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In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems
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A sys emic and model-less app oach o eal- ime op imal con ol o
unbalanced AC mic og ids domina ed by powe elec onics
J. Ca los Oli es-Campsa,b, Ál a o Rod íguez del Nozala, Juan Manuel Mau icioa,
José Ma ía Maza-O egaa,∗
aDepa men o Elec ical Enginee ing, Uni e sidad de Se illa, Camino de los descub imien os s/n, Se ille, 41092, Spain
bCITCEA-UPC, Uni e si a Poli ècnica de Ca alunya, A . Diagonal, 647, Ba celona, 08034, Spain
ARTICLE INFO
Keywo ds:
AC mic og ids
Unbalanced low ol age dis ibu ion sys ems
Cen alised hie a chical con ol
Online eedback op imisa ion
Load sha ing con ol
Seconda y ol age con ol
ABSTRACT
This pape p esen s a wo-laye hie a chical con ol sys em o powe egula ion be ween g id- o me
con e e s eeding islanded o g id-connec ed mic og ids. This wo k conside s he inhe en unsymme ic na u e
o mic og ids ha can be composed o , in addi ion o h ee-phase loads, single-phase loads. The p oposed
seconda y con ol s a egy combines he classical au oma ic gene a ion con ol (AGC) o ac i e powe and
equency egula ion wi h an online eedback op imisa ion (OFO) me hod o eac i e powe and ol age
egula ion pe phase. By coo dina ing he g id- o ming de ices using hese echniques, an op imal ope a ing
poin can be a ained ha minimises he ol age imbalance on a ge buses. This me hodology con e s a high
deg ee o obus ness o dis u bances and model inaccu acies. The e icacy o he me hodology is e alua ed
h ough he simula ion o wo case s udies. The esul s demons a e he sui abili y o he p oposed s a egy,
which le e ages he as esponses o he powe con e e s.
1. In oduc ion
Con empo a y dis ibu ion ne wo ks a e unde going a ans o ma-
i e and unp eceden ed ansi ion, e ol ing om a composi ion o
o e sized and passi e elemen s o a s essed and ac i e g id [1]. This
ans o ma ion has been d i en by he necessi y o deca bonise he elec-
ical powe sys em. Consequen ly, enewable ene gy sou ces (RES),
which a e mo e scalable han adi ional powe plan s, a e being mas-
si ely in eg a ed in o he powe ne wo k [2]. A signi ican p opo ion
o hese RES a e connec ed o he low- ol age g id h ough powe
elec onics con e e s; hese uni s a e called in e e -based esou ces
(IBRs). In his con ex , he concep o mic og id (MG) eme ged as
a lexible solu ion o he managemen o elec ical powe and he
p o ision o suppo se ices o he main g id, as desc ibed in [3]. A
MG clus e s gene a ion uni s, loads, and ene gy s o age elemen s a he
dis ibu ion le el in o a single con ollable en i y o he main g id,
enabling i o be ope a ed in ei he g id-connec ed o islanded mode.
Howe e , he in oduc ion o hese smalle and au onomous e sions
o he powe sys em is causing a ple ho a o new and exing con ol
issues o conside able p ac ical impo ance, as e idenced in [4].
The p incipal di icul ies associa ed wi h low- ol age MG con ol
can be a ibu ed o he in insic unce ain y ha a ises on bo h he
gene a ion and consump ion on s [5]. Mo eo e , gi en he low ine -
ia in sys ems domina ed by powe elec onics, i becomes especially
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (J.M. Maza-O ega).
challenging o balance ac i e and eac i e powe when conside ing
in e mi en enewable gene a ion and a iable consump ion [6]. The
challenge o p e en ing mic og id ins abili y in such ola ile condi ions
equi es a de ailed in es iga ion o he de elopmen o inno a i e con-
ol echniques unlocking he lexibili y and apid esponse capabili ies
ha g id- o ming IBRs may p o ide [7–9].
The ope a ional objec i es o a IBR-d i en mic og id can be mapped
in o a hie a chical con ol sys em wi h con ol laye s o ganised acco d-
ing o hei p io i y as explained in [10]. The p ima y con ol laye is
asked wi h main aining he MG s abili y by egula ing powe balance
ega dless o s eady-s a e ope a ing poin [8,11]. The seconda y con ol
laye is esponsible o he es o a ion o ol age and equency de i-
a ions caused by he ac ions o he p ima y laye . Finally, he e ia y
con ol laye is esponsible o he economic op imisa ion o he MG
ope a ion.
This wo k add esses he challenge o compu ing he e e ence ol -
age signal, i.e. ampli ude and equency, o he MG g id- o ming IBRs
wi hin he seconda y laye [12]. In his ega d, he high pene a ion
o RES a he esiden ial le el has a signi ican impac on he ol age
quali y in he dis ibu ion ne wo ks [13]. In pa icula , he p esence o
a bi a ily connec ed single-phase loads and dis ibu ed RES in oduces
a signi ican di icul y in main aining an adequa e ol age p o ile [14–
16]. This si ua ion is u he compounded when he MG comes o
h ps://doi.o g/10.1016/j.ijepes.2024.110443
Recei ed 5 Augus 2024; Recei ed in e ised o m 28 No embe 2024; Accep ed 24 Decembe 2024
Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
A ailable online 10 Janua y 2025
0142-0615/© 2024 Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license ( h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ ).
J.C. Oli es-Camps e al.
islanded ope a ion, as he e is an addi ional need o achie e ade-
qua e powe sha ing be ween exis ing IBR uni s [17,18], pa icula ly
challenging when unbalanced condi ions a ise [19]. In his con ex ,
he e a e se e al decen alised me hods in he li e a u e, mainly based
on he concep o i ual impedance [20,21]. Howe e , as o mally
demons a ed in [22], wi hou a communica ion in as uc u e, he
esul ing s eady-s a e ol age p o ile is no op imal in any sense.
Fo his eason, his pape conside s a cen alised communica ion
in as uc u e,1whe ein op imal con olle s, which conside he sys-
em as a whole, demons a e he mos e ec i e esul s in e ms o
pe o mance [25,26]. Mos op imisa ion-based me hodologies aking
ad an age o his cen alised app oach equi e an accu a e g id model
and gene a o pa ame e s, in addi ion o ull obse abili y o he ne -
wo k s a e [27–30]. Howe e , hese p e equisi es a e o en una ainable
in MG en i onmen s. Fu he mo e, when ol age unbalance is in ended
o be co ec ed, con ol s a egies in he li e a u e a e ypically based
on sequence decomposi ion, which equi es angle in o ma ion [31,32].
While his is a iable app oach o buses wi h con e e s, i is no
applicable o o he buses wi h loads due o he una ailabili y o his
equisi e in o ma ion. P oposals ha e eme ged sugges ing he u ilisa-
ion o Phaso Measu emen Uni s o p o ide he ol age angles [33],
al hough hei deploymen in dis ibu ion ne wo ks o mic og ids is no
cu en ly easible due o economic easons. The e o e, a me hodology
ha does no necessi a e knowledge o hese angles would elimina e
he addi ional in es men cos .
