Con ol Enginee ing P ac ice 159 (2025) 106289
A ailable online 26 Feb ua y 2025
0967-0661/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/).
Con en s lis s a ailable a ScienceDi ec
Con ol Enginee ing P ac ice
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Ac i e Ne wo k Managemen ia g id- iendly elec omobili y con ol o
cu ailmen minimiza ion✩
Mladen Čičića,∗, Ca los Vi asb, Ca los Canudas-de-Wi c, F ancisco R. Rubiob
aUni e si y o Cali o nia, Be keley, USA
bDep . de Ingenie ía de Sis emas y Au omá ica, Escuela Técnica Supe io de Ingenie ía, Uni e sidad de Se illa, Spain
cUni . G enoble Alpes, CNRS, In ia, G enoble INP, GIPSA-lab, F ance
ARTICLE INFO
Keywo ds:
Mac oscopic EV T a ic
EV cha ging con ol
Elec omobili y
Ac i e Ne wo k Managemen
Cu ailmen minimiza ion
Demand esponse
ABSTRACT
We p opose an in eg a ed powe and anspo a ion sys em con ol amewo k, combining he powe g id
model wi h a mac oscopic elec omobili y model including cha ging s a ions unde V2G ope a ion. In his
amewo k, he elec ical ehicles (EVs) ac as ene gy s o age, bu also as addi ional i ual powe g id links,
anspo ing ene gy om one poin o ano he . This new holis ic app oach is used as a basis o op imal con ol
design seeking o p o ide Ac i e Ne wo k Managemen , in o de o minimize cu ailmen o enewable ene gy
sou ces and loads a a ious po s o he ne wo k, while accoun ing o he s uc u al limi a ion o he g id
and o he cons ain s necessa y o he op imal ope a ion o he EVs. The p oposed con ol scheme is shown
o be able o ou pe o m uncoo dina ed EV cha ging in e ms o o al cu ailmen in a ious s udied scena ios.
Addi ionally, we s udy he case when public cha ging s a ions a e able o incen i ize o disincen i ise EVs
o use hem, by dynamically a ying hei cha ging p ice h oughou he day, and show ha his addi ional
con ol inpu can u he educe cu ailmen in ce ain scena ios.
1. In oduc ion
As e o s owa ds deca boniza ion o all economical sec o s become
a majo p io i y, Elec ic Vehicles (EVs) ha e s a ed o eme ge as one
o he main componen s o sus ainable anspo a ion sys ems wo ld-
wide. Since EVs a e p ojec ed o each a ound 40% o he o al lee
in he EU by 2030 (Conway, Joshi, Leach, Ga cía, & Senecal,2021),
i is becoming clea ha hei in eg a ion wi h he ci y in as uc u e
(cha ging s a ions), and he elec ical powe supply ne wo k (powe
g id) poses ye unsol ed p oblems ha will be c i ical in he coming
yea s (Eu opean Commission & Di ec o a e-Gene al o Ene gy,2019).
I is sel -e iden ha a ue deca boniza ion o he anspo a ion sec o
can only be achie ed i he ene gy sec o is likewise deca bonized
h ough a massi e inc ease o he p e alence o Renewable Ene gy
Sou ces (RES) in he powe mix. These wo ends will ha e a p o ound
impac on he powe g id ope a ions in he coming yea s (Ipakchi &
Albuyeh,2009), necessi a ing majo in as uc u e imp o emen s, bo h
on he ‘‘physical’’ side, upg ading he ansmission and dis ibu ion ne -
wo ks, and on he ‘‘cybe ’’ side, making he g id sma e (Di Sil es e,
Fa uzza, Sanse e ino, & Zizzo,2018).
✩This wo k has ecei ed suppo om Knu and Alice Wallenbe g Founda ion, he Eu opean Resea ch Council (ERC) unde he Eu opean Union’s Ho izon
2020 esea ch and inno a ion p og amme (g an ag eemen N 694209) Scale-F eeBack p ojec , and by p ojec s PID2020-115561RB-C32 g an ed by he Spanish
Minis y o Science and Inno a ion.
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (M. Čičić), [email p o ec ed] (C. Vi as), [email p o ec ed] (C. Canudas-de-Wi ), [email p o ec ed] (F.R. Rubio).
T adi ionally, powe sys em managemen ope a ed unde he i
and o ge doc ine, ocusing on p eemp i e in es men s in ne wo k
componen s (e.g., lines, cables, ans o me s) o p e en conges ion and
ol age issues wi hou equi ing con inuous moni o ing and con ol o
powe lows o ol ages. Howe e , he apid p oli e a ion o enewable
and dis ibu ed gene a ion implies subs an ial ne wo k ein o cemen
cos s (Co nélusse, Vangulick, Gla ic, & E ns ,2015;Wang, Ochoa, &
Ha ison,2009) because ene gy low may e e se, mo ing om he
dis ibu ion ne wo k o he ansmission ne wo k, and in e nal dis ibu-
ion ne wo k lows may de ia e signi ican ly om his o ical pa e ns.
Ac i e Ne wo k Managemen (ANM) s a egies ocus on add essing
conges ion and ol age issues h ough sho - e m decision-making poli-
cies (Gemine, E ns , & Co nélusse,2017), o en eso ing o cu ailing
wind o sola gene a ion o main ain sys em s abili y in quasi eal- ime.
This ield o esea ch is apidly e ol ing, wi h nume ous new concep s
being p oposed, including mic og ids and i ual powe plan s (Mancò,
Tesio, Guelpa, & Ve da,2023;Palizban, Kauhaniemi, & Gue e o,
2014).
Ne e heless, cu ailmen leads o ope a ional and economic in-
e iciencies ha a e pa icula ly was e ul in case o RES gene a ion,
h ps://doi.o g/10.1016/j.conengp ac.2025.106289
Recei ed 27 Feb ua y 2024; Recei ed in e ised o m 13 Decembe 2024; Accep ed 10 Feb ua y 2025
Con ol Enginee ing P ac ice 159 (2025) 106289
2
M. Čičić e al.
gi en hei low ope a ional cos and po en ial o educe o e all powe
sys em emissions. Explo ing ANM schemes capable o le e aging load
lexibili y is hus impe a i e o educing he eliance on gene a ion
cu ailmen when he supply is highe han he demand. On he o he
hand, i he demand is highe han he supply, he in e mi ency o he
RES also means ha i may no be possible o inc ease hei powe
gene a ion. In his case, demand esponse h ough load modula ion is
necessa y, which may lead o loss o com o o p oduc i i y (Ghaz ini
e al.,2019), wi h olling blackou s as an ex eme case. Such ac ion
may also comp omise modula ion capaci ies in he nea u u e, since
lexible loads a e o en equi ed o consume a speci ic amoun o
ene gy o e a ce ain ime pe iod (e.g. hea pumps o EV cha ging)
o keep com o o usabili y s anda ds. This makes i impo an o
he ANM o ake decisions by planning ope a ions o e a su icien ly
long ime ho izon (Gemine, Ka angelos, E ns , & Co nélusse,2013;Gill,
Kocka , & Aul ,2013;Macedo, F anco, Ride , & Rome o,2015), u he
emphasizing he need o good p edic ion models.
Al hough he cons an inc ease o he elec i ica ion o he ans-
po a ion sys ems (elec omobili y) could be seen as a se ious po en ial
s ain on he powe g id, due o la ge cha ging powe demands (A ias,
Kim, & Bae,2017;Fe nandez, San Román, Cossen , Domingo, & F ias,
2010), he massi e adop ion o EVs will no necessa ily hinde he de-
elopmen o u u e elec ic powe sys ems. In ac , he po en ial o use
hei ba e ies o ene gy s o age and con ol hei cha ging o p o ide
lexibili y (abili y o quickly modi y he demand o eac o changing
condi ions) in oduces new possibili ies o p o ide ANM (Hen y &
E ns ,2021) and ancilla y se ices (Le Floch, Ka a, & Mou a,2016;
Wenzel, Neg e e-Pince ic, Oli a es, MacDonald, & Callaway,2017).
F om he powe sys em side, a powe g id ein o ced by hese me h-
ods is able o in eg a e a highe po ion o in e mi en enewable
ene gy sou ces (RES) (B ouwe , Van Den B oek, Seeb eg s, & Faaij,
2014;Dixon, Bukhsh, Edmunds, & Bell,2020;Lund & Kemp on,2008),
making he EVs an impo an componen o powe sys em ope a ions.
As he na u al in e ace be ween he mobili y and powe ne wo ks,
he cha ging s a ions will play an essen ial ole in he elec omobili y
ecosys em. Today’s powe elec onics echnology and new DC g id
opologies, oge he wi h V2G-enabled EVs, allow cha ging s a ions
o ope a e as dynamic load/supplies p o iding ancilla y se ices o
he powe g id in he o m o equency s abiliza ion and conges ion
elie (López, Ma in, Aguado, & de la To e,2013). By con olling he
cha ging p ocess o EVs p esen a cha ging s a ions, hei ope a o s a e
able o p o ide demand esponse (Yao, Lim, & Tsai,2016), apping in o
a new sou ce o e enue in addi ion o helping he powe g id. Addi-
ionally, EV cha ging can be scheduled and coo dina ed o p o ide Vol -
VA con ol (Sabillon-An unez, Melga -Dominguez, F anco, La o a o, &
Ride ,2017), keeping he ol age le els and eac i e powe low wi hin
accep able limi s. E en when EV cha ging a es canno be di ec ly
con olled, dynamic p icing con ol may enable us o shi he cha ging
demand in space (Nie, Wang, & Cheng,2017) and ime (Moghaddam,
Ahmad, Habibi, & Masoum,2019), helping educe peak load. Apa
om ac ing as a load/supply while cha ging/discha ging a a cha ging
s a ion, EVs also e ec i ely anspo ene gy as hey mo e be ween
di e en poin s in he ne wo k (Wa anabe e al.,2023). In his sense,
he EV a ic lows on he oads could also be seen as addi ional i ual
lines in he powe g id, in addi ion o p o iding ene gy s o age (Čičić
& Canudas-de-Wi ,2024).
One o he main po en ial ba ie s o ully exploi he EVs’ po en ial
is he lack o ools and me hods o o ecas ing EV lee s’ lexibili y
in bo h ime and space. This en ails o ecas ing when and whe e EVs
mo e, how hei S a e o Cha ge (SoC) e ol es, and how hey in e ac
wi h he in as uc u e. Though some app oaches based on his o ical
da a do exis (Mo lock, Rolle, Baue , & Sawodny,2019), a model-
based amewo k is p e e able o op imal con ol pu poses. The e o e,
combining elec omobili y models wi h powe g id models will enable
he use o EVs’ lexibili y po en ial o minimize cu ailmen o RES
and loads, and imp o e he use o he exis ing powe ansmission
Table 1
Table o main no a ion. Symbols ela ed o he powe sys em a e
indica ed by ilde. Subsc ip indica o s ypically de e mine he spa ial
coo dina e and supe sc ip he empo al coo dina e.
