Ouanes, Nes ine; G andón, Ta iana González; Hei sch, Holge ; Hen ion, René
A icle — Published Ve sion
Op imizing he economic dispa ch o weakly-connec ed
mini-g ids unde unce ain y using join chance
cons ain s
Annals o Ope a ions Resea ch
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Ouanes, Nes ine; G andón, Ta iana González; Hei sch, Holge ; Hen ion, René
(2024) : Op imizing he economic dispa ch o weakly-connec ed mini-g ids unde unce ain y using
join chance cons ain s, Annals o Ope a ions Resea ch, ISSN 1572-9338, Sp inge US, New Yo k,
NY, Vol. 344, Iss. 1, pp. 499-531,
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Annals o Ope a ions Resea ch (2025) 344:499–531
h ps://doi.o g/10.1007/s10479-024-06287-9
ORIGINAL - OR MODELING/CASE STUDY
Op imizing he economic dispa ch o weakly-connec ed
mini-g ids unde unce ain y using join chance cons ain s
Nes ine Ouanes1·Ta iana González G andón2·Holge Hei sch3·René Hen ion3
Recei ed: 28 Feb ua y 2024 / Accep ed: 10 Sep embe 2024 / Published online: 25 Sep embe 2024
© The Au ho (s) 2024
Abs ac
In his pape , we deal wi h a enewable-powe ed mini-g id, connec ed o an un eliable main
g id, in a Join Chance Cons ained (JCC) p og amming se ing. In se e al u al a eas in
A ica wi h low ene gy access a es, g id-connec ed mini-g id sys em ope a o s con end wi h
ou di e en ypes o unce ain ies: o ecas ing e o s o sola powe and load; equency
and ou ages du a ion om he main-g id. These unce ain ies pose new challenges o he
classicalpowe sys em’sope a ion asks.Th eeal e na i es o heJCCp oblema ep esen ed.
In pa icula , we p esen an Indi idual Chance Cons ain (ICC), Expec ed-Value Model
(EVM) and a so called egula model ha igno es ou ages and o ecas ing unce ain ies. The
JCC model has he capabili y o gua an ee a high p obabili y o mee ing he local demand
h oughou an ou age e en by keeping app op ia e ese es o Diesel gene a ion and ba e y
discha ge. In con as , he easie o handle ICC model gua an ees such p obabili y only
indi idually o di e en ime s eps, esul ing in a much less obus dispa ch. The e en
simple EVM ocuses solely on a e age alues o andom a iables. We illus a e he ou
models h ough a compa ison o ou comes a ained om a eal mini-g id in Lake Vic o ia,
Tanzania. The esul sshow he dispa chmodi ica ions o ba e yandDiesel ese eplanning,
wi h he JCC model p o iding he mos obus esul s, albei wi h a small inc ease in cos s.
Keywo ds Join chance cons ain s ·Mini-g id ope a ion ·Un eliable main g id ·S ochas ic
o ecas ing e o s ·Sphe ical adial decomposi ion
BNes ine Ouanes
[email p o ec ed]
Ta iana González G andón
[email p o ec ed]
Holge Hei sch
[email p o ec ed]
René Hen ion
[email p o ec ed]
1Chai o Managemen Science, Humbold Uni e si y Be lin, Spandaue S aße 1, 10178 Be lin,
Ge many
2Depa men o Indus ial Economics and Technology Managemen , No wegian Uni e si y o
Science and Technology, Sen albygg 1, Gløshaugen T ondheim, No way
3Weie s ass Ins i u e Be lin, Moh ens aße 39, 10117 Be lin, Ge many
123
500 Annals o Ope a ions Resea ch (2025) 344:499–531
Index
∈TSe o ime s eps
τSubse o ime-s eps when ou ages could s a
Pa ame e s
TTo al numbe o ime s eps
c, Ma ginal cos o Diesel a ime [$/kWh]
cb, Ma ginal aging cos o ba e y a ime [$/kWh]
cg, Cos o day-ahead g id impo a ime [$/kWh]
pg, P ice o day-ahead g id expo a ime [$/kWh]
cγ, Cos unc ion o ins an aneous g id exchange [$]
pd, P ice o elec ici y sales a ime [$/kWh]
η+A e age e iciency o ba e y cha ge [%]
η−A e age e iciency o ba e y discha ge [%]
σA e age ba e y sel -discha ge a e (in e nal esis ance) [%]
κA e age du a ion o ou ages [h]
max Ra ed capaci y o diesel [kW]
bmax Maximum cha ging / discha ging a e [kW]
SOCmax Maximum s a e o cha ge o he ba e y [kWh]
SOCmin Minimum s a e o cha ge o he ba e y [kWh]
SOC0Ini ial s a e o cha ge o he ba e y [kWh]
s Fo ecas ing o sola p oduc ion a ime [kW]
gmax Maximum a ailable g id capaci y a ime [kW]
d Load o ecas ing a ime [kW]
pP obabili y o success ul islanding [-]
ωP obabili y o an ou age happening du ing one op imiza ion ho izon [-]
Random a iables
δ Load o ecas ing e o a ime [kW]
ξ Fo ecas ing e o o sola powe a ime [kW]
γ Slack g id exchange a ime [kW]
Non-nega i e decision a iables
Diesel powe a [kW]
g+
Elec ici y impo ed om he main g id a ime [kW]
g−
Elec ici y expo ed o he main g id a [kW]
b+
Ba e y cha ge a ime [kW]
b−
Ba e y discha ge a ime [kW]
SOC S a eo cha gea ime [kWh]
b
Rese e o he ba e y discha ge a ime [kW]
Rese e o he diesel a ime [kW]
1 In oduc ion
The Uni ed Na ions’ Sus ainable De elopmen Goal 7 (SDG-7) u gen ly calls o uni e sal
access ocleanene gy, a mission ha aceschallengesdespi e inc eased elec i ica ion e o s.
P ojec ions by he in e na ional ene gy agency es ima e ha by 2030, a ound 650 million
people migh s ill lack access, and 9 ou o 10 will be in Wes , Cen al, and Eas A ica
123
Annals o Ope a ions Resea ch (2025) 344:499–531 501
(IEA, 2020). Elec i ica ion ia cen alized g ids is slow and expensi e, p omp ing in e es
in decen alized mini-g id ene gy sys ems ha o e quicke deploymen and enhanced cos
compe i i eness (An onanzas-To es e al., 2021).