F om a compu a ional pe spec i e, he p ima y limi a ion o hese
echniques is he e alua ion o he nonlinea se o equa ions ha
comp ise he powe low p oblem. As a esul , cen alised op imisa ion
echniques a e ypically sol ed o line, conside ing load and gene a ion
o ecas s, and a ully known ne wo k model [34]. Then, he solu ion is
applied di ec ly o he sys em in a eed o wa d manne . A subs an ial
body o esea ch has been conduc ed wi h he objec i e o educing his
compu a ional cos . This has in ol ed he use o app oxima ions and
elaxa ions o con exi y he p oblem [35–37], model- ee o mula ions
such as he ex emum-seeking app oach [38], and he use o me a-
heu is ic algo i hms o sol e p oblems wi h an a bi a y p ecision [39,
40]. Howe e , he eed o wa d s uc u e is pa icula ly ulne able o
unce ain y and model inaccu acy.
A no el pe spec i e ha anscends hese p e iously iden i ied
sho comings and a oids he need o simpli ica ion o elaxa ion o
he model is he Online Feedback Op imisa ion (OFO) pa adigm [41].
The undamen al concep o OFO is he implemen a ion o op imisa ion
algo i hms as eedback con olle s, which a e coupled o physical
plan s h ough a closed-loop scheme. E en ually, he ope a ing poin
con e ges o he solu ion o he op imisa ion p oblem by using eal-
ime measu emen s. As a esul , i is highly obus agains dis u bances
and unce ain ies. Mo eo e , he nonlinea plan model is no e alua ed
nume ically, which signi ican ly educes he compu a ional bu den.
Fu he mo e, OFO p o ides da a p i acy because i equi es only a
s eady-s a e inpu –ou pu sensi i i ies map o he plan . I is also
wo hy o no e ha his s a egy is cu en ly unde conside a ion o
implemen a ion in he elec ical powe sys em. Ea ly esul s o i s de-
ploymen in labo a o y se ings [42,43] and exis ing powe g ids [44]
ha e al eady been published in he academic li e a u e.
P e ious esea ch on cons ained OFO o mula ions in powe sys em
applica ions has demons a ed he s abili y and con e gence o his
closed-loop app oach wi h he unde lying dynamic plan [45,46]. Addi-
ionally, an expe imen al alida ion o a se up o ol age egula ion by
eac i e powe injec ion was de eloped in [42], and he join ope a ion
o OFO and a dynamic s a e es ima o was illus a ed in [47]. I should
1Ano he app oach o g ea in e es o he con ol communi y is dis ibu ed
op imisa ion, which has he po en ial o enhance he obus ness o he con ol
laye . Howe e , a comp ehensi e e iew o hese me hodologies is beyond he
scope o his a icle. Fo u he insigh , we ecommend consul ing [23,24].
be no ed ha all o he a o emen ioned wo ks assume a s i g id
wi h a s able ol age signal a he powe con e e e minals, which
enables he g id- ollowing mode o IBRs. Howe e , his assump ion
may no longe be alid in si ua ions whe e he MG is islanded om
he main g id. The ini ial e o s o u ilise he OFO algo i hm o he
coo dina ion o g id- o ming de ices can be ound in p e ious wo k
by he au ho s. In [48], he OFO is o mula ed o he coo dina ion o
gene a o s in DC mic og ids, which was expe imen ally alida ed in a
labo a o y en i onmen [43]. Subsequen ly, a s a egy based on OFO
combined wi h a con en ional Au oma ic Gene a ion Con olle (AGC)
was p oposed o balanced AC mic og ids in [49].
This pape makes a signi ican con ibu ion o he exis ing body o
esea ch by ex ending he wo-laye hie a chical algo i hm p oposed
in [49] o conside he inhe en asymme ical2na u e o MGs. The
p ima y con ol laye o he g id- o ming de ices is based on he
i ual synch onous gene a o scheme p esen ed in [51]. The seconda y
con ol laye is di ided in o wo pa allel algo i hms. On he one hand,
an op imal comp omise solu ion be ween phase ol age egula ion and
eac i e powe sha ing is achie ed using a non-con ex OFO algo-
i hm [52]. On he o he hand, equency egula ion is conduc ed using
classical AGC o adjus he ac i e powe injec ion o each g id- o ming
IBR.
The main con ibu ions o his wo k a e summa ised as ollows:
•Ex end he wo k p esen ed in [49] o include in he OFO o -
mula ion he egula ion o unbalanced ol age on ce ain MG
a ge buses, main aining he eac i e powe sha ing be ween
gene a o s. The use o OFO allows he desi ed obus ness and
p i acy cha ac e is ics o be main ained in his laye .
•The op imisa ion p oblem is o mula ed in such a way ha i
makes use o he RMS ol age measu emen s o he a ge nodes.
The o mula ion imposes some ol age magni udes o achie e
a pe o mance as close as possible o a balance ideal scena io,
he eby elie ing he algo i hm o he necessi y o de e mine he
angle o each phase and he subsequen need o synch onise he
measu emen s.
The ollowing sec ions cons i u e he emaining con en o his
pape . Sec ion 2p esen s he issue unde examina ion and se s ou he
objec i es pu sued in his s udy. Sec ion 3p esen s he me hodology
ha has been p oposed. Fi s ly, he implemen a ion o a g id- o ming
IBR is discussed, and subsequen ly, he seconda y con ol s a egy is
de ailed. Sec ion 4p esen s he esul s o simula ions o wo di e en
MGs, which analyse he pe o mance o he p oposed algo i hm. The
subsequen sec ion, Sec ion 5, p esen s he conclusions o his s udy.
2. P oblem desc ip ion
Le us conside a h ee-phase and ou -wi e AC MG, which may
be modelled as an undi ec ed g aph, deno ed by he no a ion =
(,), whe e ep esen s he se o buses and ⊆× ep esen s
he mul ise o edges ha in e connec he a o emen ioned buses. A
ma ix e m o app op ia e size is associa ed wi h each edge in o
ep esen he (asymme ical) line impedance. Some buses, 𝑔⊆,
a e supplied by IBRs ope a ing in g id- o ming mode. I is assumed
ha IBRs a e ou -leg ol age sou ce con e e s (VSCs), which a e
capable o independen ly con olling he phase ol age ampli udes and
he equency a hei co esponding poin o in e connec ion (POI), as
demons a ed in [53]. Finally, i is conside ed ha some uncon ollable
single- and h ee-phase loads, as well as g id- ollowing IBRs, can be
connec ed o he emaining MG buses ∖𝑔.
All g id- o ming IBRs in he ne wo k a e assumed o ha e an app o-
p ia e measu emen and con ol sys em ha enables hem o emula e a
2The e minology used is ha o [50]. Balanced is used in he con ex o
signals and symme ical in he con ex o ci cui s.
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
2
J.C. Oli es-Camps e al.
synch onous gene a o . On he DC side, he de ices a e assumed o be
connec ed o a p ima y esou ce sou ce ha de e mines he maximum
amoun o s eady-s a e powe . In addi ion, IBRs ha e a communica ion
sys em wi h a cen alised con ol laye ha calcula es hei se poin s.
The aim o he cen al con olle is o coo dina e he join ope a ion o
all he MG IBRs wi hou elying on a de ailed ne wo k model. Ins ead,
he con olle elies on he da a ob ained om he ne wo k measu e-
men s, which se e as he only sou ce o in o ma ion. Fu he mo e, i
is o u mos impo ance o espec echnical and physical cons ain s o
a oid any possible comp omise and o ensu e he MG sa e ope a ion.
The echnical objec i es ha he cen alised con ol laye seeks o
achie e can be summa ised as ollows:
(i) Elimina e s eady-s a e equency de ia ions caused by he p i-
ma y con ol laye .
(ii) Main ain bus phase ol ages 𝑉ℎ,𝑎𝑏𝑐 wi hin ol age limi s o all
ℎ∈𝑔.
