Enla ged ope a ional a ea o an In e line DC powe Flow con olle ia
adap i e d oop con ol o Mul i-Te minal HVDC sys ems
Mi hamed Pou mi asgha iyan
a,*
, G.B Gha ehpe ian
b
, O iol Gomis-Bellmun
c
,
Da id Campos-Gaona
a
, Panagio is N. Papadopoulos
d
a
Depa men o Elec onic and Elec ical Enginee ing (EEE), Uni e si y o S a hclyde, Glasgow, G1 1XQ, UK
b
Depa men o Elec ical Enginee ing, Ami kabi Uni e si y o Technology, Teh an 15875-4413, I an
c
CITCEA-UPC, Uni e si a Poli `
ecnica de Ca alunya, Ba celona 08028, Spain
d
School o Elec ical and Elec onic Enginee ing, The Uni e si y o Manches e , Manches e , M60 1QD, UK
ARTICLE INFO
Keywo ds:
Adap i e D oop Con ol
HVDC
IDC-PFCs
MMCs
Op imal Powe Flow
ABSTRACT
The e ec i e pe o mance o In e line DC Powe Flow Con olle s (IDC-PFCs) in Modula Mul ile el Con e e s
(MMC)-based High Vol age Di ec Cu en (HVDC) g ids is es ic ed by 1) he cu en limi a ion o he HVDC
cables/lines, 2) he HVDC buses’DC ol ages and 3) he IDC-PFC capaci o ’s DC ol age limi . The pi o al
emedy o his issue is o u ilize an adap i e d oop con ol o he MMCs ha a ies i s d oop gain o maximize
he IDC-PFC ope a ion ange. In his pape , 3D cu es o he IDC-PFC’s impo an cha ac e is ics a e used o
assess he lexibili y o he IDC-PFC con ol. By using his app oach, a new deg ee o eedom o IDC-PFC
con ollabili y is achie ed. The pe o mance o he op imal-adap i e-d oop-con olled s a egy p esen ed in
his pape is alida ed using powe low s udies. The esul s demons a e ha a wide ope a ional a ea is
concei able o he IDC-PFC when his echnique is applied as a combina ion o MMC con e e s’d oop con ol
and IDC-PFC du y cycle.
1. In oduc ion
1.1. Mo i a ions
Fu u e meshed Modula Mul ile el Con e e (MMC)-based High
Vol age Di ec Cu en (HVDC) g ids, a ge ing he in eg a ion o
Renewable Ene gy Sou ces (RESs) in o main powe sys ems p omise
enhanced in eg a ion s abili y, wi hou conce ns ega ding eac i e
powe equi emen s [1–3]. Mo eo e , o enhance he lexibili y o he
Powe Flow (PF) o he MMC-based HVDC g ids, he In e line DC Powe
Flow Con olle s (IDC-PFCs) ha e been p oposed in he esea ch li e -
a u e [4–6]. Al hough IDC-PFCs a e a p omising echnology, he egion
o he IDC-PFC’s ope a ion is somehow limi ed. In ac , he ope a ion o
IDC-PFCs is highly dependen on he cu en limi a ion o HVDC cables/
lines, he DC ol age limi a ions o in e connec ed buses, and he IDC-
PFC capaci o ’s DC ol age bounda ies [5].
The e o e, he main goal o he p esen pape ocuses on in oducing
a new deg ee o eedom o o e come he men ioned issues o he IDC-
PFCs’ope a ion in MMC-based HVDC g ids and enhancing IDC-PFC’s
con ibu ion o DC-PF.
1.2. Li e a u e Re iew
The e is a g owing in e es in in e connec ing Vol age Sou ce Con-
e e s (VSCs) o MMCs o he pu pose o achie ing Mul i-Te minal
HVDC s uc u e (MT-HVDC) g ids, o meshed HVDC g ids. The
meshed HVDC g ids ha e many HVDC cables/lines wi h complex con ol
sys ems and ope a ion modes [7] and [8]. One majo conce n o he MT-
HVDC g ids is Powe Flow s udies. In case o poo con ol, conges ion,
and bo lenecks o e loading would occu in HVDC cables/lines.