Symbol Meaning
𝜁Node iden i ie
𝑘Time s ep
𝑆To al powe
𝛥Cu ailed powe
𝛤RES gene a ion
𝛬Load
𝑈Cha ging s a ion powe
𝜌T a ic densi y
𝜀S a e-o -Cha ge
𝜂Dis ibu ion o EVs by SoC a a cha ging s a ion
𝑣T a ic speed
𝑐Cha ging a e
𝜉Vehicle class iden i ie
𝛽Spli ing a io
𝜋Cha ging p ice
𝑟On- and o - amp lows
𝜇EV lows en e ing o exi ing he cha ging s a ion
𝑉Node ol age
𝑆Line powe
𝑢Rela i e cha ging powe con ol
𝛿Rela i e cu ailmen con ol
ne wo k. Cha ging o a la ge popula ion o EVs was modelled and
con olled in Le Floch e al. (2016) o ollow he load e e ence and
p o ide balancing se ices, neglec ing he spa ial componen o EV
mobili y. In Hen y and E ns (2021), RES cu ailmen was minimized
by means o an op imal con ol s a egy, which was also used o lea n a
compu a ionally e icien con ol law based on ein o cemen lea ning.
Ne e heless, in ha wo k s o age and cha ging o EVs was assumed o
be si ua ed a a single poin in he powe ne wo k, whe eas in eali y
bo h he EVs and he cha ging s a ions a e dis ibu ed in ime (in case o
EVs) and space, and connec ed o di e en powe g id nodes. In Zhou,
Zhang, Guo, and Sun (2021), coupled a ic and powe g id dynamics
we e conside ed, bu he a ic lows we e only desc ibed on g aph
le el. Simila ly, L , Wei, Sun, Chen, and Zang (2019) ackles conges ion
o bo h a ic and powe ne wo k, coupled h ough dynamic wi eless
cha ging, bu wi hou explici ly modelling EV a ic dynamics on he
oad. I is clea ha he coupling be ween he elec omobili y and he
powe sys em will need o be conside ed on he ope a ional le el, as
well as when planning o he u u e (Xie, Hu, & Wang,2020).
In his wo k we ackle he p oblem o p o iding ANM h ough
con ol o EV cha ging, aimed a minimizing cu ailmen o bo h RES
gene a ion and loads, while ensu ing sa e powe g id ope a ion. The
main con ibu ions a e in p oposing he gene al con ol amewo k,
applicable o a gene ic and designing p edic ion-based op imal con ol
based on a mac oscopic combined elec omobili y and powe g id
model (Čičić, Vi as, Canudas-de-Wi , Ca los & Rubio,2023) o achie e
he con ol objec i es. By adop ing a de ailed elec omobili y model,
we a e able o iden i y and le e age he complex, long- e m in luence
ha cu en EV cha ging decisions ha e on he u u e s a e o he
sys em, enabling us o go beyond models elying on andom EV a i als.
The p oposed amewo k is applicable o a gene al case o powe and
mobili y ne wo k, and we s udy i s pe o mance on a simple example.
The emainde o he pape is s uc u ed as ollows. Fi s , he con ol
a chi ec u e and p oblem a e ou lined and desc ibed in Sec ion 2. The
model is p esen ed in Sec ion 3, combining a mul i-class elec omobili y
model, desc ibing he EV a ic and cha ging, and a powe g id model.
The elec omobili y laye is in e aced wi h he powe laye h ough
he cha ging s a ions, which ac as p edic able ime- a ying ene gy
s o age, and whose ope a ion we use o con ol he o e all sys em.
In Sec ion 4we i s gi e he gene al op imal con ol o mula ion,
and hen discuss some simpli ica ions ha can be made in he speci ic
simple oad and powe ne wo k ha is conside ed he e. These h ee
Con ol Enginee ing P ac ice 159 (2025) 106289
3
M. Čičić e al.
Fig. 1. Layou o he combined elec omobili y and powe g id model used o
his s udy, oge he wi h he ou line o he con ol sys em. The con olle p o ides
Ac i e Ne wo k Managemen by aking in he in o ma ion abou he powe g id, he
elec omobili y s a e on he oad and a cha ging s a ions, and loads and RES gene a ion
a he nodes, and ou pu s cha ging a es o indi idual EVs a all cha ging s a ions and
cha ging p ice o he public cha ging s a ion.
Sec ions, 2–4desc ibe he me hodology o his wo k, and he main
no a ion used he ein is p esen ed in Table 1. Finally, in Sec ion 5 he
con ol amewo k is es ed ex ensi ely in simula ions, unde di e en
ope a ing scena ios, and he ope a ion o he con ol laws a e analysed
and discussed. The pape is concluded wi h Sec ion 6, whe e we
summa ize he esul s and gi e di ec ions o u u e wo k.
2. Ac i e ne wo k managemen ia elec omobili y con ol
In his wo k we s udy how con olled EV cha ging can be used o
p o ide ANM, using a con ol sys em a chi ec u e ou lined in Fig. 1.
The sys em is comp ised o he elec omobili y laye , consis ing o he
oad ne wo k and EV a ic, and he powe laye , consis ing o he
powe g id, loads, and gene a o s. The inal pa o he sys em a e
he cha ging s a ions, which a e o pa icula impo ance because hey
se e as he in e ace be ween he elec omobili y laye and he powe
g id, and can be used as ac ua o s o imp o e he si ua ion in he wide
powe sys em.
We ocus on h ee con ol objec i es:
1. ensu ing sa e powe g id ope a ions by keeping he line powe s
wi hin hei capaci ies, |
𝑆𝑘|≤
𝑆, and node ol ages wi hin hei
limi s,
𝑉≤
𝑉𝑘≤
𝑉,
2. minimizing o al cu ailmen |
𝛥𝑘
𝜁|o RES gene a ion
𝛤𝑘
𝜁and
loads
𝛬𝑘
𝜁a all nodes 𝜁,
3. and keeping he o e all SoC o all EVs wi hin he sys em 𝜀𝑘
a g
close o some e e ence 𝜀 e ,
a all ime ins an s 𝑘, in o de o dec easing p io i y. The i s wo
con ol objec i es all wi hin he classical pu iew o ANM, while he
hi d one yields adequa e unc ioning o he elec omobili y laye .
In case he EV cha ging demand is uncon ollable and in lexible, he
only way ANM may achie e he p ima y con ol objec i e a each ime
s ep 𝑘is by cu ailing ei he he gene a ion o he load a each node
𝜁whe e such ac ion is needed. We de ine ela i e cu ailmen
𝛿𝑘as a
con ol inpu deno ing he po ion o RES gene a ion ha needs o be
cu ailed o 0<
𝛿𝑘≤1, o he po ion o load ha needs o be cu ailed
o −1 ≤
𝛿𝑘<0. The ac ual cu ailed powe a each po is hus
𝛥𝑘
𝜁=⎧
⎪
⎨
⎪
⎩
−
𝛿𝑘
𝜁
𝛤𝑘
𝜁,
𝛿𝑘
𝜁≥0,
𝛿𝑘
𝜁
𝛬𝑘
𝜁,
𝛿𝑘
𝜁<0.
(1)
Since cu ailmen is was e ul and cos ly, i is o in e es o achie e
he p ima y con ol objec i e while cu ailing as li le o al powe as
possible, as s a ed by he seconda y con ol objec i e.
I he cha ging s a ion dynamics can be con olled, we can use hem
o p o ide lexibili y o ANM and educe, o e en elimina e he need
o cu ailmen o gene a ion and loads. By inc easing he cha ging
s a ion powe 𝑈𝑘
𝜁, we a e able o abso b mo e powe gene a ion a
node 𝜁, hus educing he need o RES cu ailmen . Con e sely, by
dec easing 𝑈𝑘
𝜁, we a e able o educe he need o load cu ailmen .
He e we adop he con en ion ha i 𝑈𝑘
𝜁>0, he cha ging s a ion is a
ne powe consume om he pe spec i e o he g id, and is using he
g id powe o cha ge he EVs. O he wise, i 𝑈𝑘
𝜁<0, he cha ging s a ion
is a ne powe p o ide o he g id, and is using he ene gy s o ed in
he ba e ies o some EVs o p o ide V2G se ices.
On he uppe le el o cha ging s a ion con ol, he powe sys em
ope a o ge s om each cha ging s a ion 𝜁 he ange o powe ha
hey can consume o gene a e, deno ed by 𝑈𝑘
𝜁and 𝑈𝑘
𝜁, espec i ely.
Based on hese limi s, he ope a o can se he no malized powe low
o (o om) each cha ging s a ion 𝜁, deno ed 𝑢𝑘
𝜁∈ [−1,1]. The ac ual
cha ging s a ion powe s 𝑈𝑘
𝜁 ep esen he con ol inpu o he lowe
le el o con ol, and a e gi en by
𝑈𝑘
𝜁=⎧
⎪
⎨
⎪
⎩
𝑢𝑘
𝜁𝑈𝑘
𝜁, 𝑢𝑘
𝜁≥0,
−𝑢𝑘
𝜁𝑈𝑘
𝜁, 𝑢𝑘
𝜁<0,
(2)
he e o e we ha e 𝑈𝑘
𝜁≤𝑈𝑘
𝜁≤𝑈𝑘
𝜁. On he lowe le el o con ol, he
di ec con ol inpu we use in his amewo k a e he cha ging a es
a each cha ging s a ion 𝜁𝑐𝑘
𝑖, se o po en ially di e en le els o EVs
wi h di e en SoC. These cha ging a es a e se in o de o con ol
he cha ging s a ion powe o achie e he e e ence cha ging s a ion
powe communica ed om he uppe le el o con ol. We calcula e he
limi s o achie able cha ging s a ion powe 𝑈𝑘
𝜁∈ [𝑈𝑘
𝜁, 𝑈𝑘
𝜁] o he uppe
le el o con ol based on he limi a ions on he cha ging a es and he
numbe and SoC o EVs a each cha ging s a ion.
Gi en ha he ange o achie able cha ging powe , and he e o e
also he amoun o lexibili y p o ided o ANM, depends di ec ly on he
numbe o EVs p esen a a cha ging s a ion, being able o in luence
when and whe e EVs cha ge can lead o an imp o emen in o e all
con ol pe o mance. In pa icula , in his wo k we assume ha we
a e able o con ol cha ging p ices 𝜋𝑘
𝜁o public cha ging s a ions. In
doing so, we can incen i ize EVs o cha ge a nodes whe e he e is oo
much powe gene a ion, abso bing he excess ene gy and educing RES
cu ailmen , by educing he cha ging p ice o hese public cha ging
s a ions. Con e sely, a cha ging p ice inc ease causes EVs o de e
cha ging, o cha ge elsewhe e.
Finally, we ackle sa is ying he h ee con ol objec i es using an
app oach simila o lexicog aphic op imiza ion (Eh go ,2005). We
eso o cu ailing RES gene a ion o load
𝛥𝑘
𝜁only i i is impossible
o sa is y he p ima y con ol objec i e by con olling cha ging s a ion
powe s 𝑈𝑘
𝜁and cha ging p ice 𝜋𝑘
𝜁. I he p ima y objec i e is sa is ied
wi h minimum cu ailmen , we con ol he cha ging s a ion powe s o
egula e he o e all SoC o he elec omobili y laye . The e e ence
SoC alue is selec ed such ha i keeps he EVs’ ba e ies ope a ing
wi hin an e icien SoC ange, while also ensu ing ha we a e able o
bo h inc ease and dec ease cha ging s a ion powe in o de o minimize
po en ial u u e cu ailmen .
