supplies han he benchma k e e ences (1000 Wh/day ins ead o he
In e na ional Ene gy Agency h eshold o 685 kWh/day [
5
]) and is economically com-
pe i i e, wi h cos s anging om 2500 o 3000 USD pe consume . The e o e, he ool
in oduced he e o elec i ica ion sys ems design has a signi ican impac on he li es yle o
he a ge popula ions, imp o ing hei li e quali y h ough access o elec ici y se ices,
while espec ing hei cus oms and social adi ions since he p oposed model accoun s
o hese la e issues. In addi ion, he elec i ica ion sys ems designed he e a e based on
en i onmen ally iendly echnologies based on enewable ene gies, hus ep esen ing a
mo emen owa ds he Sus ainable De elopmen Goals s a ed by he Uni ed Na ions [
1
].
Finally, e en hough he ma hema ical ool p oposed o elec i ica ion sys ems design is
de eloped acco ding o RAE’s ea u es, i is no limi ed o he case s udies ea ed he e.
Ra he , accoun ing o IEA’s p ojec ions ha mic og id-based sys ems using enewables
should accoun o 30% o he connec ions o be ins alled in he nex yea s [
61
], he model
is o mula ed and implemen ed in a gene ic manne o imp o e i s e sa ili y and allow i s
adap a ion o dis inc geog aphic con ex s (in pa icula , u al a eas o La in Ame ica o
A ica) and policy make s’ p io i ies.
Rega ding pe spec i es o u u e wo k, a i s guideline may be a s udy o he obus -
ness o he con igu a ions designed—sxin pa icula , accoun ing o possible a ia ions o
he ene gy and powe demands. On he o he hand, he en i onmen al impac s should also
be included in he e alua ion o such small-scale elec i ica ion p ojec s. Besides, i is wo h
ecalling ha , ypically, di e en ac o s ha play a ole in he design o elec i ica ion
sys ems a e no always known wi h ce ain y. In pa icula , he amoun o PV ene gy
collec ed (which depends on me eo ological ac o s) and use ’s ene gy/powe demands
(which a e o en di icul o es ima e) a e he pa ame e s mos equen ly conside ed as
unce ain in he specialized li e a u e. The e o e, accoun ing o his unce ain y (ei he
h ough s ochas ic p og amming o uzzy se heo y) migh allow he design o mo e obus
sys ems and cons i u es a p omising pe spec i e.

Ma hema ics 2022,10, 1226 22 o 24
Au ho Con ibu ions:
Concep ualiza ion, B.D., L.F.-M.; me hodology, B.D., G.H.; in es iga ion, B.D.,
G.H., A.P.; esou ces, F.G.; alida ion, L.F.-M., F.G., R.P.; p ojec adminis a ion: L.F.-M.; w i ing—
o iginal d a p epa a ion: G.H., A.P.; w i ing— e iewing and edi ing: A.P.; supe ision, R.P. 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 Spanish Minis y o Science and Inno a ion (RTI-2018-
097962-B-I00) and he Cen e o Coope a ion De elopmen (CCD) o he Uni e si a Poli ècnica de
Ca alunya-Ba celonaTech (UPC).
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen :
In o med consen was ob ained om all subjec s in ol ed in
he s udy.
Da a A ailabili y S a emen :
The da a used o suppo he indings o his s udy a e a ailable a
h ps://gi ioc.upc.edu/ioc/2022_ma hema ics_equado (accessed on 6 Ap il 2022).
Acknowledgmen s:
The au ho s a e e y g a e ul o he NGO Enginee ing Wi hou Bo de s (Spain)
and he Ene gy21 p og am o Uni e si y College Dublin (UCD) o all he suppo and assis ance
gi en du ing he de elopmen o his esea ch.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1.
Uni ed Na ions. T ans o ming Ou Wo ld: The 2030 Agenda o Sus ainable De elopmen . 2015. A ailable online: h ps:
//sdgs.un.o g/goals (accessed on 18 Janua y 2022).
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