Renewable-powe ed mini-g ids (MGs) as de ined by González G andón e al. (2021)and
Inensus (2014) a e hyb id elec ici y supply sys ems combining wind u bine o pho o ol aic
(PV) gene a ion ( om 10 kW o 10 MW), ene gy s o age sys ems, and (usually) a Diesel
gene a o in o low and medium ol age dis ibu ion ne wo ks. Classi ied by hei ela ionship
wi h he main powe g id, MGs come in wo o ms: islanded mini-g ids which ope a e
independen ly, de ached om he na ional (main) ne wo k and g id-connec ed mini-g ids
which link wi h he main g id and unc ion as bo h backup sys ems o dis ibu ion as well
as s andalone uni s.
This a icle ocuses on he ope a ional in icacies o g id-connec ed mini-g ids, si ua ed
wi hin he u ban and pe i-u ban a eas o Sub-Saha an A ican coun ies lacking in ene gy
access. The pa adox lies in he ac ha , despi e he p esence o cen al-g id in as uc u e, an
ala ming numbe o households, numbe ing in he hund eds o millions, s ill g apple wi h he
challenge o accessing less han ou hou s o elec ici y pe day (Rocky Moun ain Ins i u e,
2018). To coun e his issue, go e nmen s in hese coun ies a e p og essi ely adop ing g id-
connec ed mini-g ids in a eas wi h un eliable main g id supply. Ye , echnical unce ain ies
inhe en o g id-connec ed mini-g ids in his con ex s em om mul iple sou ces: s ochas ic
sola powe and demand o ecas e o s; absolu e unce ain main g id ou age onse imes;
and ou age du a ions subjec ed o s a is ical analysis. As ope a ional decisions a e aken
p io o he obse a ion o he unce ain y, i becomes impe a i e o adop sui able modeling
me hodologies ha can e ec i ely inco po a e all hese di e se o ms o unce ain y. Thus,
ou p ima y esea ch aim ocuses on add essing he ou men ioned unce ain ies h ough
modeling and algo i hmic app oaches, u ilizing Join Chance Cons ain s. Addi ionally, he
s udy aims o assess he e ec i eness o he p oposed scheduling s a egy by applying i o
a case s udy example o a mini-g id in Lake Vic o ia.
In oduced by Cha nes and Coope (1959), chance cons ain s o e an appealing ool
o dealing wi h unce ain y in he cons ain s o an op imiza ion p oblem. A classical and
undamen al in oduc ion o he heo y and nume ical ea men o chance cons ain s is p e-
sen ed by P ékopa (1995). A mode n ea men is p o ided by Shapi o e al. (2014). Since
hei in oduc ion,chance cons ain s ha e become common o economicdispa ch p oblems,
no ably in hyd o ese oi managemen (e.g., Be hold e al., 2022; Loiaciga, 1988; P ékopa
& Szán ai, 1978; anAckooije al.,2014), bu also in powe dispa ch (e.g., Hong e al., 2022;
Peña-O die es e al., 2021). Fo mini-g id o mic o-g id dispa ch, he app oach has p edom-
inan ly in ol ed he use o pu ely-de e minis ic p edic i e models (González G andón e al.,
2021; Pa isio e al., 2014). The e is inc easing in e es , howe e , in applying p obabilis ic
models in o de o ensu e su icien ly sa e sa is ac ion o demand in a highly s ochas ic en i-
onmen , no only wi h espec o enewable ene gy bu also wi h espec o ins abili ies o
he main g id. Zhao e al. (2014)andLiue al.(2017) in oduce he in e es ing concep s o
p obabili y o sel -su iciency and p obabili y o success ul islanding, espec i ely, in o de
o model he sel -su iciency o a mini-g id when isola ed om he main g id by an ou age.
These concep s ely on keeping ese es o Diesel employmen and ba e y discha ge such
ha an ou age o he main g id, o which he mini-g id is connec ed, can be su i ed based
on hese ese es wi h su icien ly high p obabili y. The sho coming o he models by Zhao
e al. (2014)andLiue al.(2017) is ha hey use indi idual (sepa a e o each ime in he
gi en in e al) chance cons ain s. While such a model is com o able o handle since he
chance cons ain can be ans o med wi hou e o in o an explici equi alen , i does no
eally e lec he wish o obus sel -su iciency. Indeed, e en i one may gua an ee ha
123
502 Annals o Ope a ions Resea ch (2025) 344:499–531
sel -su iciency holds ue a each ime indi idually wi h high p obabili y, he p obabili y o
iola ingsel -su iciencya some imemaybehighas well(o : hep obabili y o gua an eeing
sel -su iciency h oughou a gi en pe iod o ime may be small). This is why we will a he
conside join chance cons ain s in his pape which a e, howe e , mo e di icul o deal wi h.
Indeed, hey equi e he conside a ion o p obabili ies and hei sensi i i ies wi h espec o
he decision ec o unde mul i a ia e andom dis ibu ions. Fo nume ical solu ions ela ed
wi h join chance cons ain s, we shall make use o he so-called sphe ical- adial decompo-
si ion o Gaussian andom ec o s which e icien ly applies o Gaussian, Gaussian-like (e.g.
mul i a ia e log-no mal, Gaussian mix u e) o ellip ically symme ic (e.g. mul i a ia e S u-
den ) dis ibu ions and has ound a lo o applica ions bo h in ope a ions esea ch and op imal
con ol unde PDEs wi h andom coe icien s (e.g., Be hold e al., 2022; Fa shba -Shake e
al., 2020; González G andón e al., 2017; Hei sch, 2020). The possibly s iking di e ence
be ween indi idual and join chance cons ain s has been widely s udied (e.g., Van Ackooij
e al., 2010, p. 547); (Be hold e al., 2022,p.34).
In his pape we ocus on he challenges a ising om he model wi h join chance con-
s ain s. The e o e, we keep some o he modeling aspec s simple. In pa icula , we will no
include bina y decisions (simpli ied model o Diesel gene a o ), we will no adequa ely
model ealis ic ba e y ageing by means o di e en ial equa ions and we will keep all deci-
sions o be s a ic wi h espec o he un olding o unce ain y o e ime ( hus igno ing he
gain o in o ma ion based on andom obse a ions p io o decision aking). The inclusion
o all hese aspec s is subjec o cu en and u u e wo k, e.g., by González G andón e al.