The e olu ion o he sys em s a e is desc ibed by he Coupled
Elec omobili y and Powe G id Model which is p esen ed in Sec ion 3.
The ull cu en s a e is assumed o be known, and he model is used o
calcula e he p edic ion-based con ol ac ions. Since we a e in e es ed
in demons a ing he limi s o wha can be achie ed using such con ol
sys em, we s udy he idealized case whe e he cen alized con ol
au ho i y has ull con ol o e he cha ging a es (wi hin some ange)
o indi idual EVs acco ding o hei SoC, while hey a e a he cha ging
s a ions a each node. Fu he mo e, we assume ha he EVs a e cha ged
Con ol Enginee ing P ac ice 159 (2025) 106289
4
M. Čičić e al.
o discha ged wi hou losses, and ha all EVs a e connec ed o he
cha ging in as uc u e while hey a e pa ked. The EVs a e ee o lea e
cha ging s a ions acco ding o hei commu ing schedules, and he
ex en o in luence ha he con olle has on hei mo emen is limi ed
o po en ially incen i izing o disincen i ising hem o en e he public
cha ging s a ion by changing he cha ging p ice. In doing so, we hope
o be able o a ac mo e low-SoC EVs o he public cha ging s a ion by
se ing a low cha ging p ice (a which he cha ging s a ion sells powe
o he EVs) a imes when he e is an o e p oduc ion o RES, bu also
when he e is a need o ans e ene gy om hese nodes o o he nodes.
Finally, we assume ha all EV use s a e app op ia ely compensa ed o
he se ices ha hei ehicles p o ide o he g id, including o se ing
po en ial ba e y deg ada ion due o V2G discha ging, so ha hey ully
comply wi h he cen ally managed cha ging con ol.
The gene al se up s udied in his wo k is simila o he one used
in Hen y and E ns (2021), abs ac ing he si ua ion whe e people
commu e be ween home and wo k using EVs, po en ially s opping on
he way a a public cha ging s a ion. As shown in Fig. 1, we conside a
powe g id wi h h ee nodes (Home,Public cha ging s a ion, and Wo k,
in u he ex deno ed by h,p, and w, espec i ely) connec ed bo h
by powe lines and oad links. The wo conside ed oad links connec
nodes hand w, wi h a pai o o - and on- amps a he middle, exi ing
owa ds node p. This dis inc ion is adop ed in o de o ga he he
spa io empo al ope a ion o he daily commu es o a la ge majo i y o
EV use s, occasionally making use o public cha ging in as uc u e.
We assume ha he e is in e mi en RES powe gene a ion and EV
cha ging po s a all h ee nodes, and ha he e is some ime- a ying
load a nodes hand w. No e ha while o simplici y he e we conside
he case wi h one h,w, and pnode each, i is s aigh o wa d o ex end
he amewo k o he mo e ealis ic case when mul iple nodes o each
ype a e connec ed by gene ic oad and powe ne wo ks.
3. Combined elec omobili y and powe g id model
In his sec ion we p esen he combined elec omobili y and powe
g id model which is used o ep esen and simula e he o e all sys-
em. We i s in oduce he new mul i-class agg ega ed elec omobili y
model including he cha ging s a ions, hen p esen he g id model,
p opose a model o p o ile pe u ba ions, and inally combine he pa s
in o an in eg a ed model. While he combined model is adap ed o
he speci ic se up s udied in his wo k, i can eadily be adjus ed o
accommoda e o he ypes o oad and powe ne wo ks as well. The
model combines he con ibu ions om Čičić, Vi as, e al. (2023)
and Čičić, Gasnie , and Canudas-de-Wi (2023), and is included he e
in ull o comple eness.
3.1. Elec omobili y model
The dynamics o EV oad a ic a e desc ibed by a disc e e- ime
mul i-class simpli ied Coupled T a ic, Ene gy, and Cha ging (CTEC)
model, consis ing o he mac oscopic dynamics o he EVs on he oads
and a cha ging s a ions, coupled h ough amp lows. This model is
a cell-based disc e iza ion o i s PDE coun e pa (Čičić & Canudas-de
Wi ,2022),
𝜕 𝜌(𝑥, 𝑡)
𝜕 𝑡+𝜕(𝑣(𝑥, 𝑡)𝜌(𝑥, 𝑡))
𝜕 𝑥= 0,(3)
𝜕(𝜌(𝑥, 𝑡)𝜀(𝑥, 𝑡))
𝜕 𝑡+𝜕(𝜌(𝑥, 𝑡)𝑣(𝑥, 𝑡)𝜀)
𝜕 𝑥=𝜌(𝑥, 𝑡)(𝑣(𝑥, 𝑡)),(4)
𝜕 𝜂(𝜀, 𝑡)
𝜕 𝑡+𝜕(𝑐(𝜀, 𝑡)𝜂(𝜀, 𝑡))
𝜕 𝜀= 0,(5)
whe e 𝑥is he posi ion on he oad, 𝑡 he ime, 𝜌(𝑥, 𝑡) he a ic
densi y, 𝑣(𝑥, 𝑡) he a ic speed, 𝜀(𝑥, 𝑡) he mac oscopic SoC, (𝑣) he
ba e y discha ge as a unc ion o EV speed, 𝜂(𝜀, 𝑡) he dis ibu ion
o EVs a a cha ging s a ion acco ding o hei SoC 𝜀, and 𝑐(𝜀, 𝑡) he
cha ging a e o EVs, po en ially di e en o EVs wi h di e en 𝜀.
This model is also ex ended by spli ing he a ic lows in o di e en
classes (e.g., combus ion engine ehicles and EVs) which ha e he
same beha iou while d i ing on he oad, bu may ha e dis inc SoC
dynamics o beha iou a on- and o - amps.
Fig. 2. Spli ing a ios o EVs owa ds he public cha ging s a ion. As 𝜋𝑘
𝐩inc eases,
𝜁(𝜋𝑘
𝜁)dec eases and ewe EVs en e he public cha ging s a ion due o highe p ices,
op ing ins ead o cha ge a home o a wo k.
3.1.1. Agg ega e a ic densi y equa ions
Each oad link 𝑙is spli in o 𝑁𝑙
𝑥cells o leng h 𝐿𝑥,𝐿𝑥≥𝑣 𝑇, whe e
𝑣 is he ee low speed o he a ic, and 𝑇is he disc e iza ion ime
s ep, app op ia ely selec ed o ensu e nume ical s abili y. The agg ega e
mac oscopic s a e o he a ic on each oad link 𝑙is gi en by he a ic
densi y 𝜌𝑘
𝑖in each cell 𝑖a each disc e e ime ins an 𝑘, which is upda ed
acco ding o
𝜌𝑘+1
𝑖=𝜌𝑘
𝑖+𝑇
𝐿𝑥(𝑞𝑘
𝑖−−𝑞𝑘
𝑖+),(6)
𝑞𝑘
𝑖+= min {𝑣 min{𝜌𝑘
𝑖, 𝜌c },
𝜔(𝜌jam − max{𝜌𝑘
𝑖, 𝜌c }) −𝑟𝑘
on,𝑖+1
1 −𝛽𝑘
𝑖},(7)
𝑞𝑘
𝑖−=𝑞𝑘
𝑖−1+(1 −𝛽𝑘
𝑖−1) +𝑟𝑘
on,𝑖,(8)
He e, 𝑞𝑘
𝑖−deno es he a ic low en e ing cell 𝑖a i s ups eam end,
and 𝑞𝑘
𝑖+ he a ic low exi ing cell 𝑖a i s downs eam end. The on-
amp low in o cell 𝑖is deno ed 𝑟𝑘
on,𝑖 and assumed o en e i a i s
ups eam end, and he o - amp low om cell 𝑖is assumed o lea e
i a i s downs eam end, and deno ed 𝑟𝑘
o ,𝑖 =𝛽𝑘
𝑖𝑞𝑘
𝑖+, whe e 𝛽𝑘
𝑖is he
spli ing a io o mains eam a ic low owa ds he o - amp. The
a ic is assumed o ollow a iangula undamen al diag am wi h
c i ical densi y 𝜌c and jam densi y 𝜌jam, yielding conges ion wa e speed
𝜔=𝑣
𝜌c
𝜌jam−𝜌c . No e ha we omi s a ing o which oad link 𝑙we a e
e e ing o be e eadabili y.
3.1.2. Mul i-class a ic densi y equa ions
The agg ega e a ic low is spli in o some numbe o ehicle
classes 𝜉∈𝛯, wi h indi idual a ic densi ies deno ed 𝜉𝜌𝑘
𝑖. Since he e
we assume ha all ehicle classes ha e he same a ic beha iou , he
e olu ion o hei a ic densi ies is gi en by
𝜉𝜌𝑘+1
𝑖=𝜉𝜌𝑘
𝑖+𝑇
𝐿𝑥(𝜉𝑞𝑘
𝑖−−𝜉𝑞𝑘
𝑖+),(9)
𝜉𝑞𝑘
𝑖+=
𝜉𝜌𝑘
𝑖
𝜌𝑘
𝑖
𝑞𝑘
𝑖+,(10)
𝜉𝑞𝑘
𝑖−=𝜉𝑞𝑘
𝑖−1+(1 −𝜉𝛽𝑘
𝑖−1) +𝜉𝑟𝑘
on,𝑖,(11)
whe e he amp lows de ined h ough 𝜉𝛽𝑘
𝑖−1 and 𝜉𝑟𝑘
on,𝑖 a e dis inc o
each class 𝜉, wi h 𝜉𝑟𝑘
o ,𝑖 =𝜉𝛽𝑘
𝑖−1
𝜉𝑞𝑘
𝑖+. The pe -class quan i ies ela e o
he agg ega e quan i ies acco ding o
𝜌𝑘
𝑖=∑
𝜉∈𝛯
𝜉𝜌𝑘
𝑖, 𝑟𝑘
on,𝑖 =∑
𝜉∈𝛯
𝜉𝑟𝑘
on,𝑖, 𝛽𝑘
𝑖=∑
𝜉∈𝛯
𝜉𝜌𝑘
𝑖
𝜌𝑘
𝑖
𝜉𝛽𝑘
𝑖.(12)
In he s udied se up, once he ehicles each he end o he oad link
om h o wo om w o h, all o hem exi he oad and en e cha ging
s a ions wand h, espec i ely, wi h 𝛽𝑘
𝑁𝑥= 1 o bo h oad links. The
Con ol Enginee ing P ac ice 159 (2025) 106289
5
M. Čičić e al.
spli ing a ios owa ds he public cha ging s a ions 𝜉𝛽𝑘
𝑖, whe e 𝑖=𝑖in
𝐩𝐡
o he link h o w, and 𝑖=𝑖in
𝐩𝐰 o w o h, depend on he SoC o he
app oaching ehicles 𝜉𝜀𝑘
𝑖and he EV cha ging p ice 𝜋𝑘
𝐩, and a e de ined
by logis ic unc ion,
𝜉𝛽𝑘
𝑖= 1 −⎛⎜⎜⎝
1 +𝑒−
𝜉𝜀𝑘
𝑖−𝜁(𝜋𝑘
𝜁)
𝛾𝛽⎞⎟⎟⎠
−1
.(13)
One example o such unc ion, pa ame e ized by he cha ging p ice 𝜋𝑘
𝜁,
is exempli ied in Fig. 2. The logis ic unc ion models how EVs a e mo e
likely o en e a cha ging s a ion i hei SoC is low. I s in lec ion poin
is gi en by 𝜁(𝜋𝑘
𝜁), as a unc ion o he cha ging p ice 𝜋𝑘
𝜁, and i s slope
is calib a ed by some pa ame e 𝛾𝛽.