(iii) Sha e he ac i e and eac i e powe load demanded in he
ne wo k among he ins alled IBRs o p e en o e loads.
(i ) Minimise he ol age imbalance and keep each phase ampli ude
close o he gi en se poin o ce ain a ge buses 𝑡⊆.
( ) Main ain he injec ed cu en s wi hin hei he mal limi s, wi h
special a en ion de o ed o injec ion o he neu al wi e cu en .
To achie e hese objec i es, he cen al seconda y con ol laye
mus calcula e he se poin s o he ac i e powe injec ions and i -
ual elec omo i e o ce (EMF) pe phase o each IBR. Consequen ly,
when he asymme ical s uc u e o he MG is conside ed, he o iginal
o mula ion p esen ed in [49], is gene alised, since objec i es (ii) and
(iii) mus be me pe phase, and in addi ion, objec i es (i ) and ( ) a e
inco po a ed in o de o educe MG imbalance.
3. P oposed me hod
This sec ion p o ides a de ailed and comp ehensi e p esen a ion o
he me hodology p oposed o add ess he p oblem ou lined in Sec ion 2.
In o de o achie e his, he con ol a chi ec u e is p esen ed i s ,
ollowed by a de ailed desc ip ion o he g id- o ming IBR p ima y
con olle s. Finally, he seconda y con olle , which is esponsible o
coo dina ing he con ol ac ions o he IBRs in o de o achie e p ope
MG ope a ion, is ou lined. No e ha he ollowing no a ion will be used
in he sequel: lowe -case a iables e e o ins an aneous alues, while
uppe -case ones a e ela ed o he co esponding RMS alues.
3.1. Con ol a chi ec u e
The p oposed hie a chical wo-laye con ol a chi ec u e, designed
o ul il he objec i es ou lined in he p e ious sec ion, is illus a ed in
Fig. 1.
In o de o elucida e he unc ioning o he con olle , le us con-
side a h ee-phase ou -wi e (3P-4W) VSC connec ed o he main g id
h ough a coupling il e wi h impedance 𝐙𝑠. The VSC is ope a ing in
g id- o ming mode and is capable o syn hesising phase- o-neu al ol -
ages, i.e. 𝑣𝑠,𝜒 wi h 𝜒∈ {𝑎𝑛, 𝑏𝑛, 𝑐 𝑛}, and he neu al- o-g ound ol age
𝑣𝑠,𝑛𝑔 a he POI. Fou cu en s a e injec ed in o he POI o se he ou pu
ol age, i.e. 𝑖𝑠,𝜒 wi h 𝜒∈ {𝑎, 𝑏, 𝑐 , 𝑛}. The cha ac e is ics o hese cu en s
will depend on Ki chho ’s laws acco ding o he ne wo k a iables.
The VSC lowes con ol le el is esponsible o ensu ing compliance
wi h he POI ol age se poin s (phase ampli udes and equency) wi h
a cha ac e is ic ime esponse in he o de o ens o milliseconds [53].
The p ima y con ol laye is implemen ed a local le el wi hin
each IBR, and is he e o e designed o acili a e apid esponses ha
con ibu e o main ain a s able MG ope a ion. This wo k adop s he
me hodology o emula ing a synch onous gene a o which enables IBRs
o main ain a connec ion be ween a i ual in e nal equency and he
ac i e powe exchanged wi h he g id. Since he IBR con ains a model
o a synch onous gene a o , he se poin s o his con ol laye a e he
ac i e powe e e ence, 𝑝𝑐, and inc emen s o he phase ampli udes
o he in e nal EMFs, 𝛥𝐯𝑟. No e ha he EMF ampli ude can be se
independen ly in each phase, being no equi ed o pe o m a balanced
ope a ion.
The seconda y con ol is designed a he sys em le el in a cen alised
manne . This laye ecei es measu emen s om all he g id- o ming
IBRs and a leas one MG a ge bus whe e he ol age imbalance is
egula ed. I s ask is o coo dina e he g id- o ming IBRs o e u n he
g id o a p ope ope a ing poin a e a dis u bance. This objec i e is
achie ed wi h wo independen s a egies o he calcula ion o he
ac i e powe and in e nal EMF e e ences o each g id- o ming IBR.
No e ha his laye will calcula e one ol age e e ence pe IBR phase.
Fu he de ails on each con ol laye a e gi en in he ollowing
subsec ions.
3.2. P ima y i ual synch onous gene a o con ol
The IBR p ima y con ol is based on he P opo ional-In eg al Vi -
ual Synch onous Gene a o (PI-VSG) [51]. The ins an aneous ac i e
powe injec ed by he IBR is calcula ed om he h ee-phase cu en s
and phase- o-neu al ol ages, 𝐢𝑠and 𝐯𝑠, as:
𝑝𝑒=𝐯⊤
𝑠𝐢𝑠.(1)
No e ha his ins an aneous ac i e powe may ha e oscilla o y
e ms because ol ages and cu en s a e unbalanced. The e o e, a low-
pass il e is applied o elimina e any oscilla o y componen esul ing
in he a e age ac i e powe 𝑝𝑒𝑓 .
As s a ed abo e, he PI-VSG con ol equi es he ac i e powe
se poin de ined by he seconda y con ol laye , 𝑝𝑐. This signal is passed
h ough a low-pass il e in o de o p e en undesi ed ansien s. The
i ual mechanical powe o he IBR is ob ained by adding o his
il e ed powe se poin , 𝑝𝑐 𝑓, a d oop e m as:
𝑝𝑚=𝑝𝑐 𝑓+1
𝑘𝑑
(𝜔⋆−𝜔𝑣),(2)
whe e 𝑘𝑑is he d oop cons an and 𝜔𝑣and 𝜔⋆ ep esen he i ual
angula speed and i s e e ence, espec i ely.
The ac i e powe e o 𝜖𝑝, i.e. di e ence o he i ual mechanical
powe 𝑝𝑚and he il e ed POI powe 𝑝𝑒𝑓 , is ed in o a PI con olle o
upda e he i ual angula speed, 𝜔𝑣. Finally, he in e nal angle o he
i ual o o , 𝜃𝑣, is ob ained by in eg a ing he i ual o o speed.
No e ha his pape p oposes an independen pe -phase con ol o
he IBR by using a i ual h ee-phase sys em associa ed o each IBR
phase [54]. Fo his pu pose, i ual quad a u e signals a e compu ed
o he co esponding ac ual phase magni udes, ol ages, o cu en s.
This enables he gene a ion o a magni ude in he i ual s a iona y
e e ence ame o each phase o a ing a 𝜔𝑣. These o a ing magni udes
can be ans o med in o he i ual synch onous e e ence ame by
applying he co esponding ans o ma ion using he angle 𝜃𝑣.
Once he phase ol ages and cu en s a e ans o med in o hei co -
esponding i ual synch onous ame, a i ual impedance is applied o
de e mine he e e ence IBR e minal ol ages. This is accomplished by
conside ing he il e ed ol age inc emen s compu ed by he seconda y
con olle , as de ailed in he nex subsec ion. Finally, he Pa k an i-
ans o ma ion is applied o compu e he phase- e minal ol age o he
IBR. I is no ewo hy ha he i ual quad a u e signal associa ed wi h
each phase is dis ega ded, as i co esponds o he a i icially c ea ed
quad a u e signal.
3.3. Seconda y con ol
The seconda y con ol laye has wo main unc ions. Fi s , i mus
es o e he sys em equency o he e e ence alue dic a ed by he
e ia y con ol. Secondly, i mus egula e he bus ol ages wi hin he
egula o y limi s and balance hem as much as is easible.