(2022).
This a icle makes wo key con ibu ions, namely o he ealm o applied join chance
cons ain p og amming and o he ad ancemen o SDG-7. Mo e p ecisely, ou analysis
in ol es he ollowing s eps:
•We base ou inpu da a on eal powe measu emen da a ob ained om an ope a ing MG
in Tanzania and use o ecas s o hei sola PV gene a ion and he elec ici y demand o
he connec ed households.
•We p opose a model o an economic scheduling s a egy o a MG connec ed o a weak
main g id wi h a speci ied high eliabili y le el. This model u ilizes join chance con-
s ain s, aking in o accoun a ious unce ain ies, including: 1) o ecas ing e o s in
enewable powe gene a ion, 2) o ecas ing e o s in demand p o iles, and unce ain ies
ela ed o 3) equency and 4) du a ion o ou ages.
•We compa e he p oposed JCC model wi h h ee al e na i es. We con as esul s wi h
an Indi idual Chance Cons ain , Expec ed-Value Model and a (mains eam) so-called
egula model ha igno es ou ages and o ecas ing e o s.
2 Mini-g id opology and modelling assump ions
Figu e1illus a es he g id-connec ed mini-g id con igu a ion. This a angemen includes a
sola PV gene a ion uni , one ba e y ene gy-s o age sys em (BESS), one Diesel gense , and
one connec ion o an un eliable main-g id.
The PV and BESS a e in eg a ed in o he sys em ia wo dis inc in e e s linked o he
Al e na ing Cu en (AC) bus. In e ac ions wi h he main g id occu s a he Poin o Common
Coupling (PCC). A he PCC, a ans o me modi ies he ol ages, ansla ing hem om
he lowe le el o he mini-g id o he highe -le el o he p ima y g id ( e e ed o he ea e
as main g id o highe -le el g id). In he examined con ex o his pape , his connec ion o
123
Annals o Ope a ions Resea ch (2025) 344:499–531 503
Fig. 1 Weakly G id-connec ed Mini-g id Topology
he main g id is un eliable, wi h ou ages occu ing equen ly due o ailu es in he main
g id. This could be an o e loading o he main g id, an uncu ailed excess gene a ion o
echnical aspec s due o ailu es o con ol s a egies, physical componen s o ansmission
lines o e hea ing. Las ly, he Diesel gense , ac ing as a supplemen a y componen o b idge
any elec ici y supply gaps, gene a es AC cu en di ec ly, injec ing i in o he AC bus.
Since he MG is balanced h ough an AC bus, bo h ac i e and eac i e powe s will low
o he loads. In his pape , o simplici y pu poses, we assume ha he loads o all connec ed
households main ain a pu ely esis i e na u e. We, he e o e, conduc all ou modelling using
only ac i e powe low igu es, neglec ing all eac i e componen s.
Fu he on, we neglec any empe a u e e ec s on he MG componen s and ope a ion,
especially o he ba e y s o age sys em. The la e is modelled as one la ge synch onized
uni ha can ei he be cha ged o discha ged, hus neglec ing speci ic dynamics be ween he
ba e y packs. Addi ionally, we model cha ging and discha ging losses in he ba e y in e e
as well as he ba e y’s in e nal esis ance as single, a e age e iciency pa ame e s.
In e ms o he lexibili y o dispa ch, we assume ha he Diesel gense is able o un
be ween0 and amaximalcapaci y,wi hou any amping cons ain s o es ic ionson minimal
es ing and unning imes. Mo eo e , we ollow he p emise ha he MG ope a o always
main ains a con inuous uel supply. Addi ionally, we assume ha he sola PV ou pu can be
con olled wi h he help o a Cha ge Con olle (CC). In case o an excess o PV gene a ion
and an ou age p e en ing any expo o he main g id, he PV ou pu can be cu ailed as
enough such ha he o e all local powe balance can be wi hheld.
To model he demand-side, we agg ega eallelec ici yloadsin oone node, hus neglec ing
any line losses ha occu a he local dis ibu ion le el. Finally, we assume ha he MG
ope a o hasaccess o ime se ies o ecas ing o sola powe gene a ion and agg ega edemand
o he o hcoming24o 48-hou imein e al. Fu he mo e, we assume ha , whenan ou age
o hemaing idoccu s, hepowe exchangea hePCCd ops o0, husp e en inganyplanned
impo s o expo s. Ou ages can occu wi h a gi en p obabili y a any a bi a y momen , he
p ecise iming o which emains unce ain. We u he assume ha he e is s a is ical e idence
abou he a e age du a ion o an ou age om he main-g id (Klugman e al., 2021). Table 1
123
504 Annals o Ope a ions Resea ch (2025) 344:499–531
Table 1 Mini-g id unce ain ies
Sou ce o unce ain y Assump ions
Agg ega e demand Access o a o ecas ing ime se ies o he
o hcoming 48h
Sola PV gene a ion Access o a o ecas ing ime se ies o he
o hcoming 48h
Timing o main g id ou ages Unknown
Du a ion o main g id ou ages Access o s a is ical e idence as an
exogenous pa ame e (Klugman e al.,
2021)
summa izes he unce ain ies p esen a he examined MG and assump ions made abou hei
modeling.