3.1.3. Mul i-class SoC equa ions
The mac oscopic SoC o each class is deno ed 𝜉𝜀𝑘
𝑖, and i s e olu ion
is de ined by
𝜉𝜌𝑘+1
𝑖
𝜉𝜀𝑘+1
𝑖=𝜉𝜌𝑘
𝑖(𝜉𝜀𝑘
𝑖+𝜉𝑑𝑘
𝑖𝑇)+𝑇
𝐿𝑥(𝜉𝜙𝑘
𝑖
−
−𝜉𝜙𝑘
𝑖+),(14)
𝜉𝜙𝑘
𝑖
−
= (𝜉𝑞𝑘
𝑖
−
−𝜉𝑟𝑘
on,𝑖)(𝜉𝜀𝑘
𝑖
−
1+𝜉𝑑𝑘
𝑖
−
1𝑇)+𝜉𝑟𝑘
on,𝑖
𝜉𝜀𝑘
on,𝑖,(15)
𝜉𝜙𝑘
𝑖+=𝜉𝑞𝑘
𝑖+(𝜉𝜀𝑘
𝑖+𝜉𝑑𝑘
𝑖𝑇),(16)
whe e he ba e y discha ge a e 𝜉𝑑𝑘
𝑖depends on cell 𝑖 a ic speed,
𝜉𝑑𝑘
𝑖=𝜉(𝑣𝑘
𝑖), 𝑣𝑘
𝑖=𝑞𝑘
𝑖+
𝜌𝑘
𝑖
,(17)
and is de ined by discha ge unc ions 𝜉(𝑣), which may be di e en o
each class 𝜉.
3.1.4. Cha ging s a ions equa ions
The elec omobili y model is comple ed by he cha ging s a ion
dynamics. The SoC space o each cha ging s a ion 𝜁, is spli in o 𝑁𝜀
bins o leng h 𝐿𝜀,𝑁𝜀𝐿𝜀= 1. Each bin 𝑖= 1,…, 𝑁𝜀co esponds o
a ange o SoC [(𝑖− 1)𝐿𝜀, 𝑖𝐿𝜀], and he s a e o he cha ging s a ion is
gi en by he numbe o EV in each bin 𝜁𝜂𝑘
𝑖.
We allow he a e o cha ging 𝜁𝑐𝑘
𝑖 o each cha ging s a ion 𝜁and
o each SoC le el 𝑖, o a y in ime wi hin some ange |||𝜁𝑐𝑘
𝑖|||≤𝐶,
wi h 𝐿𝜀≥𝐶 𝑇 equi ed o nume ic s abili y. The cha ging s a ion s a e
upda e is hus gi en by
𝜁𝜂𝑘+1
𝑖=𝜁𝜂𝑘
𝑖+𝑇
𝐿𝜀(𝜁𝑐𝑘
𝑖−1
𝜁𝜂𝑘
𝑖−1 −|||𝜁𝑐𝑘
𝑖|||𝜁𝜂𝑘
𝑖−𝜁𝑐𝑘
𝑖+1𝜂𝑘
𝑖+1)…
… +𝑇(𝜁𝜇𝑘
in,𝑖 −𝜁𝜇𝑘
ou ,𝑖),
(18)
whe e 𝜁𝑐𝑘
𝑖= max{0,𝜁𝑐𝑘
𝑖}, and 𝜁𝑐𝑘
𝑖= min{0,𝜁𝑐𝑘
𝑖}, and 𝜁𝜇𝑘
in,𝑖 and 𝜁𝜇𝑘
ou ,𝑖 a e
he lows o ehicles o SoC le el 𝑖en e ing and exi ing he cha ging
s a ion, espec i ely.
The lows be ween he oad and he cha ging s a ion 𝜁a e
𝑟𝑘
on,𝑖ou
𝜁
=
𝑁𝜀
∑
𝑖=1
𝜁𝜇𝑘
ou ,𝑖, 𝑟𝑘
on,𝑖ou
𝜁
𝜀𝑘
on,𝑖ou
𝜁
=
𝑁𝜀
∑
𝑖=1
𝜁𝜇𝑘
ou ,𝑖(𝑖− 1)𝐿𝜀,(19)
𝑟𝑘
o ,𝑖in
𝜁
=
𝑁𝜀
∑
𝑖=1
𝜁𝜇𝑘
in,𝑖, 𝑟𝑘
o ,𝑖in
𝜁
𝜀𝑘
𝑖in
𝜁
=
𝑁𝜀
∑
𝑖=1
𝜁𝜇𝑘
in,𝑖(𝑖− 1)𝐿𝜀,(20)
whe e 𝑖in
𝜁and 𝑖ou
𝜁a e he cells whe e he o - and on- amp connec ing
he oad o he cha ging s a ion 𝜁a e, espec i ely, ensu ing bo h he
ehicles and he ene gy a e conse ed. In his wo k, we assume ha
mul iple classes o ehicles can en e a single cha ging s a ion 𝜁,
𝜁𝜇𝑘
in,𝑖 =∑
𝜉∈𝛯
𝜁 ,𝜉 𝜇𝑘
in,𝑖,(21)
and ha all ehicles exi ing he cha ging s a ion a e o he same class
𝜉ou
𝜁, he e o e 𝜉𝑟𝑘
on,𝑖ou
𝜁
=𝑟𝑘
on,𝑖ou
𝜁
wi h 𝜉=𝜉ou
𝜁, and 𝜉𝑟𝑘
on,𝑖ou
𝜁
= 0 o 𝜉≠𝜉ou
𝜁.
The lows o each class en e ing and exi ing he cha ging s a ions a e
u he de e mined by
𝜁 ,𝜉 𝜇𝑘
in,𝑖 =
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
⎛⎜⎜⎝
𝑖−
𝜉𝜀𝑘
𝑖in
𝜁
𝐿𝜀⎞⎟⎟⎠
𝜉𝑟𝑘
o ,𝑖in
𝜁
, 𝑖− 1≤
𝜉𝜀𝑘
𝑖in
𝜁
𝐿𝜀
< 𝑖, 𝜉∈𝛯in
𝜁,
⎛⎜⎜⎝
𝜉𝜀𝑘
𝑖in
𝜁
𝐿𝜀
−𝑖+ 2⎞⎟⎟⎠
𝜉𝑟𝑘
o ,𝑖in
𝜁
, 𝑖− 2≤
𝜉𝜀𝑘
𝑖in
𝜁
𝐿𝜀
< 𝑖− 1, 𝜉∈𝛯in
𝜁,
0,o he wise,
(22)
𝜁𝜇𝑘
ou ,𝑖 ∈⎡⎢⎢⎣
0,⎛⎜⎜⎝
1
𝑇−|||𝜁𝑐𝑘
𝑖|||
𝐿𝜀⎞⎟⎟⎠
𝜁𝜂𝑘
𝑖+
𝜁𝑐𝑘
𝑖
−
1
𝐿𝜀
𝜁𝜂𝑘
𝑖
−
1−
𝜁𝑐𝑘
𝑖+1
𝐿𝜀
𝜁𝜂𝑘
𝑖+1⎤⎥⎥⎦
,(23)
whe e 𝛯in
𝜁deno es he se o ehicle classes ha en e cha ging s a ion
𝜁, and he exi ing low 𝜁𝜇𝑘
ou ,𝑖 depends on he pa icula beha iou al
logic o each cha ging s a ion. We assume ha each a ic class 𝜉en e s
a mos one cha ging s a ion 𝜁 om each cell 𝑖, bu i may en e mul i-
ple cha ging s a ions in case o - amps leading o hem a e in di e en
cells 𝑖in
𝜁1≠𝑖in
𝜁2.
In pa icula , we de ine he lows o EVs exi ing each cha ging s a-
ions acco ding o hei depa u e demand p o ile 𝜁𝜇𝑘
o ,
𝜁𝜇𝑘
ou ,𝑖 =⎧
⎪
⎨
⎪
⎩
min {𝜁𝜇𝑘
o ,(1
𝑇−|||𝜁𝑐𝑘
𝑖|||
𝐿𝜀)𝜁𝜂𝑘
𝑖}, 𝑖=𝑖𝑘
𝜁,
0, 𝑖≠𝑖𝑘
𝜁.
(24)
Fo 𝜁=𝐡and 𝜁=𝐰,𝜁𝜇𝑘
o is de ined ex e nally h ough a depa u e
demand p o ile 𝜁𝜇𝑘
o ha ep esen lows o EVs s a ing hei commu e
om home o wo k and om wo k o home. Fo 𝜁=𝐩, we assume
ha he EVs s ay a he cha ging s a ion o 1h, so he depa u e de-
mand will depend on he low en e ing he cha ging s a ion in he pas .
A each sampling ime, we andomly selec he SoC o he depa ing
EVs o each cha ging s a ion 𝑖𝑘
𝜁, wi h p obabili ies o each SoC le el
p opo ional o he numbe o ehicles cu en ly a cha ging s a ion 𝜁.
3.2. Powe g id model
He e we desc ibe he dynamics o he al e na ing cu en (AC) powe
g id ha is employed o model he in e ac ion be ween a con en ional
g id and he elec omobili y laye . To a oid con usion ega ding no a-
ion con en ions, and educe abuse o no a ion, we indica e he pa am-
e e s and a iables o he powe g id model by ilde.
Th ee-phase AC powe sys ems a e he mos widely used in elec i-
cal enginee ing oday. They consis o h ee al e na ing cu en s, each
phase shi ed by 120 deg ees, p o iding a con inuous and balanced
low o elec ici y. Phaso desc ip ion simpli ies analysis by ep esen -
ing hese cu en s as o a ing ec o s in complex plane, aiding in cal-
cula ions o ol age, cu en , and powe in balanced ne wo ks. In his
wo k, he powe ansmission segmen depic ed in Fig. 1is assumed o
be accu a ely desc ibed as such h ee-phase balanced ne wo k, hus we
adop i s equi alen single-phase ep esen a ion in he ollowing.
The dis ibu ion g id is usually desc ibed as a g aph equi alen model
whe e nodes ep esen g id buses (e.g. gene a ion o load poin s, sub-
s a ions, e c.) and edges ep esen ansmission lines, powe ans o m-
e s and phase shi e s connec ing he buses. The powe g id is hus
modelled as a di ec ed g aph
𝐷(
,
)whe e
deno es he se o
nodes,
⊆
×
is he se o o ien ed edges, whe e each link
𝑙∈
connec s buses 𝑛 and 𝑚,
𝑙= (𝑛, 𝑚),𝑛, 𝑚 ∈
. Fo powe line modelling,
he con en ional 𝛱model is employed, as depic ed in Fig. 3, whe e
each powe line can be de ined by i e pa ame e s: a se ies esis ance
𝑟
𝑙, a se ies eac ance 𝑥
𝑙, a cha ging suscep ance
𝑏
𝑙, a ap a io magni-
ude 𝜏
𝑙and phase shi
𝜃
𝑙. These magni udes allow o de ine he link
se ies admi ance 𝑦
𝑙, shun admi ance 𝑦sh
𝑙, and he complex ap a io
o ans o me s,
𝑡
𝑙as
𝑦
𝑙=1
𝑟
𝑙+𝑗 𝑥
𝑙
, 𝑦sh
𝑙=𝑗
𝑏
𝑙
2,
𝑡
𝑙=𝜏
𝑙𝑒𝑗
𝜃
𝑙,
𝑙∈
,(25)
Con ol Enginee ing P ac ice 159 (2025) 106289
6
M. Čičić e al.
Fig. 3. 𝛱 ansmission line model.
whe e 𝑗deno es he imagina y uni . See Zimme man, Mu illo-Sánchez,
and Thomas (2011) o a mo e de ailed explana ion o he b anch con-
en ion employed.