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
3
J.C. Oli es-Camps e al.
Fig. 1. P oposed hie a chical con ol a chi ec u e.
The s a egy p oposed in his a icle employs wo con ol s ages ha
ope a e in pa allel. A PI con ol s uc u e pe o ms he ask o an Au-
oma ic Gene a ion Con olle (AGC) by dispa ching powe se poin s o
he g id- o ming IBRs. In u n, an op imal s a egy based on he Online
Feedback Op imisa ion (OFO) echnique is esponsible o managing he
a ge bus ol ages.
3.3.1. Ac i e powe - equency egula ion
The objec i e o his s age is o es o e he powe sys em equency
o he e e ence alue by balancing gene a ion and load wi hin he MG.
A PI-based con olle ecei es he sys em equency and de e mines
he equi ed ac i e powe o es o e he equency o i s s eady-s a e
e e ence alue, hus main aining he powe balance. This con olle
can be o mula ed in a disc e e o m as ollows:
𝜖𝜔[𝑛] =𝜔⋆−𝜔𝐶 𝑂 𝐼[𝑛],
𝜉[𝑛+ 1 ] =𝜉[𝑛] +𝛥𝑡 𝐾𝑝
𝑇𝑖
𝜖𝜔[𝑛],
𝑃𝑟[𝑛] =𝐾𝑝𝜖𝜔+𝜉[𝑛],
(3)
whe e 𝐾𝑝,𝑇𝑖a e he p opo ional gain and in eg a ion ime o he
PI con olle , espec i ely, 𝛥𝑡 is he communica ion pe iod o he sec-
onda y con ol laye and 𝑛is he disc e e- ime s ep. In his wo k,
he sys em equency is app oxima ed by calcula ing he equency a
he cen e o ine ia (𝜔𝐶 𝑂 𝐼). This alue is compu ed as he weigh ed
a i hme ic mean o he i ual o o speeds o all g id- o ming IBRs.
Once 𝑃𝑟has been de e mined using (3), he g id- o ming IBRs a e
dispa ched acco ding o hei a ed powe .
Gi en he limi ed amoun o da a equi ed o his p ocess and he
as dynamics o he VSCs, i is p oposed o execu e his con ol cycle
a ime in e als o one second.
3.3.2. Reac i e powe - ol age egula ion
This pa o he seconda y con ol is conce ned wi h he manage-
men o he MG ol age p o ile. This is achie ed by adjus ing he
in e nal phase EMF e e ences o he g id- o ming IBRs o main ain
app op ia e ol age le els on some p e iously selec ed a ge buses.
Vol age egula ion in asymme ical sys ems ep esen s a signi ican
challenge. One o he main obs acles is ha e en when he loads
Fig. 2. Equi alence be ween he balanced ol age sys em and an equila e al iangle.
(asymme ical) in a sys em a e supplied wi h balanced ol ages, he
cu en consump ion will ine i ably show an imbalance. Consequen ly,
he ol age d op ac oss he eede will also be unbalanced, and he
sys em ol ages a o he poin s will e ain his imbalance. Fo his
eason, he au ho s o his pape a bi a ily speci ied he a ge buses
on which he egula ion is pe o med. The ques ion o de e mining
he numbe and loca ion o buses o be con olled is ela ed o he
op imal loca ion o he g id- o ming IBRs o ensu e sys em s abili y.
The e o e, in his pape , his ask is conside ed as pa o a ne wo k
planning p oblem a he han an ope a ional one, whe e he objec i e
is o o mula e a con olle ha is no dependen on he a ge buses.
This wo k aims o elimina e he need o decompose unbalanced ol -
age sys ems using he Fo escue ans o m, which in ol es measu ing
he ol age angle o each phase. This a oids he need o synch onous
measu emen s o co ec ly calcula e he ela i e angles in case se e al
MG a ge buses a e selec ed. As a esul , his wo k only conside s
he phase- o-neu al, phase- o-phase, and neu al- o-g ound ol ages.
The e o e, o achie e he highes possible balance in he 3P-4W MG,
i is necessa y o main ain he phase- o-neu al and he phase- o-phase
ol ages as close as possible o hei nominal alues while he neu al-
o-g ound ol age close o ze o. Ensu ing ha all moduli ha e he same
leng h is equi alen o gene a ing a balanced ol age sys em. Fig. 2
illus a es he ans o ma ion o a balanced ol age sys em in o an
equila e al iangle.
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Fu he mo e, i is in ended o main ain an equal dis ibu ion o
he loads be ween he MG IBRs. Consequen ly, a his s age, i is
also conside ed ha an app op ia e pe -phase eac i e powe sha ing
be ween he g id- o ming IBRs mus be ca ied ou . In o de o acili a e
he calcula ion when di e en VSC sizes a e conside ed, all a iables
a e exp essed in a p.u. sys em ela i e o each de ice.
In o de o achie e an ope a ing poin ha sa is ies he a o emen-
ioned objec i es, he p oblem is o mula ed wi hin a ma hema ical
op imisa ion amewo k. This app oach allows o he inco po a ion
o cons ain s o ensu e ha ope a ional limi s a e no exceeded. To
o mula e his p oblem, conside he se o buses wi h gene a ion, 𝑔,
and he se o a ge buses, 𝑡.
The con ol ac ions a e he in e nal FEM a ia ions pe phase o
all o he g id- o ming IBRs: 𝐮ℎ= [𝛥𝑣𝑟,𝑎, 𝛥𝑣𝑟,𝑏, 𝛥𝑣𝑟,𝑐]⊤
ℎ∀ℎ∈𝑔. The
endogenous a iables, i.e. MG measu emen s, a e composed o he
phase- o-neu al, neu al- o-g ound and phase- o-phase ol ages mea-
su ed a he a ge buses: 𝐲𝑘= [𝑉𝑎𝑛, 𝑉𝑏𝑛, 𝑉𝑐 𝑛, 𝑉𝑛𝑔 , 𝑉𝑎𝑏, 𝑉𝑏𝑐 , 𝑉𝑐 𝑎]⊤
𝑘∀𝑘∈
𝑡, as well as he phase- o-neu al and neu al- o-g ound ol ages,
cu en and ac i e and eac i e powe injec ions o he g id- o ming
IBRs: 𝐲ℎ= [𝑉𝑎𝑛, 𝑉𝑏𝑛, 𝑉𝑐 𝑛, 𝑉𝑛𝑔 , 𝐼𝑎, 𝐼𝑏, 𝐼𝑐, 𝐼𝑛, 𝑃𝑎, 𝑃𝑏, 𝑃𝑐, 𝑄𝑎, 𝑄𝑏, 𝑄𝑐]⊤
ℎ∀ℎ∈𝑔.
F om he con ol pe spec i e, i is assumed ha he plan main ains
a nonlinea ela ionship be ween all hese a iables in s eady-s a e
and ha hey a e a ec ed by a numbe o exogenous a iables, 𝐰,
e.g. powe consump ion o g id- ollowing IBRs wi h no comple e powe
egula ion capabili y:
𝐲=𝐡(𝐮,𝐰).(4)
whe e 𝐡(⋅)encompasses all nonlinea unc ions ha de ine he s eady-
s a e beha iou o he plan . In gene al, hese equa ions encompass, bu
a e no limi ed o, he equa ions ha desc ibe he powe low p oblem
and he impac o he con olle s implemen ed in he VSCs.