3 Ma hema ical models
This sec ion de ines he ma hema ical model used o he abo e desc ibed MG opology. As
men ioned be o e, ou aim is o ind he op imal dispa ch o a weakly-connec ed MG which
allows one o su i e a ailu e o he main g id by means o local ene gy supply (ba e y,
Diesel). In o de o achie e his goal, in addi ion o he planning o a egula dispa ch o
s ongly connec ed MGs, ese es o Diesel and ba e y a e included in o he model. These
can hen be employed in he islanded mode o he MG in o de o mee he local powe
demand. Since dispa ch has o be planned p io o obse ing andomness in demand, sola
adia ion and ou ages, me hods o s ochas ic op imiza ion come in o play. Inspi ed by he
wo k o Zhao e al. (2014)andLiue al.(2017), we ind he applica ion o p obabilis ic o
chance cons ain s o be mos app op ia e he e. They p o ide he possibili y o ind an op imal
dispa ch including ese es o Diesel and ba e ies such ha he local demand in an islanded
mode caused by an ou age is me wi h a gi en p obabili y ( ypically close o one). Howe e ,
as men ioned in he in oduc ion, hese pape s handle p obabili y indi idually in ime. While
his app oach eases d as ically he compu a ional ea men o chance cons ain s, i does no
gua an ee obus demand sa is ac ion h oughou he whole c i ical ime in e al. The e o e,
ou main con ibu ion is de o ed o model he MG dispa ch p oblem wi h espec o join
chance cons ain s. This ype o cons ain s canno di ec ly be con e ed in o explici linea
cons ain s as in he indi idual case, bu equi es o deal wi h p obabili y unc ions and hei
de i a i es unde mul i a ia e dis ibu ions.
This sec ion is o ganized as ollows: a e in oducing he used nomencla u e, we p esen
he objec i e unc ion and he (common) de e minis ic cons ain s o he op imiza ion p ob-
lem. Then, we in oduce ou addi ional join chance cons ain s wi h espec o decisions on
keeping ese es. We shall assume ha an ou age may occu a mos once a day wi h a gi en
p obabili y. I s ini ial ime is supposed o be uni o mly dis ibu ed o e he day. As o he
du a ion o he ou age, we ei he ix i (e.g., as a s a is ical mean om a ailable his o ical
da a) o conside i o be an ex e io pa ame e which can be a ied in di e en compu a ions.
Nex , we mo e on o a model wi h indi idual chance cons ain s as used by Zhao e al. (2014)
and Liu e al. (2017), in o de o illus a e he gain in obus ness by using join chance con-
s ain s. Finally, o he sake o comple eness, we also men ion wo u he simpli ica ions
o he model: i s , he expec ed- alue model in which we eplace all andom pa ame e s by
123
Annals o Ope a ions Resea ch (2025) 344:499–531 505
hei expec ed alues, such ha s a is ical in o ma ion is educed o he i s momen . Sec-
ond, he widely-used egula model, which in addi ion o he p e ious simpli ica ion, also
comple ely igno es he possibili y o ou ages (Bea h e al., 2023; Elegeonye e al., 2023;
González G andón e al., 2021;Kuma ,2021; Kuma and Pahuja, 2021).
3.1 Objec i e unc ions
In his subsec ion, we will in oduce he objec i e unc ion used ac oss he ma hema ical
models s a ing wi h he cases o a s ongly-connec ed and a weakly-connec ed mini-g id.
S ongly-connec ed mini-g id
We i s p esen he objec i e unc ion o a s ongly-connec ed mini-g id, i.e., whe e ou ages
do no occu . In such a mini-g id, i he ope a o aces o ecas ing e o s in sola and load
p edic ions, he impac is negligible om a physical s andpoin . The sys em can be adap ed
by managing expo s and impo s. Howe e , economically, i ’s impo an o no e ha ins an-
aneous g id exchanges come wi h a highe cos compa ed o day-ahead exchanges.
Fo suchap oblem,wewan omaximize hep o i o heMGdispa ch,which isequi alen
o minimizing he ne ope a ional cos s minus ne e enue. The summa ion e m in he
objec i e (1) e lec s he nega i e p o i o he MG o e he nominal ime ho izon T,plus he
du a ion o an ou age κ, as he ou age could s a a he end o he nominal ime ho izon.
0(, b+,b−,g+,g−):=
T+κ
=1
c, · +cb, ·(b+
+b−
)−pd, ·d +cg, g+
−pg, g−
+E[cγ, (γ )].(1)
The aim is o maximize he p o i o he MG dispa ch, i.e., minimize he unc ion 0,whe e
e enues om elec ici y sales a e sub ac ed om he ma ginal cos s o ope a ion. The
ma ginal cos s consis o he uel cos s c, o he Diesel gense , o he ba e y li e ime
dep ecia ion cos s cb, due o he discha ge-cha ge cycling, and o he cos s cg, o impo -
ing om he main g id. Sola powe is assumed o ha e ze o ma ginal cos s and he e o e
doesn’ appea in he objec i e unc ion. Thus a , main enance cos s a e no explici ly con-
side ed. They could be le elized and included in he cos coe icien s o each echnology.
Howe e , o his pape , u he conside a ions abou he main enance app oach a e omi ed.
The e enues a e gene a ed om selling he elec ici y a exogenous p ice o pd, o he con-
nec ed households consuming elec ici y a he MG le el, as well as om expo ing excess
elec ici y a he exogenous p ice o pg, o he main g id. The andom a iable
γ =γ (z,ξ,δ):= d +δ −s −ξ − −b−
+b+
−g+
+g−
,∀ =1,...,T+κ,
wi h z:= (, b−,b+,g−,g+),
(2)
isde inedas hecompensa ingac iones ablishing heloadbalancea e applying hedecisions
zon he employmen o Diesel , ba e y b and g id exchange g on a day-ahead app oach
and obse ing he andom a iables sola ene gy s +ξ and demand d +δ (gi en o ecas
plus andom de ia ion in bo h cases). A posi i e alue o γ ep esen s a lack o ene gy o be
compensa ed by ins an aneous impo om he main g id (a a p ice ha may be subs an ially
highe han o he impo on he day ahead ma ke ). A nega i e alue o γ e e s o an excess
o ene gy which has o be ins an aneously expo ed o he g id, again a ce ain cos s (so ha ,
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506 Annals o Ope a ions Resea ch (2025) 344:499–531
con a y o he day ahead ma ke he e is no ewa d o he expo ). As γ and, hence, i s
associa ed cos s, a e andom, we minimize he expec a ion o hese cos s.
All cos s and p ices a e de ined as ime- a iable pa ame e s, hence he index o each o
hese pa ame e s.