Acco dingly, DC-PFCs, which a e equi alen o he Flexible AC T ans-
mission Sys em (FACTS) de ices and ha e he same asks, a e in o-
duced. DC-PFCs can manipula e he DC Powe Flow (DC-PF) equa ions
and b ing abou lexibili y and con ollabili y in powe /cu en low.
Gene ally, DC-PFCs a e known in h ee ca ego ies se ies, cascaded, and
in e line. Among hese h ee ca ego ies, since he IDC-PFCs ha e a
simple con ol sys em, s uc u e, and economic ad an ages, hey a e
unde hea y a en ion [8]. The IDC-PFCs injec DC ol age in se ies in o
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (M. Pou mi asgha iyan).
Con en s lis s a ailable a ScienceDi ec
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems
jou nal homepage: www.else ie .com/loca e/ijepes
h ps://doi.o g/10.1016/j.ijepes.2024.110430
Recei ed 7 Sep embe 2024; Recei ed in e ised o m 10 No embe 2024; Accep ed 15 Decembe 2024
Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
A ailable online 20 Decembe 2024
0142-0615/© 2024 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/ ).
hei in e connec ed HVDC cables/lines by se ing app op ia e con ol
a iables (du y cycle) and consequen ly a y he cu en /powe o hei
in e connec ed HVDC cables/lines.
The e a e se e al me hods o powe sha ing and cu en con ol in
MT-HVDC g ids apa om he p esence o DC-PFCs. One e ec i e way
ha enables he ope a o s o app op ia ely u ilize he capaci y o con-
e e s, is adap i e d oop con ol. In [9], a gene alized d oop-con ol
s a egy has been p oposed ha has he abili y o ope a e unde h ee
possible modes (cons an powe , cons an ol age, and d oop-con olled
ol age-powe ) acco ding o he demand o ope a o s wi h a ional
powe sha ing. The gene ic me hod o [9] pa es he way o easy
maneu e abili y o e h ee di e en ope a ion modes. The au ho s o
[10], ha e p oposed a scheme o he d oop con ol o VSCs ha no only
ac s o each a easonable powe -sha ing bu also a oids con e e s’
powe and DC ol age limi iola ions. The s udied me hod o [10] is
adap i e in which d oop alues o he con e e s a e se au oma ically o
each a s able and sa e ope a ion o he sys em. Mo eo e , in [11], a
d oop con ol s a egy has been in oduced ha conside s equency
de ia ion and powe sha ing acco ding o he cha ac e is ics o he
ol age-cu en - equency ela ionship. Also, in [12], he au ho s ha e
de eloped a new coo dina ed-p edic i e d oop con ol ha a oids High
Vol age Ride-Th ough (HVRT) by ollowing an op imal d oop coe i-
cien . Fu he , he pe o mance o d oop echniques in e ms o powe
sha ing and ansien s abili y, ha e been e alua ed in [13].
The combina ion o d oop con ol o MMCs/VSCs wi h he IDC-PFCs
has some impo an ad an ages, which ha e ne e been s udied in he
p e ious d oop-con olled s a egies. Manipula ing d oop alues o
hose con e e s in which an IDC-PFC is connec ed, can emo e o a
leas mi iga e he limi a ions o he IDC-PFC. Mo e impo an ly,
emo ing hese limi a ions would pa e he way o each a mo e lexible,
widened, and con ollable ope a ion o he IDC-PFCs and he g id as
well. The a o emen ioned es ic ions a e deal wi h pa ially in [5] by
using only one deg ee o eedom ( he du y cycle), howe e , a comple e
me hod o o e coming IDC-PFC limi a ions has no ye been p oposed
in he open li e a u e. The e o e, his pape aims o o e come hose
es ic ions wi h he help o adap i e d oop con ol o IDC-PFC’s in e -
connec ed con e e s. Consequen ly, an adap i e-op imal d oop con ol
s a egy is in oduced, which conside s he men ioned limi a ions and
adap s he con e e s’d oop gains acco dingly o widen he IDC-PFCs’
ope a ion capabili ies.