All hese a iables a e subjec o limi s ha ep esen he cons ain s
o he op imisa ion p oblem:
•On he a ge buses, all measu ed ol ages mus be bounded:
𝑉𝑘≤𝑉𝑘,𝜒 ≤𝑉𝑘
𝑉𝑛𝑔 ≤𝑉𝑘,𝑛𝑔 ≤𝑉𝑛𝑔 ∀𝑘∈𝑡, 𝜒∈ {𝑎𝑛, 𝑏𝑛, 𝑐 𝑛},
𝑉′
𝑘≤𝑉𝑘, 𝜒 ≤𝑉′
𝑘𝜒 ∈ {𝑎𝑏, 𝑏𝑐 , 𝑐 𝑎}
(5)
whe e he unde sco e and o e sco e symbols indica e he lowe
and uppe limi s, espec i ely. The use o p imes se es o iden i y
ha he limi s a e phase- o-phase ol age.
•On he gene a ion buses, all he phase ol ages and cu en s a e
cons ained wi hin he co esponding echnical limi s, whe eas
he ac i e powe is limi ed by he a ailable powe o he p ima y
esou ce, 𝑃𝑝𝑟𝑖𝑚:
𝑉𝑘≤𝑉𝑘,𝜒 𝑛≤𝑉𝑘,
𝑉𝑛𝑔 ≤𝑉𝑘,𝑛𝑔 ≤𝑉𝑛𝑔,∀ℎ∈𝑔, 𝜒∈ {𝑎, 𝑏, 𝑐}
𝐼ℎ,𝜒 ≤𝐼ℎ, 𝐼ℎ,𝑛 ≤𝐼𝑛,
∑
𝜒
𝑃ℎ,𝜒 ≤𝑃ℎ,𝑝𝑟𝑖𝑚,
(6)
The se o cons ain s (5)–(6) is linea and can he e o e be w i en
compac ly as3:
𝐛−𝐀𝐲 ≥𝟎,(7)
whe e 𝐛con ains he limi alues o he a iables, whe eas 𝐀con ains
he linea ope a ions be ween a iables.
3Conside one a iable wi h uppe and lowe bounds as ollows: 𝑥𝑙 𝑜𝑤
1≤
𝑥1≤𝑥𝑢𝑝
1. This can be ew i en in wo exp essions 𝑥1≤𝑥𝑢𝑝
1and −𝑥1≤−𝑥𝑙 𝑜𝑤
1.
Finally, he a o emen ioned cons ain s can be exp essed in ma ix o m as:
[1 −1]⊤𝑥1≤[𝑥𝑢𝑝
1−𝑥𝑙 𝑜𝑤
1]⊤.
The objec i e unc ion ep esen ing he ol age egula ion and e-
ac i e powe sha ing p oblem can be o mula ed as ollows:
𝜙(𝐲) =∑
𝑘∈𝑡
𝛽𝑘∑
𝜒∈{𝑎𝑛,𝑏𝑛,𝑐 𝑛}
𝜒∈{𝑎𝑏,𝑏𝑐 ,𝑐 𝑎}
((𝑉𝑘,𝜒 −𝑉⋆
𝑘)2+ (𝑉𝑘,𝑛𝑔 −𝑉⋆
𝑘,𝑛𝑔)2+ (𝑉𝑘, 𝜒 −𝑉′⋆
𝑘)2)
+∑
ℎ,𝑗∈𝑔
𝛾ℎ,𝑗 ∑
𝜒∈{𝑎,𝑏,𝑐}
(𝑄ℎ,𝜒 −𝑄𝑗 ,𝜒 )2,
(8)
whe e 𝛽𝑘and 𝛾ℎ,𝑗 a e weigh ing a iables o de e mine he ela i e
impo ance o each ac o . Eq. (8) can be s a ed in a mo e compac
manne as:
𝜙(𝐲) = (𝐲−𝐲⋆)⊤𝐐(𝐲−𝐲⋆),(9)
whe e 𝐐agg ega es all weigh s.
Finally, he op imisa ion p oblem can be o mula ed by combining
he se s o Eqs. (4),(7) and (9) as ollows:
min
𝐮𝜙(𝐲),
𝑠.𝑡. 𝐛−𝐀𝐲 ≥𝟎,(10)
𝐲−𝐡(𝐮,𝐰) =𝟎.
In o de o add ess (10) in eal ime, his pape p oposes he
implemen a ion o a closed-loop OFO scheme. This app oach o e s
mul iple ad an ages a bo h he compu a ional and da a-p i acy le els,
as p e iously discussed in Sec ion 1. In pa icula , i is p oposed o use
he algo i hm p esen ed in [52]. In his scheme, he con ol ac ions (𝐮)
a e upda ed ollowing he g adien descen di ec ion [55]:
𝐮+=𝐮+𝛼
𝝈(𝐲),(11)
whe e supe sc ip +indica es he upda ed alue o he nex i e a ion,
𝛼is a non-nega i e ixed s ep-size, and
𝝈(𝐲)is he di ec ion ec o
calcula ed as ollows:
𝝈(𝐲) ∶= a g min
𝐰‖𝐰+𝐇⊤∇𝜙(𝐲)⊤‖2
2
𝑠.𝑡. 𝐛−𝐀(𝐲+𝛼𝐇𝐰)≥𝟎,(12)
whe e 𝐇co esponds o he sensi i i y ma ix ha ela es con ol
ac ions, 𝐮, and measu emen s, 𝐲, a ound an ope a ional poin . Conse-
quen ly,
𝝈can be de ined as he p ojec ion o he con ol ac ions ha
minimise he objec i e unc ion on o he easible egion o he con ol
ac ions.
I should be no ed ha his o mula ion does no equi e any knowl-
edge o he ne wo k opology o i s pa ame e s, which makes he algo-
i hm highly obus . Fu he mo e, i is no necessa y o ha e knowledge
o he in e nal con ol models o he VSCs in ol ed in he egula ion
o p ope ly ope a e he p oposed con ol s a egy. Thus, in e nal laye s
can be main ained as black-box models, he eby ensu ing he p o ec ion
o he indus ial p ope y o he co esponding manu ac u e s. Addi-
ionally, i is also impo an o no e ha he e is no need o demand
measu emen s o p edic ions due o he algo i hm’s abili y o adap o
MG dynamic changes.
Finally, i should be no ed ha his pape is based on he assump-
ion ha he sensi i i y ma ix 𝐇is known o can be de e mined
expe imen ally [56].
4. Pe o mance assessmen
This sec ion p esen s he esul s o nume ical simula ions o e i-
dence he dynamic pe o mance o he p oposed con ol a chi ec u e
on wo di e en MGs. The is sys em is a4-bus se up comp ising
wo g id- o ming IBRs supplying symme ical and asymme ical loads.
This simple sys em is employed o analyse he undamen al a ibu es
o he con olle . Simula ions a e hen pe o med on one MG based
on he opology o he CIGRE Eu opean LV benchma k dis ibu ion
ne wo k [57], demons a ing he e ec i eness o he con ol scheme
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
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Fig. 3. One-line diag am o he simple es case.
in a la ge sys em.