Weakly-connec ed mini-g id
He e, we in oduce he in icacies associa ed wi h a weakly-connec ed MG, i.e. wi h po en ial
ou ages in he main g id and adjus he objec i e unc ion acco dingly. S a ing wi h he ini ial
objec i e unc ion (1), we adap he objec i e unc ion 0 o accoun o he case o possible
g id ou ages. We in oduce he ollowing de ini ion:
1(, b+,b−,g+,g−, , b):=
T+κ
=1
c, · +cb, ·(b+
+b−
)−pd, ·d
+ω
T
T
τ=1T+κ
=1, =τ,...τ+κcg, g+
−pg, g−
+E[cγ, (γ (z,ξ,δ))]+
τ+κ
=τ
c,
+cb, b−
+(1−ω)
T+κ
=1cg, g+
−pg, g−
+E[cγ, (γ (z,ξ,δ))].(3)
The i s e m is de ined in analogy o (1) including again a summa ion o e he Diesel and
ba e ycyclingcos sminus he e enuesgene a edbyselling heelec ici y o local cus ome s
a a gi en p ice. Now, conside ing he possibili y o g id ou ages, we delinea e wo scena ios
o he cos s and e enues associa ed wi h g id exchange: one when an ou age occu s and
ano he when i does no . These scena ios a e a icula ed in he second and hi d e ms o
(3). We make he assump ion ha , a mos , one ou age may occu du ing he designa ed day
ahead, and he p obabili y o i s occu ence is deno ed as ω. In he absence o an ou age du ing
he speci ied day (p obabili y equals 1 −ω), he sole addi ional cos s s em om ading wi h
he main g id ( hi d e m). Con e sely, when an ou age occu s, suppose a ime τ, henin
he in e al [τ,...,τ +κ], ep esen ing he du a ion o he ou age, no ading cos s wi h
he main g id a e incu ed (due o he los connec ion). Howe e , addi ional cos s a ise om
u ilizing ese e capaci ies
, b−
o Diesel gene a ion and ba e y discha ge o mee he
local demand in he islanded mode (second e m).
We assume ha τis uni o mly dis ibu ed, indica ing ha an ou age may ini ia e a any
ime wi h equal p obabili y. Consequen ly, in he second e m o (3), we compu e he a e age
cos s ac oss all po en ial s a ing imes o he ou age. In con as , he du a ion κo he ou age
is ega ded as he s a is ical mean de i ed om his o ical da a o ea ed as an exogenous
pa ame e o he p oblem, capable o assuming mul iple alues.
3.2 De e minis ic cons ain s
The physical cons ain s ep esen well-known o mula ions o mini-g id economic dispa ch
models, consis ing in capaci y cons ain s o he MG componen s, a powe balancing con-
s ain o MG supply and demand, in e - empo al cons ain s o he s o age s a e o cha ge,
as well as conside a ions o con e sion and dis ibu ion losses exp essed h ough a e age
e iciency pa ame e s. These will be discussed in mo e de ail in he ollowing.
Capaci y cons ain s o he Diesel gense : i is assumed ha he Diesel gense can ope a e
be ween 0 and i s maximal capaci y max. The e o e, he sum o he decision a iables o
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Annals o Ope a ions Resea ch (2025) 344:499–531 513
Expec ed- alue cons ain s igno ing o ecas ing e o s
Nex , we u he simpli y he model by main aining he assump ion o a weakly-connec ed
mini-g id, while comple ely a oiding p obabilis ic cons ain s. Ins ead, we equi e ha he
cons ain s o success ulislandingholdona e age.Speci ically, hisexpec ed- alue dispa ch
model does conside ou ages bu only conside s he expec ed alue o o ecas e o s o bo h
sola p oduc ion and o demand in (20). As a esul , he sys em o cons ain s desc ibing
success ul islanding is gi en by:
d −s − −b−
+b+
≤ b
+
,∀ =1,...,T+κ. (24)
Obse e ha (24) ep esen s he same linea inequali y sys em as (23) bu wi h di e en
igh -hand side.
We no e he e ha we choose o de ine he sys em o cons ain s (23)and(24) o =
1,...,T+κso ha we can apply he objec i e unc ion de ined in (3) ac oss all models
ha conside ou ages, hus allowing o a compa abili y be ween he esul s o he simpli ied
models (ICC and expec ed- alue models) and he JCC model.
Regula dispa ch igno ing ou ages and o ecas ing e o s
The inal model we in oduce is he simples and mos basic, as i igno es bo h o ecas ing
e o s and he occu ence o ou ages om he main-g id. We e e o his as he egula model
due o i s widesp ead use in he li e a u e, whe e he ope a ion o economic dispa ch o mini-
g ids o en o e looks unce ain ies (Bea h e al., 2023; Elegeonye e al., 2023; González
G andón e al., 2021;Kuma ,2021; Kuma and Pahuja, 2021). These models ypically adop
a de e minis ic app oach, ocusing solely on he expec ed alue o he o ecas ing ime se ies,
dis ega ding bo h hei andom e o s and he unce ain y o iming o blackou s. The i s
simpli ica ion in his model is he elimina ion o he las e m in he objec i e unc ion 0
p esen ed in (3), which accoun s o he andom ins an aneous exchange wi h he g id. The
second simpli ica ion conce ns he assump ion o a s ongly connec ed mini-g id, whe e
ou ages a e dis ega ded. Consequen ly, he ese e a iables o Diesel and ba e y sys ems,
which a e included in o he models o ensu e he eliabili y o supply du ing blackou s, a e
also excluded om his model.
Unde hese simpli ica ions, we ob ain he ollowing minimiza ion objec i e unc ion:
2(, b+,b−,g+,g−)=
T+κ
=1c, · +cb, ·(b+
+b−
)−pd, ·d +cg, g+
−pg, g−
.(25)
We also adjus he powe balancing cons ain om (9) as ollows:
s + +b−
−b+
+g+
−g−
,=d ∀ =1,...,T+κ. (26)
Fu he mo e, since he egula model doesn’ include any ese e conside a ions, we adjus
he box de e minis ic cons ain s o he Diesel and ba e y discha ge a iables as shown
in (27)-(28), and emo e he cons ain o he po en ial s a e o cha ge p esen ed in (13).
Meanwhile, he cons ain s (7), (8)and(10)-(12) emain unchanged.