3.2.1. Powe ne wo k equa ions
The complex powe lowing om bus 𝑚 o bus 𝑛 is deno ed by
𝑆(𝑚, 𝑛),
and may be exp essed as he p oduc o he complex ol age a bus 𝑚,
𝑉𝑚, and he complex conjuga e o he cu en lowing be ween buses
𝐼(𝑚, 𝑛). Assuming a gene ic 𝛱powe line model wi h asymme ic shun
admi ances 𝑦sh
(𝑚, 𝑛)and 𝑦sh
(𝑛, 𝑚), and applying Ki chho ’s cu en law, he
ne cu en lowing in o nodes A and B (see Fig. 3) equals he ne powe
ou o hem, yielding
𝐼(𝑚, 𝑛)=
𝐼sh
(𝑚, 𝑛)
𝑡∗
(𝑚, 𝑛)
+𝑦sh
(𝑚, 𝑛)
𝑉𝑚
|
𝑡(𝑚, 𝑛)|2,(26)
𝐼(𝑛, 𝑚)= −
𝐼sh
(𝑚, 𝑛)+𝑦sh
(𝑚, 𝑛)
𝑉𝑛.(27)
Applying Ohm’s Law, we ha e
𝐼sh
(𝑚, 𝑛)=𝑦(𝑚, 𝑛)(
𝑉𝑚
𝑡(𝑚, 𝑛)−
𝑉𝑛), and he co e-
sponding complex powe lows a e hus
𝑆(𝑚, 𝑛)=𝑦∗
(𝑚, 𝑛)(|
𝑉𝑚|2
|
𝑡(𝑚, 𝑛)|2−𝑉𝑚𝑉∗
𝑛
𝑡(𝑚, 𝑛))+ (𝑦sh
(𝑚, 𝑛))∗|𝑉𝑚|2
|
𝑡(𝑚, 𝑛)|2,(28)
𝑆(𝑛, 𝑚)=𝑦∗
(𝑛, 𝑚)(|
𝑉𝑛|2−𝑉𝑛𝑉∗
𝑚
𝑡∗
(𝑚, 𝑛))+ (𝑦sh
(𝑛, 𝑚))∗|𝑉𝑛|2.(29)
A con enien o mula ion o he powe low p oblem in he se up a
hand is he Bus Injec ion Model (BIM) ha pa ame e izes he p oblem
in e ms o node ol ages and powe injec ed a nodes. In ou p oblem,
a any gi en bus we can spli he injec ed powe in o ou componen s:
dis ibu ed RES gene a ion
𝛤𝜁𝑛 , loads
𝛬𝜁𝑛 , EV cha ging s a ion powe
𝑈𝜁𝑛 , and cu ailmen o gene a ion o load
𝛥𝜁𝑛 . He e we use he no a ion
𝜁𝑛 o emphasize he connec ion be ween he elec omobili y node 𝜁𝑛
and powe sys em bus 𝑛. Fo hose buses 𝑛 ha ha e no elec omobili y
nodes associa ed o hem, we ake
𝛤𝜁𝑛 =
𝛬𝜁𝑛 =𝑈𝜁𝑛 =
𝛥𝜁𝑛 = 0.
Applying Tellegen’s Theo em, we de ine he ne injec ed powe low
a bus 𝑛 as
𝑆𝑛 =
𝛤𝜁𝑛 +
𝛬𝜁𝑛 +𝑈𝜁𝑛 +
𝛥𝜁𝑛 ,(30)
which can equi alen ly be exp essed in e ms o b anch powe s
𝑆𝑛 =∑
(
𝑘, 𝑛)∈
𝑆(
𝑘, 𝑛)+∑
(𝑛,
𝑘)∈
𝑆(𝑛,
𝑘),(31)
whe e he ne injec ed powe is exp essed as he sum o he incoming
( i s e m) and ou going (second e m) powe lows o and om he
co esponding adjacen buses o bus 𝑛.
F om Eqs. (30) and (31) and using Eqs. (26)–(29) he inal BIM
o mula ion can be ob ained
𝛤𝜁𝑛 +
𝛬𝜁𝑛 +𝑈𝜁𝑛 +
𝛥𝜁𝑛 =⋯
…∑
(
𝑘, 𝑛)∈
(𝑦∗
(𝑛,
𝑘)(|
𝑉𝑛|2
|
𝑡(𝑛,
𝑘)|2−
𝑉𝑛𝑉∗
𝑘
𝑡(𝑛,
𝑘))+ (𝑦sh
(𝑛,
𝑘))∗|𝑉𝑛|2
|
𝑡(𝑛,
𝑘)|2)+⋯
…∑
(𝑛,
𝑘)∈
(𝑦∗
(
𝑘, 𝑛)(|
𝑉𝑛|2−
𝑉𝑛𝑉∗
𝑘
𝑡∗
(
𝑘, 𝑛))+ (𝑦sh
(𝑛,
𝑘))∗|𝑉𝑛|2).
(32)
3.2.2. Powe low solu ion
The s anda d powe low p oblem in ol es sol ing o he se o
ol ages
𝑉𝑚 a buses 𝑚 ∈
and complex line powe lows
𝑆(𝑚, 𝑛)a
b anches (𝑚, 𝑛) ∈
o a ne wo k co esponding o a speci ied pa e n
o powe injec ion a buses a a gi en ime ins an 𝑘. Exp ession (32)
yields complex- alued equa ions ha allows o sol e o ol ages a
buses assuming he injec ed powe a he nodes (po en ially pa ially
cu ailed dis ibu ed RES gene a ion and loads, and EV cha ging s a ion
powe ) a e known. B anch powe lows can hen be de e mined om
exp essions (28)–(29).
This o mula ion yields |
|complex- alued equa ions, ha can be
posed as 2|
|quad a ic eal alued equa ions wi h 2|
| eal unknowns.
By con en ion, a single bus is ypically chosen as a e e ence (slack)
bus o se e he oles o bo h a ol age angle e e ence and a eal
powe slack. The ol age angle a he e e ence bus has a known alue,
bu he eal powe gene a ion a he slack bus is aken as unknown o
a oid o e speci ying he p oblem. Unknowns o he p oblem will hus
be ac i e and eac i e powe s a he slack bus oge he wi h complex
ol age le el a he emaining |
|− 1buses.
These equa ions a e sol ed a e e y ime s ep 𝑘o he p oblem p o-
iding equilib ium powe s and ol ages a he bus le el. The sol e o
choice employed in his wo k is based on a s anda d New on’s me hod
(Tinney & Ha ,1967) using a pola o m and a ull Jacobian upda ed
a each i e a ion. Each New on s ep in ol es compu ing he misma ch,
o ming he Jacobian based on he sensi i i ies o hese misma ches
o changes in and sol ing o an upda ed alue o he unknowns by
ac o izing his Jacobian.
3.3. P o ile pe u ba ion model
Finally, we model he e ec s o unce ain y by in oducing pe u -
ba ions o he a ious p o iles. We assume ha he u u e alues o de-
pa u es om 𝐡and 𝐰cha ging s a ions 𝜁𝜇𝑘
o ,𝜁∈ {𝐡,𝐰}, and RES gen-
e a ion
𝛤𝑘
𝜁and load
𝛬𝑘
𝜁a all nodes, 𝜁∈ {𝐡,𝐩,𝐰}, a e only known ap-
p oxima ely, wi h only hei cu en ac ual alues assumed o be known
exac ly. We deno e he expec ed alues o hese p o iles by
𝑓𝑘, whe e
𝑓𝑘is 𝜁𝜇𝑘
o ,
𝛤𝑘
𝜁, o
𝛬𝑘
𝜁, and de ine hei ac ual alues as
𝑓𝑘=𝑎𝑘
𝑓
𝑓𝑘,(33)
whe e 𝑎𝑘
𝑓is he andom mul iplica i e pe u ba ion o 𝑓a ound i s
p o ile
𝑓a each ime s ep 𝑘.
Each o he s a ed p o iles is pe u bed by i s own ealiza ion o 𝑎𝑘
𝑓,
wi h an excep ion o RES gene a ion p o iles, which a e pe u bed by
he same 𝑎𝑘
𝛤 o all nodes 𝜁. We de ine 𝑎𝑘
𝑓as a weigh ed sum o wo
andom walks, one o wa d in ime and he o he backwa ds in ime,
𝑎𝑘
𝑓= 1 +
2𝜎max
𝑓
√𝑘end + 1(𝑘end −𝑘
𝑘end − 1𝑊𝑘
𝑓++𝑘− 1
𝑘end − 1𝑊𝑘end+1−𝑘
𝑓−),(34)
whe e 𝑊𝑘
𝑓+and 𝑊𝑘
𝑓−a e independen Gaussian andom walks wi h
no mally dis ibu ed s eps,
𝑊1
𝑓±∼(0,1), 𝑊𝑘+1
𝑓±−𝑊𝑘
𝑓±∼(0,1),(35)
Con ol Enginee ing P ac ice 159 (2025) 106289
7
M. Čičić e al.
and 𝜎max
𝑓is a design pa ame e deno ing he maximum s anda d de ia-
ion o 𝑎𝑘
𝑓. This pe u ba ion is de ined sepa a ely o each day,
𝑘= 1,…, 𝑘end, whe e 𝑘= 1and 𝑘=𝑘end co espond o he beginning
and he end o he day, espec i ely. This way, he pe u ba ion has he
highes a iance a he middle o he day, and is ze o a i s beginning
and a i s end.
As an in eg al pa o p edic ion-based con ol, a he cu en ime
𝑘, we a e equi ed o p edic he u u e e olu ion o he ull sys em
s a e un il some p edic ion ho izon 𝑘+𝐻. While he u u e alues o
he elec omobili y s a e on he oad and a he cha ging s a ions can be
p edic ed well using he elec omobili y model, he pe u bed p o iles
a e p edic ed using hei known cu en alue 𝑓𝑘and he expec ed
p o ile
𝑓𝑘+ℎ,ℎ= 1,…, 𝐻,
𝑓𝑘+ℎ|𝑘≈
𝑓𝑘+ℎ
𝑓𝑘𝑓𝑘, ℎ= 1,…, 𝐻 .(36)
3.4. Combined model
We deno e he encapsula ed ull s a e o he mobili y model a ime
𝑘as 𝑀𝑘,
𝑀𝑘=(𝜉𝜌𝑘
𝑙 ,𝑖|𝑙∈ {𝐡𝐰,𝐰𝐡}, 𝑖= {1,…, 𝑁𝑥}, 𝜉∈ {𝐡∕𝐰,𝐩},…
𝜉𝜀𝑘
𝑙 ,𝑖|𝑙∈ {𝐡𝐰,𝐰𝐡}, 𝑖= 1,…, 𝑁𝑥, 𝜉∈ {𝐡∕𝐰,𝐩},…
𝜁𝜂𝑘
𝑖|𝑖∈ {1,…, 𝑁𝜀}, 𝜁∈ {𝐡,𝐰,𝐩𝐡𝐰,𝐩𝐰𝐡}),(37)
cap u ing he a ic densi y 𝜉𝜌𝑘
𝑙 ,𝑖 and SoC 𝜉𝜀𝑘
𝑙 ,𝑖 o all ehicle classes 𝜉
and all cells 𝑖on bo h oad links 𝑙, and he numbe s o cha ging ehicles
𝜁𝜂𝑘
𝑖a each SoC le el 𝑖a each cha ging s a ion 𝜁.