In o de o e alua e he ol age imbalance on he 𝑗 h bus, his s udy
uses he ollowing indica o om [58]:
𝜓𝑉𝑗= 100 max
𝜒∈{𝑎𝑛,𝑏𝑛,𝑐 𝑛}(|𝑉𝑗 ,𝜒 −𝑉𝑗 ,𝑎𝑣𝑔|
𝑉𝑗 ,𝑎𝑣𝑔 ),
whe e 𝑉𝑎𝑣𝑔 is he a e age o he RMS phase- o-neu al ol ages. A
simila indica o is p oposed o measu ing he eac i e powe sha ing
pe phase:
𝜓𝑄𝜒= 100 max
ℎ∈𝑔(|𝑄ℎ,𝜒 −𝑄𝑎𝑣𝑔 ,𝜒 |
𝑄𝑎𝑣𝑔 ,𝜒 )∀𝜒∈ {𝑎, 𝑏, 𝑐}.
A no able dis inc ion is ha he eac i e powe sha ing indica o as-
sesses he disc epancy in he amoun o eac i e powe injec ed by each
IBR in o a phase, a he han he imbalance be ween phases.
In bo h case s udies, he sensi i i y ma ix 𝐇is calcula ed using a
pe u b-and-obse e app oach [48, Appendix B]. This equi es coo di-
na ion be ween he g id- o ming IBRs. In his wo k, i is assumed ha
he coo dina ion is done h ough a oken ha can only be held by one
de ice a a ime. The p ocess is as ollows:
1. The cen al con olle eques s he alue o he con ol ac ions
o each g id- o ming VSC and he alue o he measu emen s o
in e es , hus es ablishing a base case: 𝐮0,𝐲0.
2. The cen al con olle gi es he oken o he 𝑘 h IBR, which
pe o ms a pe u ba ion on i s con ol ac ion: 𝛥𝑢𝑘
3. The cen al con olle eco ds he a ia ion o he con ol ac ion,
he e ec on he measu emen s o in e es and de e mines he
inc emen s wi h espec o he base case: 𝛥𝑢𝑘,𝛥𝐲=𝐲−𝐲0. The
ela ionship be ween ou pu j and inpu k de e mines he 𝑘, 𝑗- h
alue o he 𝐇-ma ix: 𝐻𝑗 ,𝑘 =𝛥𝑦𝑗
𝛥𝑢𝑘
This p ocedu e ends when all g id- o ming IBRs in he ne wo k ha e
ecei ed he oken. In he e en o a new g id- o ming de ice being
connec ed o he sys em, he seconda y laye mus be in o med so ha
i s con ibu ion o equency and ol age egula ion can be conside ed.
The sensi i i y de e mina ion p ocess mus hen be epea ed in o de
o quan i y he upda ed ma ix alues. This app oach ensu es ha no
pa ame e s a e se manually, he eby minimising human in e en ion
in he p ocess.
4.1. Illus a i e example: simple es case
The sys em comp ises wo g id- o ming IBRs ha eed he wo
ends o a low- ol age ou -wi e eede . Fo simplici y and wi hou
loss o gene ali y, i is assumed ha bo h IBRs ha e iden ical a ed
powe wi h e e ence ol ages equal o he MG a ed ol age. The e
a e wo buses loca ed along he eede . A h ee-phase symme ical load
is connec ed o one o hem, while a single-phase load be ween he
phase 𝑎and he neu al wi e is connec ed o he o he . Fig. 3shows
he one-line diag am o he desc ibed sys em, and Table 1p o ides
a summa y o he main MG pa ame e s. The cables ha comp ise he
eede co espond o he UG3/3-ph cables wi h he impedance ma ix
de ailed in [57]. I is impo an o highligh ha he physical geome y
o he o e head lines gi es ise o dispa a e induc i e e ec s be ween
all phases. Mo eo e , he longes eede sec ion is he one be ween he
loads, which p esen s a signi ican challenge i powe sha ing be ween
g id- o ming IBRs is pu sued.
Table 1
Pa ame e s o he simple es case.
Pa ame e Value
Ra ed ph- o-ph ol age (RMS) 400 V
Leng h o line 1–250 m
Leng h o line 2–380 m
Leng h o line 3–450 m
Powe load 𝑆212 +𝚥7kVA/phase
Powe load 𝑆320 +𝚥7kVA/phase
The balanced e sion o he con ol algo i hm was he subjec o
a p e ious analysis in [49]. In pa icula , ha wo k co e ed he AGC
pe o mance, he con e gence o he OFO o a model-based OPF and i s
pa ame isa ion. The e o e, he aim o he nex subsec ions is o discuss
all he speci ic di e en ial ac o s o he p esen ed me hodology wi h
espec o he p e ious balance e sion.
4.1.1. Deg ee o imbalance and compa ison wi h he OPF wi h comple e
in o ma ion
In his ini ial es , he e icacy o he algo i hm in egula ing he
ol age o a a ge bus while main aining he eac i e powe sha ing pe
phase o he IBRs is e alua ed. The p ima y objec i e is o demons a e
he capaci y o he algo i hm o adap and con ol a ge buses wi h
di e en deg ees o imbalance in powe consump ion. The benchma k
sys em is ini ia ed in a s eady s a e, wi h no equency de ia ion, he
phase EMFs o bo h IBRs a e equal o 1 pu, and he cen alised OFO
algo i hm is disabled. A his ope a ing poin , he complex ol ages
𝑎𝑛, 𝑏𝑛, 𝑐 𝑛and 𝑛𝑔 on he load buses a e:
𝑉2=[210.7∠-5.9◦234.9∠-123.2◦225.3∠119.6◦9.5∠34.7◦]⊤𝑉 ,
𝑉3=[211.3∠-6.1◦231.6∠-123.5◦220.8∠119.4◦8.6∠45.1◦]⊤𝑉 ,
whe eas he IBR phase eac i e powe s a e:
𝑄1=[7.24 2.17 1.87]⊤k a , 𝑄4=[8.23 4.76 4.50]⊤k a .
A he ime ins an 𝑡= 5s, he cen alised OFO-based ol age
egula ion algo i hm is enabled wi h he ollowing pa ame e s: 𝛼=
1.2 × 10−4,𝛽ℎ= 400,𝛾ℎ,𝑗 = 100,∀ℎ, 𝑗∈𝑔. The a ge ol age modules
a e se o 230 V o phase- o-neu al, 400 V o phase- o-phase and 0
V o neu al- o-g ound. No e ha he communica ion pe iod o he
seconda y laye is one second. The a ge bus, i.e. he MG bus whe e
he ol age imbalance is con olled, is he bus 2 om 𝑡= 5s o
𝑡= 60 s and bus 3 om his ins an un il he end o he simula ion.
No e ha his a ge bus change causes a small ansien in he ac i e
powe , a ibu able o he esis ance o he cables. Howe e , his e ec
is conside ed negligible o he AGC. The e o e, he esul s o his es
ocus on he egula ion capabili ies o he cen alised OFO algo i hm.
Fig. 4(a) illus a es he phase- o-neu al ol age e olu ion o bo h buses
(solid line) and he compa ison wi h he same model-based OPF esul
(do ed line).
When bus 2 is a ge bus, he OFO-based algo i hm b ings he
sys em o an ope a ing poin cha ac e ised by he ollowing condi ions:
𝑉2=[230.6∠-4.8◦232.8∠-123.5◦229.8∠119.0◦7.6∠33.1◦]⊤𝑉 ,
𝑉3=[231.4∠-5.0◦229.1∠-122.8◦225.4∠119.1◦6.5∠47.3◦]⊤𝑉 ,
𝑄1=[7.44 3.51 3.31]⊤k a , 𝑄4=[7.60 3.43 3.24]⊤k a .