Capaci y cons ain s o he Diesel gense
0≤ ≤max,∀ =1,...,T+κ. (27)
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514 Annals o Ope a ions Resea ch (2025) 344:499–531
Capaci y cons ain s o he ba e y discha ge
0≤b−
≤bmax,∀ =1,...,T+κ. (28)
4 Case s udy
Ou case s udy ocuses on an ope a ional mini-g id in Chi ule, an island o Lake Vic o ia,
Tanzania. Cu en ly, he MG emains isola ed, lacking any linkage o he main-g id. Howe e ,
conside ing he esea ch ques ion a hand and an icipa ing a u u e whe e many MGs in sub-
Saha an A ica will be connec ed o weakly-connec ed main g ids, we assume ha he MG,
o he pu poses o his s udy, is connec ed o a main g id cha ac e ized by un eliabili y and
equen ou ages.
In he ollowing, we p esen he physical and mone a y pa ame e s p ese o his case
s udy. Fo he andom pa ame e s used in his case s udy, we e e o Sec .3.3 whe e we
de i e he pa ame e s cha ac e izing he andom componen s o ou model, i.e., he mean and
(co)- a iances o he dis ibu ion unc ions esul ing om he p esen ed o ecas ing models.
Physical pa ame e s
The physical pa ame e s pe inen o his con ex a e de ailed in Table 2, while he p ice and
cos pa ame e s a e p esen ed in Table 3. In conjunc ion wi h he pa ame e s lis ed in hese
ables, i is assumed ha he nominal ime ho izon is 24h, and he ime s ep is se a one hou .
All pa ame e s, excluding hose pe aining o he main g id connec ion, a e de ined based
on da a om he MG ope a o in Tanzania. Fo he pa ame e s ela ed o he g id connec ion,
we use heu is ic and model-in o med alues. Fo ins ance, o he maximum a ailable g id
capaci y gmax, we assume a high alue meaning ha in case o g id a ailabili y, he g id can
always gua an ee he sa is ac ion o he local elec ici y demand. Fo he a e age du a ion
o an ou age κ, we ini ially se i s alue o 3h, based on empi ical alues collec ed om
main-g ids in simila geog aphies (Klugman e al., 2021). The κpa ame e will be a ied,
u he on, o examine i s e ec on he ob ained solu ions. Fo he p obabili y o an ou age
ω, we assume ha ou ages occu on nine days ou o en, i.e., ω=0.9 o magni y he e ec
o ou ages on he daily planning o he dispa ch.
Mone a y pa ame e s
The mone a y pa ame e s, p esen ed in Table 3a e also based on indica ions by he MG
ope a o o he ma ginal cos s o he uel, he elec ici y sale p ice o local cus ome s, and
scala aging cos s. The ma ginal aging cos o he ba e y is he mone a y alue e lec ing he
cos o ba e y aging due o an addi ional powe uni cha ged o discha ged om he ba e y
Xu e al. (2018). Fo he cos o impo ing om he g id and he p ice o expo ing o i ,
we use alues de i ed om expe in e iews wi h weakly-connec ed MGs in sub-Saha an
A ica. The eby, we assume ha p ices and cos s di e be ween nigh and day, e lec ing
ma ke condi ions, i.e., ha impo ing om he g id cos s mo e du ing he nigh han du ing
he day, since a nigh , he main g id is o e all mo e s essed due o he absence o (cheap)
enewable-ene gy gene a ion (e.g., sola ), and ha selling o he g id du ing he day is mo e
aluable due o he highe demand.
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Annals o Ope a ions Resea ch (2025) 344:499–531 515
Table 2 Physical pa ame e s
Pa ame e Uni Value Desc ip ion
Ba e y sys em η+% 95 A e age ba e y cha ging e iciency
η−% 95 A e age ba e y discha ging
E iciency
σ% 0 Ba e y sel -discha ge a e
bmax kW 10 Maximal ba e y cha ging/dis-
Cha ging a e
bcap kWh 100 Maximal ba e y capaci y
SOCmax % 90 Maximal ba e y s a e o cha ge
SOCmin % 20 Minimal ba e y s a e o cha ge
SOC( =0)% 35 Ini ial ba e y s a e o cha ge
Diesel gense max kW 5 Ra ed Diesel gense capaci y
Main g id connec ion gmax kW 100 Maximum a ailable g id capaci y
κ- 3 A e age du a ion o ou ages
ω- 0.9 P obabili y o an ou age occu ing
du ing one day
Sola inpu s kW see Fig.3Fo ecas ing o maximum sola
P oduc ion
Me a-pa ame e s p% 90 P obabili y o Success ul islanding
d kW see Fig.2Fo ecas ed elec ici y load
Table 3 Cos & p ice pa ame e s
Pa ame e Uni Values Desc ip ion
c, e/kWh 0.35 Ma ginal cos o Diesel
cb, e/kWh 0.0055 Ma ginal cos o ba e y
cg, e/kWh 0.55 (nigh ), 0.15 (day) Ma ginal cos o g id impo
cγ, e/kWh 0.85 (nigh ), 0.45 (day) Cos o ins an aneous g id exchange
pg, e/kWh 0.08 (nigh ), 0.13 (day) P ice o g id expo
pd, e/kWh 0.55 P ice o elec ici y sales o cus ome s
5 Resul s
In his sec ion, we p esen he esul s o he op imiza ion p oblem o he egula (pu ely-
de e minis ic), expec ed, ICC and JCC models and compa e hem o in es iga e he e ec s
achie ed due o he di e en o mula ions and ways o handling he unce ain ies. Fo hese
esul s, we use he p e iously p esen ed inpu pa ame e s and ime se ies.
Op imal dispa ch
Fi s , we look in o igu es 4-7showcasing he op imal dispa ch o each componen
( espec i ely-colo ed ba s), in o de o mee he o ecas ed elec ici y demand p o ile (as
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516 Annals o Ope a ions Resea ch (2025) 344:499–531
Fig. 4 Economic dispa ch in he egula model. A ows nea he x-axis show an inc ease o a dec ease in he
p ice/cos o g id expo /impo
a con inuous black line) gi en he o ecas ed sola powe p oduc ion (as a con inuous yellow
line).
To emphasize he p ice signaling ha is implied by he chosen mone a y pa ame e s o
g id exchange (see Table 3), we show using op and down a ows nea he x-axis o he igu es
he changes in g id impo cos (da k blue) and g id expo p ice (ligh blue). A op a ow
indica es an inc ease in he cos pa ame e , while a down a ow indica es a dec ease. In his
sense, he cos o g id impo d ops o i s se lowe le el a 9 a.m. and inc eases again o i s
se highe le el a 6 p.m. each day, while he p ice o g id expo inc eases a 9 a.m. and
d ops a 10 p.m.