The elec omobili y s a e is upda ed acco ding o (9),(14), and (18),
as de ined in Sec ion 3.1, and we w i e join ly
𝑀𝑘+1 =(𝑀𝑘, 𝑈𝑘, 𝜋𝑘
𝐩),(38)
whe e we deno e by 𝑈𝑘 he collec ion o powe 𝑈𝑘
𝜁alloca ed o all
physical cha ging s a ions, 𝜁∈ {𝐡,𝐰,𝐩}, ha indi ec ly de e mine he
e olu ion o 𝜁𝜂𝑘
𝑖 h ough (18) by in luencing he cha ging a es, and 𝜋𝑘
𝐩
is he public cha ging s a ion cha ging p ice.
Powe g id equa ions a disc e e ime s ep 𝑘a e gi en by a s a ic
mapping, and can be w i en compac ly as
[
𝑉𝑘
𝑆𝑘]=
(
𝛤𝑘,
𝛬𝑘, 𝑈𝑘,
𝛥𝑘),(39)
yielding bus ol ages
𝑉𝑘and line powe s
𝑆𝑘a ime 𝑘, as a unc ion
o he EV powe injec ed o e ie ed a cha ging s a ions 𝑈𝑘, loads
𝛬𝑘, dis ibu ed RES gene a ion
𝛤𝑘, and cu ailmen
𝛥𝑘, o each po
𝜁∈ {𝐡,𝐰,𝐩}. These need o be kep wi hin sa e ope a ing limi s o en-
su e p ope unc ioning o he powe g id.
4. Con ol design
Wi h he Coupled Elec omobili y and Powe G id Model in oduced,
we a e now able o o mula e con ol laws ha ollow he con ol a chi-
ec u e and achie e he con ol objec i es ou lined in Sec ion 2. As dis-
cussed, he con ol can be spli in o wo le els: he uppe le el, which is
asked wi h achie ing he h ee s a ed con ol objec i es by con olling
he e e ence cha ging s a ion powe s and he public cha ging p ice,
and he lowe le el, which is asked wi h ensu ing ha he ac ual powe
o he cha ging s a ion ollows his e e ence by con olling he cha g-
ing a es. We i s desc ibe he lowe le el con ol, hen o mula e he
gene al op imal con ol p oblem, and inally, discuss he app oxima-
ions app op ia e o he speci ic s udied scena io ha a e employed o
make he p oblem ac able.
4.1. Cha ging s a ion powe con ol
Gi en a dis ibu ion o EVs a a cha ging s a ion 𝜁a ime 𝑘acco d-
ing o hei SoC, 𝜁𝜂𝑘
𝑖, he cu en o al powe o each cha ging s a ion
𝜁is gi en by
𝑈𝑘
𝜁=
𝑁𝜀
∑
𝑖=1
𝜁𝑐𝑘
𝑖
𝜁𝜂𝑘
𝑖𝐵 ,(40)
whe e 𝐵is he a e age EV ba e y capaci y. Since he lowe le el o
con ol has no in luence o e EV a i als and depa u es, i may only
achie e o al powe e e ence acking by se ing he cha ging a es 𝜁𝑐𝑘
𝑖
o EVs a di e en SoC le els 𝑖. Fo each cha ging s a ion we de ine
he maximum cha ging a e 𝐶𝜁≥0and maximum discha ging a e
𝐶𝜁≤0pe EV, yielding 𝐶𝜁≤𝜁𝑐𝑘
𝑖≤𝐶𝜁, espec ing he nume ic s abili y
equi emen s |𝐶𝜁|≤𝐶and |𝐶𝜁|≤𝐶.
I can be seen ha he choice o 𝜁𝑐𝑘
𝑖 ha makes 𝑈𝑘
𝜁 ollow some
e e ence is no unique. Ins ead, we may adop a cha ging a e scheme
ha also achie es some o he desi able ou comes. In o de o ensu e
ha he EVs ha e an adequa e SoC o con inue commu ing, as well
as o bes u ilize he powe a ailable o he cha ging s a ion, we
employ a hie a chical cha ging scheme, whe e ehicles wi h lowe SoC
ha e a highe cha ging p io i y. We spli he ehicles in o wo g oups
acco ding o hei SoC: low-SoC EVs, wi h 0≤𝜀 < 𝜀bnd, and high SoC
EVs, wi h 𝜀bnd < 𝜀≤1. In case no enough powe is alloca ed o cha ge
he low-SoC EVs, o i he uppe le el o con ol eques s powe o be
p o ided o he g id, we u ilize he ene gy s o ed in ba e ies o EVs
wi h highe SoC o sa is y he powe demands. Gi en he bounda y
be ween hese g oups o EVs 𝜀bnd, we de ine 𝑖bnd as he highes 𝑖 o
which 𝜀𝑖< 𝜀bnd,𝑖∈ {1,…, 𝑁𝜀}. The numbe o EVs cu en ly in each
g oup is hen gi en by
𝜁𝜂𝑘
lo =
𝑖bnd
∑
𝑖=1
𝜁𝜂𝑘
𝑖,𝜁𝜂𝑘
hi =
𝑁𝜀
∑
𝑖=𝑖bnd+1
𝜁𝜂𝑘
𝑖.(41)
We assign a di e en cha ging a e o each g oup o EVs, deno ed 𝜁𝑐𝑘
low,
and 𝜁𝑐𝑘
high, and gi en by
𝜁𝑐𝑘
𝑖=⎧
⎪
⎨
⎪
⎩
𝜁𝑐𝑘
lo, 𝑖= 1,…, 𝑖bnd,
𝜁𝑐𝑘
hi, 𝑖=𝑖bnd + 1,…, 𝑁𝜀,
(42)
and he cha ging s a ion powe simpli ies o
𝑈𝑘
𝜁=(𝜁𝑐𝑘
lo
𝜁𝜂𝑘
lo +𝜁𝑐𝑘
hi
𝜁𝜂𝑘
hi)𝐵 .(43)
While he cha ging a es o high-SoC EVs may ake any alue wi hin
he allowed ange, 𝐶𝜁≤𝜁𝑐𝑘
hi ≤𝐶𝜁, we es ic he cha ging a es o low-
SoC EVs o be nonnega i e, 0≤𝜁𝑐𝑘
lo ≤𝐶𝜁, in o de o keep hei SoC
high enough o con inue commu ing. Gi en he cu en p o ided o
demanded powe 𝑈𝑘
𝜁, he cha ging a es o low-SoC and high-SoC EVs
a e se o
𝜁𝑐𝑘
lo = max {0,min {𝑈𝑘
𝜁−𝜁𝜂𝑘
hi𝐶𝜁
𝜁𝜂𝑘
lo
, 𝐶𝜁}},(44)
𝜁𝑐𝑘
hi =⎧
⎪
⎨
⎪
⎩
𝐶𝜁,𝜁𝜂𝑘
lo𝐶𝜁> 𝑈𝑘
𝜁−𝜁𝜂𝑘
hi𝐶𝜁,
min {𝑈𝑘
𝜁−𝜁𝜂𝑘
lo𝐶𝜁
𝜁𝜂𝑘
hi
, 𝐶𝜁},𝜁𝜂𝑘
lo𝐶𝜁≤𝑈𝑘
𝜁−𝜁𝜂𝑘
hi𝐶𝜁,
(45)
espec i ely. This cha ging scheme esul s in a ange o cha ging s a ion
𝜁powe consump ion o p oduc ion [𝑈𝑘
𝜁, 𝑈𝑘
𝜁],
𝑈𝑘
𝜁=(𝜁𝜂𝑘
lo +𝜁𝜂𝑘
hi)𝐶𝜁𝐵 ,(46)
𝑈𝑘
𝜁=𝜁𝜂𝑘
hi𝐶𝜁𝐵 ,(47)
which a e communica ed o he uppe le el o con ol.
Con ol Enginee ing P ac ice 159 (2025) 106289
8
M. Čičić e al.
4.2. Op imal con ol o mula ion
We now o mula e ANM aking elec omobili y in o accoun as a
eceding ho izon op imal con ol p oblem
minimize
𝑢𝜁,𝜋𝐩,
𝛿𝜁
𝐽(𝑘, 𝑢𝜁, 𝜋𝐩,
𝛿𝜁)
subjec o Mobili y and g id dynamics (38)–(39),
|
𝑆𝑘+ℎ
𝑙|≤𝑆
𝑙,
𝑙∈
, ℎ= 0,…, 𝐻 ,(48)
𝑉𝑖≤|
𝑉𝑘+ℎ
𝑖|≤
𝑉𝑖, 𝑖∈
−
slack, ℎ= 0,…, 𝐻 ,(49)
− 1≤𝑢𝜁≤1, 𝜁∈ {𝐡,𝐩,𝐰}(50)
𝜋≤𝜋𝐩≤𝜋 ,(51)
− 1≤
𝛿𝜁≤1, 𝜁∈ {𝐡,𝐩,𝐰}(52)
He e, he decision a iables a e ela i e cha ging powe s 𝑢𝜁, public
cha ging p ices 𝜋𝐩, and ela i e cu ailmen s
𝛿𝜁, all gi en as s acked
ec o s o cu en and u u e con ol inpu s, o e an 𝐻-s ep ho izon,
𝑢𝜁=⎡⎢⎢⎢⎢⎢⎣
𝑢𝑘
𝜁
𝑢𝑘+1
𝜁
⋮
𝑢𝑘+𝐻
𝜁
⎤⎥⎥⎥⎥⎥⎦
, 𝜋𝐩=⎡⎢⎢⎢⎢⎢⎣
𝜋𝑘
𝐩
𝜋𝑘+1
𝐩
⋮
𝜋𝑘+𝐻
𝐩
⎤⎥⎥⎥⎥⎥⎦
, 𝛿𝜁=⎡⎢⎢⎢⎢⎢⎣
𝛿𝑘
𝜁
𝛿𝑘+1
𝜁
⋮
𝛿𝑘+𝐻
𝜁
⎤⎥⎥⎥⎥⎥⎦
.(53)
Bo h he ela i e cha ging powe 𝑢𝑘
𝜁and ela i e cu ailmen 𝛿𝑘
𝜁a e
no malized con ol ac ions, allowing o accommoda e he cons ain s
on he ime- a ying mobili y ese e powe s, loads, and RES gene -
a ion in a mo e con enien o m. F om he side o he powe g id,
he op imiza ion p oblem is subjec o cons ain s on he maximum
appa en powe |
𝑆𝑖𝑗 |(48), and ol age |
𝑉𝑖|de ia ion a buses (49). I
is s aigh o wa d o e o mula e his p oblem o a mo e gene al se up
wi h mul iple 𝐡,𝐩, and 𝐰nodes, connec ed by a mo e elabo a e oad
ne wo k and powe g id.