Likewise, he load ol ages and he IBR eac i e powe s when he
a ge bus is he bus 3 a e he ollowing:
𝑉2=[230.1∠-4.9◦236.2∠-123.4◦233.9∠119.4◦7.8∠33.1◦]⊤𝑉 ,
𝑉3=[230.9∠-5.0◦232.7∠-122.8◦229.6∠119.2◦6.8∠46.5◦]⊤𝑉 ,
𝑄1=[7.49 3.53 3.25]⊤k a , 𝑄4=[7.58 3.41 3.28]⊤k a .
Fig. 4(b) illus a es he e olu ion o he con ol ac ions compu ed by
his laye . The ansi ion o he op imum is smoo h, he eby a oiding
se e e ansien s. Finally, Table 2collec s he indica o s’ alues a he
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
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Fig. 4. Simple es case: simula ion esul s compa ing he p oposed con olle (solid lines) wi h classical OPF (do ed lines). Colou s iden i y he phases 𝑎( ed), 𝑏(g een), and 𝑐
(blue).
Table 2
Value o indica o s o simple es : e ec o a ge bus.
𝑉2,𝑎𝑣𝑔 [V] 𝜓𝑉 ,2[%] 𝑉3,𝑎𝑣𝑔 [V] 𝜓𝑉 ,3[%] 𝜓𝑄,𝑎[%] 𝜓𝑄,𝑏[%] 𝜓𝑄,𝑐 [%]
ic 223.64 5.78 221.23 4.68 6.42 37.34 41.3
𝑏𝑡=2 231.06 0.73 228.66 1.42 1.00 1.18 0.94
𝑏𝑡=3 233.41 1.41 231.05 0.69 0.60 1.79 0.52
h ee s eady s a e ins an s indica ed as: ic (ini ial condi ions), 𝑏𝑡= 2
(bus 2is a ge ed), and 𝑏𝑡= 3(bus 3is a ge ed).
The esul s indica e ha he a e age ol age o he a ge bus
ends owa ds i s nominal alue and ha he ol age unbalance is
conside ably educed, i espec i e o he imbalance o he load con-
nec ed o he a ge bus and he asymme ical impedances o he
cables. Fu he mo e, imp o ing he ol age quali y on he a ge bus
yields bene i s o he o he bus as well. The pe -phase eac i e powe
sha ing be ween he g id- o ming IBRs is enhanced simul aneously
since he eac i e powe indica o s a e signi ican ly educed. Finally,
i is no ewo hy ha he sys em is d i en o an op imal ope a ing
poin , he eby achie ing he same esul as he classical OPF. No e ha
he classical OPF is e alua ed o line and equi es an accu a e model
(including opology and impedance alues) o he dis ibu ion ne wo k
and a o ecas o pe u ba ions (loads and non-con olled gene a o s).
On he con a y, he OFO-based algo i hm is un online wi h eal- ime
measu emen s and he knowledge o he ma ix 𝐇iden i ied by he
pe u ba ion-and-obse e me hodology.
4.1.2. Pe o mance wi h cons ained a iables
The objec i e o his es is o analyse he pe o mance o he OFO-
based algo i hm when an ope a ional limi is eached. To acili a e he
compa ison o he esul s, wo ime-domain simula ions a e conduc ed:
one in which he ope a ing limi s a e elaxed ( o p e en hem om
being eached) and one in which he ope a ing limi s a e igh ened. In
pa icula , i is equi ed ha he phase cu en s injec ed by he g id-
o ming IBR o bus 1 in s eady s a e do no exceed a alue o 0.6 pu.
Hence, his cons ain is inco po a ed in o he seconda y con ol laye ,
bu no in o he p ima y laye s o he basic con ol o he VSC, hus
allowing o ansien exceedance o his alue.
The sys em ini ia es in s eady-s a e, wi h he a o emen ioned ini ial
condi ions. Once again, he e is no de ia ion in equency, and he
phase EMF o bo h IBRs a e equal o 1 pu. The OFO-based cen alised
con olle is ac i a ed a 𝑡= 5s wi h bus 2 as a ge bus, and he
ollowing pa ame e s: 𝛼= 1.2 × 10−4,𝛽ℎ= 400,𝛾ℎ,𝑗 = 100,∀ℎ, 𝑗∈𝑔.
The ol age e e ences o he a ge buses a e he same as in he es
p esen ed in 4.1.1.
Table 3
Value o indica o s o simple es : cons ained a iables.
𝑉2,𝑎𝑣𝑔 [V] 𝜓𝑉 ,2[%] 𝜓𝑄,𝑎[%] 𝜓𝑄,𝑏[%] 𝜓𝑄,𝑐 [%]
Relaxed 231.25 2.31 1.40 2.31 1.91
Cons ained 233.62 3.33 24.92 28.47 53.08
Once he con olle is enabled, a 𝑡= 5s he sys em con e ges o a
poin ha does no each he limi . A 𝑡= 60 s, he e is an inc ease in he
powe consumed in he single-phase load connec ed o bus 2, eaching
𝑆2,𝑎 = 24 +𝚥11 kVA. Fig. 5illus a es he beha iou o he injec ed
cu en s in each scena io. Each phase is ep esen ed in a subplo . The
solid lines ep esen he s ongly cons ained case, while he uzzy lines
co espond o he elaxed case. The ed colou is used o iden i y he
g id- o ming IBR injec ions o bus 1, and he blue colou is used o
iden i y he g id- o ming IBR injec ions o bus 4.
As illus a ed in Fig. 5, in he elaxed case, he s eady-s a e cu en
o phase 𝑎o he IBR on bus 1 is obse ed o each a alue o
0.7 pu. The e o e, o educe he cu en injec ed in o phase 𝑎when
he cons ain is applied, i is necessa y o he IBR o bus 4 o inc ease
i s con ibu ion in phase 𝑎and al e he cu en s o he o he phases.
This has an e ec on bo h he ol age and he eac i e powe . Upon
he occu ence o he load change, a ansien o e cu en (wi h espec
o 0.6 pu) is obse ed o he IBR on bus 1. I is no ewo hy ha he
con olle is able o e u n he ope a ing poin o a easible egion
wi hin a single i e a ion.
The indica o s e alua ed in he inal s a e o his es , p esen ed
in Table 3, demons a e he consequences o his al e a ion, which
g ea ly a ec s he eac i e powe sha ing. This ac is consis en wi h
he weigh ings speci ied in he algo i hm.
4.2. La ge case s udy: CIGRE benchma k ne wo k
The ne wo k used o es he algo i hm on a ealis ic scale sys em
is ha p esen ed in he CIGRE Eu opean LV benchma k dis ibu ion
ne wo k [57], wi h ce ain modi ica ions. The ne wo k p ese es he
opology o he p e ious wo k [49]. I is a adial ne wo k comp ising
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
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Fig. 5. Simple es case. E olu ion o he RMS alue o injec ed cu en s by he g id- o ming IBRs in bo h cons ained and uncons ained scena ios (cons ain s:
𝐼𝑠,1,𝑎𝑏𝑐 = 0.6p.u.).
Fig. 6. MG one-line diag am based on he CIGRE Eu opean LV benchma k dis ibu ion
ne wo k [57] including he conside ed g id- o ming IBRs.
41 buses and 8g id- o ming IBRs dis ibu ed ac oss 3 eede s ( esiden-
ial, indus ial, and comme cial), as depic ed in Fig. 6. I should be
no ed ha he MG ope a es islanded om he main g id.
A 24-hou simula ion is conduc ed o e alua e he po en ial impac
ha he p oposed OFO-based con ol algo i hm may ha e on a la ge
ime and spa ial scale. The p oposed daily load cu es, as p oposed
in [57, Sec ion 7], ha e been used wi h upda es e e y 15 min. Ini ially,
a andom dis ibu ion o he o al a ed bus load is made be ween he
h ee phases and hen i e ol es ollowing he a o emen ioned daily
load cu es.