The dispa ch o he egula model is shown in Fig.4. The igu e shows ha he demand is
mainly me h ough discha ging he ba e y o he i s wo mo ning hou s and om 10 p.m.
o midnigh . In he ea ly mo ning hou s s a ing om 3 a.m., we no ice an employmen o he
Diesel gense . La e , s a ing om 9 a.m., as he p ices o main g id elec ici y impo ge
cheape (see Table 3), elec ici y is also impo ed om he main g id o a ew hou s. Du ing
he day, he demand is me by using he sola PV gene a ion wi h a peak a ound 3 p.m. The
sola PV powe is i s used o mee he elec ici y demand di ec ly. Any emaining gene a ion
om he sola sys em is ei he sold o he main g id o addi ional e enue gene a ion o used
ocha ge heba e yin p epa a ion o he nigh hou s.This op imaldispa ch leads oexpec ed
p o i s o 84.46 e o he op imiza ion ho izon T+κ, i.e., 27h.
Looking in o he expec ed- alue model, whe e he model igno es o ecas ing e o s bu
ac o in he possibili y o main g id ou ages occu ing wi h a ce ain p obabili y, he esul ing
dispa ch shown in Fig.5di e s om he egula model dispa ch, mainly by using he Diesel
gense mo e s ongly ( h oughou he i s eigh hou s o he day) and using mo e o he
excess sola PV gene a ion o cha ge he ba e y in he middle o he day. Since he inc eased
Diesel employmen adds o he ope a ional cos s and he dec eased expo ing o he main
g id educes he gene a ed p o i s, he o e all expec ed p o i s d ops o 69.06 e o he
op imiza ion ho izon o 27h.
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Annals o Ope a ions Resea ch (2025) 344:499–531 517
Fig. 5 Economic dispa ch o he expec ed- alue model. A ows nea he x-axis show an inc ease o a dec ease
in he p ice/cos o g id expo /impo
The expec ed op imal p o i s d op e en mo e o he ICC model, whe e we use indi idual
p obabilis ic cons ain s o ensu e a ce ain p obabili y o success ul islanding while consid-
e ing bo h main g id ou ages and o ecas ing e o s. Fo a p obabili y o 90%, he op imal
p o i s a e 63.73 e. In he esul ing op imal dispa ch shown in Fig.6, he op imize hedges
agains o ecas ing e o s and main g id ou ages by s a ing o cha ge he ba e y soone (i.e.,
a 1, 2, 5 and 6 a.m.) wi h elec ici y impo ed om he main g id. Addi ionally, mo e o
he excess elec ici y is used o cha ge he ba e y han o expo o he main g id and mo e
Diesel gene a ion occu s du ing he nigh hou s in o de o educe he elec ici y discha ged
om he ba e y keeping i as a ese e. This new dispa ch clea ly leads o highe Diesel
and main g id impo cos s and lowe main g id expo e enues. O e all, compa ed o he
expec ed- alue model, he p o i s wi h he ICC model d op by me ely 5 e, all while ensu ing
a high p obabili y in he o de o 90% indi idually a each ime s ep o success ully mee he
local demand in an islanded mode and compensa e o o ecas ing e o s.
S ill, as explained in ea lie sec ions, he ICC model gua an ees he equi ed le el o islanding
p obabili y only o he single ime s ep when an ou age s a s and doesn’ co e he en i e
du a ion o he ou age. Fo his, we look in o he dispa ch esul s o he JCC model, shown in
Fig.7. The igu e showcases he same e ec s as o he ICC model, only la ge in scale, i.e.,
he dispa ch shows mo e ba e y cha ging h ough main g id impo s du ing he nigh and
excess sola PV powe du ing he day, as well as less ba e y discha ging du ing he nigh
hou s eplaced by Diesel gene a ion.
As a consequence, he expec ed p o i s d op o 59.87 e, ye wi h a gua an ee ha he
minig id su i es islanding and p obable o ecas ing e o s wi h a p obabili y o p=90%
o he whole du a ion κ=3 o he main g id ou age. Fu he in his sec ion, we analyze
he in luence o he pa ame e s pand κon he expec ed p o i s and achie able success ul
islanding p obabili ies.
One concluding ema k ega ding he op imal dispa ch shown in he igu es abo e, is
ha , excep o he egula model which doesn’ include any andom a iables, he planned
dispa ch om he MG componen s do no balance ou o he o ecas ed demand gi en he
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Fig. 6 Economic dispa ch o he ICC model. A ows nea he x-axis show an inc ease o a dec ease in he
p ice/cos o g id expo /impo
Fig. 7 Economic dispa ch o he JCC model. A ows nea he x-axis show an inc ease o a dec ease in he
p ice/cos o g id expo /impo
o ecas ed sola powe gene a ion. This can be explained by he de ini ion o he balancing
cons ain as shown in (9). Wi h his equa ion, we don’ impose he expensi e condi ion ha
o ecas ed demand be me wi h he espec i e dispa ch om he MG componen s. Ins ead,
we use o he powe balance he eal demand and he eal sola powe gene a ion, as well
as he a iable γ as a andom a iable o co e he ins an aneous g id exchange (see (2)and
he linked explana ion in ha sec ion o he pape ). To showcase his e ec in a clea e way,
we p esen , in Fig.8, he ne planned dispa ch om all he componen s in he JCC model in
each ime s ep, he o ecas ed demand and one ealiza ion o he eal demand. In addi ion,
we showcase he ins an aneous dispa ch, composed by one ealiza ion o he sola powe
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Annals o Ope a ions Resea ch (2025) 344:499–531 519
Fig. 8 Planned and ins an aneous dispa ch o MG componen s, and o ecas ed and eal demand in he JCC
model wi h cγ, =0.85e/kWh (nigh ) and 0.45 e/kWh (day)
o ecas ing e o ξand he andom g id exchange γ. The igu e shows clea ly he de ia ions
be ween he planned dispa ch and he o ecas ed demand, which a e balanced ou wi h he
ins an aneous dispa ch, e.g, a ime s eps 5 and 7 as a nega i e de ia ion om he o ecas ed
demand ansla ing in o an ins an aneous need o mo e g id impo , and in mos o he ime
s eps, e.g., a imes 9 o 24 as a posi i e de ia ion, ansla ing in o an ins an aneous need o
mo e g id expo o gene a ion cu ailmen .