The op imiza ion p oblem is a icula ed as cons ained minimiza-
ion o he cos unc ion
𝐽(𝑘, 𝑢𝜁, 𝜋𝐩, 𝛿𝜁) =
𝐻
∑
ℎ=0(∑
𝜁∈{𝐡,𝐩,𝐰}(𝑤ℎ
𝛿|
𝛥𝑘+ℎ
𝜁|)+𝑤ℎ
𝜀(𝜀𝑘+ℎ
a g −𝜀 e )2).(54)
The i s pa o he cos unc ion co esponds o o al cu ailmen , and
is minimized o 𝛿𝑘+ℎ= 0, in case his can be achie ed wi hou iola ing
cons ain s (48)–(51). The second pa co esponds o de ia ion o he
a e age o e all SoC 𝜀𝑘
a g om i s e e ence alue 𝜀 e . These wo pa s
a e weighed by 𝑤ℎ
𝛿and 𝑤ℎ
𝜀, including an exponen ial o ge ing ac o ,
and we ha e 𝑤ℎ
𝛿≫ 𝑤ℎ
𝜀, i.e., educing cu ailmen is p io i ized o e
egula ing he a e age SoC, as ou lined in Sec ion 2.
While i is able o cap u e a wide a ie y o scena ios, his op i-
miza ion p oblem is e y di icul o sol e in i s exac o m, due o
he s ongly nonlinea na u e and high o de o he unde lying model,
as well as he la ge numbe o decision a iables in case he con ol
ho izon is aken o be app op ia ely long. The e o e, in o de o make
he p oblem ac able, enabling us o implemen he p oposed ANM
con ol, we need o adop some simpli ica ions and app oxima ions, as
well as o decompose he p oblem in o subp oblems, all o which will
be discussed in he emainde o his sec ion.
4.3. App oxima e o mula ion
Fi s , in o de o ease he nume ic bu den, we sol e he op imiza-
ion p oblem a a ime scale di e en han ha o he elec omobili y
model. We ake he op imiza ion sampling pe iod
𝑇as a mul iple o
elec omobili y sampling pe iods 𝑇, and deno e he op imiza ion ime
s ep
𝑘. The esul ing ela i e cha ging powe con ol 𝑢
𝑘
𝜁is hen applied
o he elec omobili y laye o e a numbe o ime s eps,
𝑢𝑘
𝜁=𝑢⌊𝑘𝑇
𝑇⌋+1
𝜁,(55)
whe e ⌊⋅⌋deno es ounding down. The cha ging p ice 𝜋𝐩is upda ed a
a lowe a e s ill, wi h ime s ep
𝑇𝜋 aken as a mul iple o
𝑇, in o de o
u he educe he compu a ional bu den, bu also o imp o e ealism
by allowing he EV d i e s mo e ime o eac o he change.
We i s s udy minimizing cu ailmen a he cu en ime s ep
𝑘,
which can only be a ec ed by he cu en cha ging con ol 𝑢
𝑘
𝜁. In ou
case, due o he simple a chi ec u e o he g id, i is enough o only
conside he powe lines di ec ly connec ed o 𝐡,𝐩, and 𝐰. Fo each
po , cons ain (48) simpli ies o
|
𝛤
𝑘
𝜁+
𝛬
𝑘
𝜁+𝑈𝛿|≤𝑆𝜁,(56)
whe e 𝑈𝛿is some pu ely eal, posi i e o nega i e, cha ging powe .
In his case, i is easy o ind he ange o 𝑈𝛿 o which he e is no
cu ailmen , 𝑈𝛿≤𝑈𝛿≤𝑈𝛿, assuming |Im{
𝛤
𝑘
𝜁+
𝛬
𝑘
𝜁}|< 𝑆𝜁. The e o e, i
[𝑈𝛿, 𝑈𝛿] ∩ [𝑈
𝑘
𝜁, 𝑈
𝑘
𝜁]≠∅, whe e 𝑈
𝑘
𝜁and 𝑈
𝑘
𝜁a e he cu en minimum and
maximum cha ging powe a 𝜁, espec i ely, i is possible o achie e
ze o cu ailmen i 𝑢
𝑘
𝜁is es ic ed o
max{𝑈
𝑘
𝛿, 𝑈
𝑘
𝜁}
𝑈
𝑘
𝜁
=𝑢
𝑘
𝜁≤𝑢
𝑘
𝜁≤𝑢
𝑘
𝜁=
min{𝑈
𝑘
𝛿, 𝑈
𝑘
𝜁}
𝑈
𝑘
𝜁
,(57)
possibly igh ening cons ain (50).
O he wise, i 𝑈
𝑘
𝛿> 𝑈
𝑘
𝜁, we eplace cons ain (50) by equali y con-
s ain 𝑢
𝑘
𝜁= 1, RES gene a ion will need o be cu ailed, since he e a e
no enough EVs ha can abso b i , and
𝛿
𝑘
𝜁can be ound by sol ing
|
𝛿
𝑘
𝜁
𝛤
𝑘
𝜁+
𝛬
𝑘
𝜁+𝑈
𝑘
𝜁|≤𝑆𝜁,0<
𝛿
𝑘
𝜁≤1.(58)
Con e sely, i 𝑈
𝑘
𝛿< 𝑈
𝑘
𝜁, we eplace (50) by 𝑢
𝑘
𝜁= −1, load will need o
be cu ailed, since he e a e no enough EVs ha can e u n powe o
he g id, and
𝛿
𝑘
𝜁can be ound by sol ing
|
𝛤
𝑘
𝜁−
𝛿
𝑘
𝜁
𝛬
𝑘
𝜁+𝑈
𝑘
𝜁|≤𝑆𝜁,−1 ≤
𝛿
𝑘
𝜁<0.(59)
We may use he cu en ela i e cha ging powe s 𝑢
𝑘
𝜁a hose po s
whe e he e is no cu ailmen a he cu en s ep o egula e he a e age
SoC a he nex ime s ep 𝜀
𝑘+1
a g o he e e ence alue 𝜀 e , igno ing he
SoC a he subsequen s eps by se ing 𝑤ℎ
𝜀= 0 o ℎ >1. We o mula e
a simple myopic p opo ional con olle ha achie es his,
𝑢
𝑘
𝜁= max {𝑢
𝑘
𝜁,min {𝑢
𝑘
𝜁,−𝐾𝑝(𝜀
𝑘
𝑇
𝑇
a g −𝜀 e )}},(60)
wi h limi s 𝑢
𝑘
𝜁and 𝑢
𝑘
𝜁gi en by (57). This con ol scheme will be used
in simula ions as a simple benchma k o compa ison wi h he app ox-
ima e op imal con ol laws. No e ha his con olle essen ially igno es
he dynamics o he elec omobili y laye , elying solely on he ba e y
s o age o he EVs ha a e cu en ly p esen a each cha ging s a ion
o a emp o educe cu en cu ailmen . I can he e o e be seen as
a simpli ied p oxy o he mo e con en ional ANM app oaches elying
solely on s a iona y ene gy s o age, albei wi hou aking ad an age o
load and gene a ion p edic ions.
Since cu ailmen a any po is highly unlikely o change sign du -
ing he p edic ion ho izon ( om RES cu ailmen o load demand e-
sponse, o ice e sa), we adop a wo-s ep-long con ol ho izon, e -
ec i ely assuming ha he ela i e cha ging a es ake some cons an
alue 𝑢>
𝑘
𝜁a e he cu en ime s ep
𝑘,
𝑢
𝑘+ℎ
𝜁=𝑢>
𝑘
𝜁, ℎ= 1,…, 𝐻 .(61)
Ins ead o inding he exac
𝛿
𝑘+ℎ
𝜁 o ℎ= 1,…, 𝐻, we calcula e he p e-
dic ed u u e cu ailmen app oxima ely, using Bol zmann ope a o
smoo h maximum,
𝛥
𝑘+ℎ
𝜁=|||
𝛤
𝑘+ℎ
𝜁+
𝛬
𝑘+ℎ
𝜁+𝑈
𝑘+ℎ
𝜁|||
1 +𝑒−𝛾𝛥|||
𝛤
𝑘+ℎ
𝜁+
𝛬
𝑘+ℎ
𝜁+𝑈
𝑘+ℎ
𝜁|||
,(62)
Con ol Enginee ing P ac ice 159 (2025) 106289
9
M. Čičić e al.
wi h he u u e cha ging powe aken as
𝑈
𝑘+ℎ
𝜁= max {𝑢>
𝑘
𝜁,0}𝑈(
𝑘+ℎ−1)
𝑇
𝑇+1
𝜁+ max {−𝑢>
𝑘
𝜁,0}𝑈(
𝑘+ℎ−1)
𝑇
𝑇+1
𝜁,(63)
o ℎ= 1,…, 𝐻.
Al hough he cu en ela i e cha ging powe does a ec u u e 𝑈𝑘
𝜁
and 𝑈𝑘
𝜁, and will hus a ec
𝛥
𝑘+ℎ
𝜁,ℎ= 1,…, 𝐻, his in luence is in
p ac ice low enough ha i can be neglec ed when inding 𝑢>
𝑘
𝜁 ha
minimize app oxima e u u e cu ailmen . I is enough o conside 𝑢>
𝑘
𝜁
aking is ex eme alues o ze o, [𝑢>
𝑘
𝐡𝑢>
𝑘
𝐩𝑢>
𝑘
𝐰] ∈ {−1,0,1}3, and selec
he u u e ela i e cha ging powe s ha minimize cu ailmen .
Nex , we upda e he cha ging p ice 𝜋𝑘
𝐩a a lowe a e han he
ela i e cha ging powe s, wi h ime s ep
𝑇𝜋 aken as a mul iple o
𝑇, in
o de o educe he compu a ional bu den, bu also o imp o e ealism
by allowing he EV d i e s mo e ime o eac o he change. Since he
e ec s o changing he p ice only mani es in he u u e, i is enough
o e alua e
𝐽𝜋(
𝑘, 𝜋𝐩;𝑢>
𝑘
𝜁) =
𝑇𝜋
𝑇
∑
ℎ=1 ∑
𝜁∈{𝐡,𝐩,𝐰}
𝑤ℎ
𝛿|
𝛥
𝑘+ℎ
𝜁|,(64)
o wo cases o u u e ela i e cha ging powe s,
𝑢>
𝑘
𝐩= 1, 𝑢>
𝑘
𝐡=𝑢>
𝑘
𝐰= ±1.(65)
This simpli ica ion can be made because in he s udied case he e is
only a single cha ging s a ion wi h only gene a ion and no load a i s
node. The cha ging p ice is upda ed o he minimum alue o which
he cu ailmen is ze o o a leas one o he cases o 𝑢>
𝑘
𝜁, o i his is
no achie able, i is se o i s minimum alue 𝜋.