The con ol ac ions upda es compu ed by he seconda y laye a e
communica ed e e y one second. The OFO con olle pa ame e s a e
he ollowing: 𝛼= 5 × 10−7,𝛽ℎ= 5 × 104,𝛾ℎ,𝑗 = 102and he a ge buses
a e R16 and C07. Wi h his choice, one a ge bus is loca ed a one
end o he eede , while he o he a ge bus is loca ed a an in e me-
dia e poin . The e o e, wo loca ions wi h di e en cha ac e is ics a e
assessed a he same ime. Finally, no e ha his pa ame isa ion o he
seconda y con olle p io i ises he ol age powe quali y wi h espec
o he eac i e powe sha ing o he IBRs.
The objec i e o his s udy is o e alua e he OFO capabili ies
in coo dina ing g id- o ming IBRs in unbalanced ne wo ks. Figs. 7
and 8p esen s he esul s ob ained o he ol age indica o s in he
a ge buses and he eac i e powe sha ing esul s, espec i ely. The
s a is ical da a on he le side o Fig. 7 ep esen he a e age ol age o
he a ge buses, measu ed du ing he 24-hou es , wi h he OFO con-
olle (blue) and wi hou i ( ed). The alue o he ol age unbalance
indica o , compu ed on he same buses, is p esen ed on he igh side
o Fig. 7. Whe eas, Fig. 8p o ides a s a is ical ep esen a ion o he
4-indica o calcula ed o each phase o he exis ing IBRs conside ing
all he 24-hou simula ion.
I is well-known ha he objec i es o con olling ol age and eg-
ula ing eac i e powe a e inhe en ly con adic o y. No e ha ol age-
ela ed objec i es ha e been de ined wi h g ea e weigh in he OFO
pa ame e s, gi en ha in he absence o a egula o , unde ol age
and unbalance condi ions abo e he pe mi ed h eshold a e obse ed.
Consequen ly, o main ain a high-quali y ol age a he a ge buses,
he eac i e powe sha ing de e io a es in some phases when compa ed
o he base case whe e he OFO is no implemen ed. Howe e , hanks
o he ma hema ical op imisa ion amewo k, i is gua an eed ha he
eac i e powe de ia ion be ween phases is he minimum equi ed o
imp o e he ol age.
The eac i e powe injec ion equi emen s a e modi ied depending
on he loca ion o he IBRs. Pa icula ly, hose IBRs loca ed a he end
o he eede end o injec mo e eac i e powe han hose loca ed
nea he seconda y subs a ion. As p e iously iden i ied in [49], he e
is a pe iod o ime du ing he day when he eac i e powe exchange
be ween he eede s is no easible due o ol age limi a ions, esul ing
in he sac i ice o he eac i e powe sha ing capabili y o he IBRs.
Despi e he sligh de e io a ion in eac i e powe sha ing, i should
be no ed ha his does no cause o e loads on any IBR. Fu he mo e,
he bene i s in ol age quali y o he a ge buses a e e iden . I should
be no ed ha he OFO con olle is capable o aising he a e age
ol ages o hei nominal alue o 230 V and main aining a low le el o
imbalance be ween he phase ol ages as shown in Fig. 7. O pa icula
in e es is he case o bus R16, whe e he OFO con olle success ully
a oids he 3% ol age unbalance speci ied in some s anda ds o weak
g ids [59].
5. Conclusion
This pape has p esen ed a con ol scheme o he coo dina ion o
g id- o ming IBRs wi hin an AC MG, aking in o accoun he e ec s o
unbalanced consump ion and asymme ic ci cui s. The me hodology is
ounded upon a hie a chical wo-laye con ol a chi ec u e. The p i-
ma y con ol laye is implemen ed locally on each g id- o ming IBR and
se es o emula e he pe o mance o a synch onous gene a o , o e ing
ac i e powe sha ing and equency and ol age suppo . The se poin s
o each g id- o ming IBR a e calcula ed by he seconda y con ol laye ,
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
8
J.C. Oli es-Camps e al.
Fig. 7. Vol age indica o s ob ained om 24-hou CIGRE benchma k ne wo k es .
Fig. 8. Reac i e powe indica o ob ained om 24-hou CIGRE benchma k ne wo k es .
which is based on wo independen s ages. Fi s ly, a con en ional AGC
is esponsible o de e mining he ac i e powe se poin s equi ed o
egula e he equency o he g id. Secondly, an OFO-based con olle
is employed o egula e he phase ol ages o he buses and he eac i e
powe sha ing be ween he con ollable IBRs in an op imal manne .
The p oposed con ol scheme o e s nume ous ad an ages o MGs
domina ed by powe elec onics:
1. The sys em exploi s he apid esponse o he VSCs, wi h al-
go i hms equi ing minimal compu a ional esou ces and no
ex ensi e communica ion channels, hus enabling quick upda es
o he se poin s.
2. I is no a p e equisi e o possess an accu a e ne wo k model,
which is ad an ageous in he con ex o MGs, whe e he p ecise
con igu a ion is o en unknown.
3. The sys em is designed o each an op imal ope a ing poin by
upda ing con ol ac ions based on sys em measu emen s a he
han consump ion o ecas s. The eedback na u e o he sec-
onda y laye algo i hms ensu es a high obus ness o he sys-
em o unce ain y, as he sys em is able o adap o changing
condi ions.
4. An op imisa ion p oblem o mula ion o unbalanced ol age
egula ion based on phase- o-neu al and phase- o-phase ol age
measu emen s ci cum en s he necessi y o de e mine he angle
o each phase, and u he mo e, elimina es he equi emen o
synch onous measu emen s.
5. The p ima y con ol laye , which is esponsible o main aining
a s able MG ope a ion, has been de eloped in a decen alised
ashion. Consequen ly, in he e en o a communica ion ailu e,
he g id- o ming IBRs would be capable o main aining he MG
powe ed, hus ensu ing he con inued s abili y o he sys em.
6. I does no equi e knowledge o he in e nal con ol laye s o he
VSCs, which ensu es ha he manu ac u e s’ indus ial p ope y
is main ained.
In o de o demons a e he e icacy and adap abili y o he p o-
posed con ol a chi ec u e, he pape has p esen ed wo es cases. The
esul s demons a e ha he con ol scheme is capable o main aining
he a ge bus ol age a a nea -nominal alue wi h minimal unbal-
ance. Addi ionally, he solu ion exhibi s an app op ia e eac i e powe
sha ing be ween he g id- o ming IBRs.
The au ho s iden i y a p ospec i e a enue o u he esea ch in
de eloping a dis ibu ed communica ion scheme o he con olle ,
which would enhance he con olle communica ion aul ole ance and
scalabili y.
CRediT au ho ship con ibu ion s a emen
J. Ca los Oli es-Camps: W i ing – o iginal d a , So wa e, Me hod-
ology, In es iga ion, Fo mal analysis, Concep ualiza ion. Ál a o Ro-
d íguez del Nozal: So wa e, Fo mal analysis, Concep ualiza ion. Juan
Manuel Mau icio: W i ing – e iew & edi ing, Supe ision, So wa e,
Me hodology, Fo mal analysis, Concep ualiza ion. José Ma ía Maza-
O ega: W i ing – e iew & edi ing, Me hodology, Concep ualiza ion.
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 165 (2025) 110443
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