An immedia e esul o his modeling logic is ha he amoun o powe ha is balanced
ou using he ins an aneous g id componen γ a ies depending on he p ede e mined cos
o γ , deno ed as cγ, : when he he cos associa ed o he ins an aneous g id exchange cγ,
and modeled in he objec i e unc ion using E[cγ, (γ )]is highe , less powe is balanced
using he expensi e, ins an aneous g id exchange γand he planned dispa ch ma ches he
o ecas ed demand mo e. In Appendix B, we showcase he e ec o a ying cγ, on he
esul ing dispa ch.
Op imal ba e y cycle
Thep esen ed di e encesindispa ch can also beexaminedin e mso op imal ba e ycycling
s a egy, shown in Fig.9-12 as cha ging and discha ging powe s (in di e en shades o g een)
and SOC de elopmen (as a con inuous black line).
Compa ed o he cases o he egula and expec ed- alue models, he chance cons ain
models showcase a s a egy o cha ge he ba e y o e nigh , which is unde s andably mo e
signi ican in hecaseo heJCCmodel.Consis en ly, hechancecons ain smodelsshowcase
less discha ging o he ba e y in he nigh hou s, his being a he lowes o he JCC model.
Looking close a he ange o ba e y cycling, we no e ha o he egula and he expec ed-
alue model, he s a e o cha ge is, as expec ed, cycled o lowe le els han o he ICC and
JCC models. In he o me wo, he s a e o cha ge eaches i s p ese minimum alue o 20
%, while, o he ICC, i only goes sligh ly below 30% and, o he JCC, i emains sligh ly
abo e 30%.
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520 Annals o Ope a ions Resea ch (2025) 344:499–531
Fig. 9 Ba e y cycle o he egula model
Fig. 10 Ba e y cycle o he expec ed- alue model
Op imal ese es
Finally, we ake a look a he ese es planned o he en i e op imiza ion ho izon. S a ing
wi h he ese es in he case o he expec ed- alue model shown in Fig.13, we no e ha
he planned diesel ese es co e almos en i ely he planned g id impo (see Fig5), hus
ensu ing ha he minig id succeeds in i s islanding p o ided ha he e is an ou age du ing
hese hou s ha would hinde he impo o powe .
Fig.14 p esen s he planned op imal ese es o he case o an ICC model. Again he e, we
no e ha he ese es a e planned a each ime s ep o planned g id impo , as shown in Fig.6.
Du ing he peak o sola gene a ion in he hou s 12-19, whe e excess elec ici y is eco ded,
no ese es a e planned.
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Annals o Ope a ions Resea ch (2025) 344:499–531 521
Fig. 11 Ba e y cycle o he ICC model
Fig. 12 Ba e y cycle o he JCC model
Fo he case o a JCC model, Fig.15 shows ha mo e ba e y and Diesel ese es a e
planned in all ime s eps and e en in mo e hou s o he day han in he ICC case, i.e., only in
ime s eps 13-18, no ese es a e p esen .
The planned ese es ansla e in o addi ional possible p o i s ha a e held back, e.g., h ough
a oided g id expo e enues o inc eased Diesel deploymen cos s, hus explaining he g ad-
ual wo sening o he objec i e unc ion alue om he egula o he JCC case (see Table
4). Howe e , in he ollowing, we p o e he inc eased eliabili y o he MG ope a ion ha is
achie ed by he JCC model, jus i ying he small d op in p o i abili y showcased so a .
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522 Annals o Ope a ions Resea ch (2025) 344:499–531
Fig. 13 Rese es o he expec ed- alue dispa ch model
Fig. 14 Rese es o he ICC dispa ch model
Inc eased eliabili y
Wi h he ICC and JCC model, we a e capable o handling many unce ain ies in he MG
ope a ion,as explainedin Sec .2, hus inc easing he eliabili yo mee ing helocal elec ici y
demand. The eby, we handle di e en ypes o unce ain y: on he one hand, he unce ain ies
abou he o ecas ing o he sola PV gene a ion and he elec ici y demand, on he o he hand,
he unce ain ies abou an ou age o he main g id. The unce ain ies abou when and how
long a main g id ou age could occu enla ges he unce ain ies abou he o ecas ing e o s:
p o ided ha he g id is eliable and doesn’ expe ience any ou ages du ing one day, he
main g id could le el up any o ecas ing e o s and ensu e ha he powe balance is always
me . Ne e heless, in he p esence o an un eliable g id, no only a e o ecas ing e o s
123
Annals o Ope a ions Resea ch (2025) 344:499–531 529
Fig. 20 Planned and ins an aneous dispa ch o MG componen s, and o ecas ed and eal demand in he JCC
model wi h cγ, =0.65 e/kWh (nigh ) and 0.25 e/kWh (day)
Fig. 21 Planned and ins an aneous dispa ch o MG componen s, and o ecas ed and eal demand in he JCC
model wi h cγ, =1.35 e/kWh (nigh ) and 0.95 e/kWh (day)
Acknowledgemen s We hank INENSUS GmbH o hei commi men o open science by sha ing eal powe
measu emen da a om one o hei mini-g id subsidia ies in Tanzania. The second au ho is hank ul o he
suppo by he Fonda ion Ma héma ique Jacques Hadama d (FMJH) P og am Gaspa d Monge in op imiza ion
and ope a ions esea ch including suppo o his p og am by Élec ici é De F ance (EDF) No P-2022-0022.
The hi d au ho is hank ul o he suppo by he Deu sche Fo schungsgemeinscha (DFG) in he Collabo a-
i e Resea ch Cen e CRC/T ans egio 154, Ma hema ical Modelling, Simula ion and Op imiza ion Using he
Example o Gas Ne wo ks, P ojec B04. The ou h au ho acknowledges he suppo by he DFG ExC 2046
MATH+: Be lin Ma hema ics Resea ch Cen e unde p ojec AA4-10.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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