Finally, he app oxima ed op imiza ion p oblem is
minimize
𝑢
𝑘
𝜁
𝐽(𝑘, 𝑢
𝑘
𝜁;𝑢>
𝑘
𝜁, 𝜋
𝑘
𝐩)(66)
subjec o Mobili y and g id dynamics (38)–(39),
𝑢
𝑘
𝜁≤𝑢
𝑘
𝜁≤𝑢
𝑘
𝜁,(67)
wi h he con ol inpu cons ain s esul ing om minimizing cu ail-
men o he cu en s ep (57), and he cos unc ion
𝐽(𝑘, 𝑢
𝑘
𝜁;𝑢>
𝑘
𝜁, 𝜋
𝑘
𝐩) =𝑤𝜀(𝜀
𝑘
𝑇
𝑇
a g −𝜀 e )2+
𝐻
∑
ℎ=1 ∑
𝜁∈{𝐡,𝐩,𝐰}(𝑤ℎ
𝛿|
𝛥
𝑘+ℎ
𝜁|).(68)
The e o e, in his o mula ion we only di ec ly op imize he cu en
ela i e cha ging powe s 𝑢
𝑘
𝜁, wi h he u u e ela i e cha ging powe s
𝑢>
𝑘
𝜁and he public cha ging p ice 𝜋
𝑘
𝐩 ha pa ame ize he p oblem,
a ec ing he model dynamics (38)–(39) and he cos unc ion.
C ucially, since 𝑢>
𝑘
𝜁,𝜁∈ {𝐡,𝐰,𝐩}, is se o educe u u e cu ailmen ,
in case he e is no cu ailmen a some po 𝜁,𝑢
𝑘
𝜁can be op imized o
u he educe possible u u e cu ailmen a o he po s
𝜁. This is only
possible due o he elec omobili y model cap u ing he lows o EVs
om one po o ano he , ca ying hei SoC wi h hem. In ce ain sim-
ple cases, he op imal con ol is s aigh o wa d o de i e. Fo example,
in case he e is no cu en cu ailmen a any po a ime
𝑘, and he
model p edic s u u e RES gene a ion will be necessa y a 𝐩a ime
𝑘𝛿,
he op imal solu ion needs o cause he u u e 𝑈
𝑘𝛿
𝐩, and consequen ly,
also 𝜁𝜂𝑘𝛿
lo +𝜁𝜂𝑘𝛿
hi , o inc ease. In o de o his o be achie ed, he op imal
cu en cha ging a es will be 𝑢
𝑘
𝐡=𝑢
𝑘
𝐰= −1, leading o a lowe SoC o
EVs en e ing he oad om 𝐡and 𝐰,𝜉𝜀𝑘
on,𝑖,𝑖∈ {𝑖ou
𝐡, 𝑖ou
𝐰}, which in u n
will cause 𝜉𝛽𝑘
𝑖in
𝐩
o inc ease, and mo e ehicles o en e 𝐩. Con e sely, in
case he model p edic s u u e load cu ailmen will be necessa y a 𝐰
a ime
𝑘𝛿, he op imal solu ion needs o inc ease he u u e 𝑈
𝑘𝛿
𝜁, and
consequen ly, also 𝜁𝜂𝑘𝛿
hi , leading o 𝑢𝜁
𝐡= 1being op imal. Such e ec s
a e shown in simula ions.
In summa y, he p ocedu e o inding he cu en app oxima e op-
imal con ol is as ollows:
Table 2
Simula ion pa ame e s and hei alues.
Symbol Meaning Value
𝐿 o al To al oad leng h 25 km
𝑁EV To al numbe o EVs 4400
𝑁𝑥Numbe o oad cells 14
𝐿𝑥Road cell leng h 1.79 km
𝑇Elec omobili y ime s ep 60 s
𝑣 F ee low speed 100 km/h
𝜌c C i ical densi y 30 eh/km
𝜌jam Jam densi y 360 eh/km
𝐿𝜀SoC cell leng h 0.1
𝑁𝜀Numbe o SoC cells 11
0Ba e y discha ge cons an e m −1.40 ⋅10−1 1/h
1Ba e y discha ge linea mul iplie −1.85 ⋅10−3 1/km
2Ba e y discha ge quad a ic mul iplie −1.02 ⋅10−6 h/km2
𝐵EV ba e y capaci y 60 kWh
𝐶
𝐡∕𝐰Maximum cha ging a e a 𝐡and 𝐰0.05 1/h
𝐶
𝐩Maximum cha ging a e a 𝐩2.5 1/h
𝛾𝛽Spli ing a io pa ame e 10
0P ice sensi i i y cons an e m 1.2
1P ice sensi i i y linea mul iplie −0.2
𝜎max
𝑓Maximum p o ile pe u ba ion SD 0.2
𝑆Powe line 𝐡,𝐩, and 𝐰capaci ies 30 MVA
𝑇Con ol ime s ep 0.5 h
𝐻Con ol ho izon 8
𝜀bnd Bounda y SoC 0.3
𝜀 e Re e ence SoC 0.5
1. i ⌊𝑘
𝑇
𝑇𝜋⌋= 0, upda e he public cha ging p ice 𝜋
𝑘
𝐩,
2. ind [𝑢>
𝑘
𝐡𝑢>
𝑘
𝐩𝑢>
𝑘
𝐰] ∈ {−1,0,1}3 ha esul s in minimal cu ail-
men , and
3. ind 𝑢
𝑘
𝜁by sol ing (66).
As discussed ea lie , i a some ime
𝑘cu ailmen canno be a oided a
some node 𝜁, cons ain s (67) degene a e o an equali y cons ain o
ha 𝑢
𝑘
𝜁, and only he emaining ela i e cha ging powe s a e op imized,
hus educing he compu a ional load.
5. Simula ion esul s and discussion
We comple e he s udy o he p oposed ANM sys em using elec-
omobili y con ol by pu ing i o he es in simula ions. A e in-
oducing he gene al simula ion se up, we conside h ee g oups o
simula ions. In he i s g oup, we execu e a single simula ion un o
wo di e en scena ios whe e we assume he gene a ion, load, and mo-
bili y p o iles a e known. The aim o hese simula ions is o illus a e
in de ail how elec omobili y con ol can be used o ANM in he ideal,
ull-in o ma ion case. The scena ios a e one wo k week long each, one
se in summe and one in win e , wi h co espondingly di e en RES
gene a ion and load p o iles. The second g oup o simula ions consis s
o 20 simula ion uns o asumme and win e wo k day wi h pe u bed
gene a ion, load, and mobili y p o iles, wi h he aim owa ds demon-
s a ing obus ness o unce ain y wi h mo e s a is ical signi icance. The
hi d g oup o simula ions s udy a la ge ne wo k s uc u e, wi h he
Home and Wo k po s spli in o mul iple nodes, demons a ing how
he app oach can be scaled up. No e ha in all cases he ehicles de-
cide whe he o no hey en e he public cha ging s a ion depending
on hei SoC, acco ding o (13), which is only di ec ly in luenced by
cha ging p ice con ol.
5.1. Simula ion se up
The o e all s uc u e o he simula ed sys em is in oduced in Sec-
ion 2, and a lis o all ele an simula ion pa ame e s and hei alues
is gi en in Table 2. The simula ed oad ne wo k consis s o wo links
o leng h 𝐿 o al, connec ing 𝐡and 𝐰in bo h di ec ions. The s a e o he
EV a ic e ol es acco ding o he elec omobili y model p esen ed in
Con ol Enginee ing P ac ice 159 (2025) 106289
16
M. Čičić e al.
allowing us o p edic he imes when signi ican con ol ac ion will be
needed, and o adjus he cha ging con ol acco dingly o p eemp his.
We compa e h ee cases o con ol: simple myopic con ol ha only
eac s o he cu en s a e, and wo cases o p edic ion-based eceding
ho izon con ol, wi h o wi hou public cha ging s a ion p ice con ol.
The p oposed con ol laws a e es ed in simula ions on summe
and win e scena ios, wi h o wi hou pe u ba ions o he a ious
powe and mobili y p o iles, as well as using a mo e complex powe
ne wo k whe e he Home and Wo k po s a e spli in o mul iple
sub-po s. The esul s show ha we a e able o signi ican ly educe
cu ailmen h ough EV cha ging con ol, e en i a simple myopic
con olle is used. Bo h p edic ion-based con olle s a e able o u he
educe cu ailmen , emphasizing he need o good p edic ion models.
The imp o emen is especially p onounced in case we a e able o
con ol he public cha ging p ice, indica ing ha p o iding incen i es
o EV d i e s can play an impo an ole in implemen ing g id- iendly
elec omobili y.
Al hough in his wo k we mainly ocus on a ai ly simplis ic sys-
em s uc u e, wi h h ee elec omobili y nodes, he o e all con ol
amewo k, including he models and con ol app oaches, lends i sel o
ex ensions o a mo e gene al case. These ex ensions include la ge and
mo e complex oad and elec ical ne wo ks, mo e ealis ic beha iou al
and echnological models o EVs and hei d i e s, and mo e in ica e
con ol objec i es, e.g., aking in o accoun balancing egula ion ma ke
dynamics. While a simple model o EV d i e s’ sensi i i y o cha ging
p ice is included in he model, and he cha ging p ice is used as
he mos s aigh o wa d addi ional con ol inpu , he ac ual incen i e
mechanisms ha could be applied in eali y a e s ill unclea . The e
emains much wo k o be done on modelling he impac ha hese
incen i es ha e on eal-wo ld EV d i e s in he nowadays la gely
hypo he ical scena io o high EV ma ke pene a ion a e.
While in his wo k we use a de ailed mac oscopic elec omobili y
model, in o de o cap u e he complex in e ac ions be ween he EVs
and he powe g id, i may be possible o achie e simila esul s using a
simpli ied model. One po en ial app oach is o ep esen he coupling
be ween pai s o cha ging s a ions as EV-implemen ed i ual powe
lines. In his case, he pa ame e s and limi a ions o such EV i ual
powe lines can be iden i ied by s udying he ull sys em, ep esen ed
by he de ailed model. The acqui ed abs ac simpli ied ep esen a ion
could hen be used o con ol, signi ican ly imp o ing he scalabili y
o he sys em. Such analysis emains as u u e wo k.
The e a e se e al p ac ical obs acles o implemen ing app oaches
simila o he ones discussed in his wo k he eal wo ld. Ve y ew
places cu en ly ha e a la ge po ion o EVs, se e ely limi ing he
impac ha any EV cha ging con ol scheme could ha e. The cha ging
in as uc u e is s ill no su icien ly de eloped, and is la gely incapable
o accep ing V2G powe lows. The incen i e s uc u es ha would
ai ly compensa e EV owne s o po en ial incon enience and addi-
ional ba e y wea a e no ye in place, and he speci ics o he business
case a e unclea . The i s s eps owa ds eal-wo ld es ing and alida-
ion o he app oach should likely ocus on small-scale isola ed sys ems,
whe e i migh be easie o con ol all he in ol ed componen s, such
as in islanded mic og ids.
CRediT au ho ship con ibu ion s a emen
Mladen Čičić: W i ing – e iew & edi ing, W i ing – o iginal d a ,
Visualiza ion, So wa e, Me hodology, In es iga ion, Fo mal analysis,
Concep ualiza ion. Ca los Vi as: W i ing – e iew & edi ing, W i ing
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Wi : Supe ision, Concep ualiza ion. F ancisco R. Rubio: Supe ision,
Concep ualiza ion.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
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