1.3. Resea ch Objec i es and con ibu ions
The con ibu ions o his pape a e as ollows:
•In sec ion II, an adap i e-d oop-con olled s a egy is p oposed in
which no only he DC ol age and powe limi s o he MMCs/VSCs
a e conside ed bu also akes he limi a ions o he IDC-PFC in o
accoun . Hence, he d oop gains o he MMCs/VSCs a e se in he
p esence o IDC-PFCs wi hin he DC ne wo k. By add essing hese
limi a ions, he du y cycle o he IDC-PFC can be swung ex ensi ely
leading o a lexible ope a ion.
•In sec ion III, he in luence o a iable d oop con ol on he beha io
o an IDC-PFC in a h ee- e minal CIGRE HVDC es g id is e alua ed.
He e, he 3D cha ac e is ics o he IDC-PFC wi h wo au onomous
deg ees o eedom (du y cycle o he IDC-PFC and he a iable
d oop) a e analyzed. A e wa ds, he esul s a e compa ed o hose o
he [5], and he bene i s o he p oposed s a egy a e discussed.
•In sec ion V, an Op imal-Adap i e-D oop-Con olled Powe Flow
(OADC-PF) s udy is conduc ed in a s eady-s a e si ua ion. Op imal
esul s o de e mining adap i e d oop alues mean he MMCs/VSCs
a e used o a ain a success ul IDC-PFC limi a ion emo al.
•Finally, in sec ion VI, dynamic and OADC-PF case s udies a e
p o ided.
2. The IDC-PFC unde adap i e d oop con ol
2.1. Ope a ion P inciples o IDC-PFC
The e a e se e al ecen publica ions ega ding new opologies o
IDC-PFC. Howe e , in his pape , he basic opology p esen ed in [14], is
chosen o ocus undamen ally on he e ec ha he a iable d oop gain
has on he beha io o he IDC-PFC, as well as ob aining i s 3D cha ac-
e is ics in he p esence o du y cycle (D) and he new deg ee o eedom
(KD oop) which p o ides an adap able gain o he d oop con olle .
Fig. 1 (a) illus a es he opology o he IDC-PFC, which is buil up o
a educed dual H-b idge and a DC capaci o . The DC capaci o o he
IDC-PFC can ope a e unde posi i e o nega i e DC ol age, and i s alue
(EC) is dependen on he ol ages o i s in e connec ed buses and he
con ol a iable se ing D. The IDC-PFC is loca ed h ough HVDC cable
(mas e cable) and HVDC cable u(sla e cable) ha injec s
p ede e mined-compensa ing ol ages Us, = (1−D)ECand Us,u= − DEC
in se ies wi h he in e connec ed HVDC cables and u, espec i ely, as
shown in Fig. 1 (b). The pe o mance o he IDC-PFC in ol es
exchanging powe be ween he wo (mas e and sla e) HVDC cables.
Thus, he mo e compensa ing ol age injec ion inc eases h ough he
main HVDC cable , he mo e powe /cu en lowing h ough he main
HVDC cable inc eases, and ice e sa.
The IDC-PFCs a e conside ed mino -sized con e e s in compa ison
wi h VSCs o MMCs. The e o e, he losses o he IDC-PFCs a e sligh
(0.002 % losses o he MMCs’)[15]. Hence, in his esea ch, he IDC-
PFCs a e ega ded o be lossless.
The HVDC cables a e modeled as
π
−Lump model, and he pa ame-
e s Z ,Zu,Y , and Yua e he se ies impedance o HVDC cable , se ies
impedance o HVDC cable u, shun admi ance o HVDC cable , and
shun admi ance o HVDC cable u, espec i ely. Mo eo e , o he
s eady-s a e analysis o he g id, he cable pa ame e s a e R ,G ,Ru, and
Guwhich ep esen he HVDC cables ’s and u’s esis ance and shun
conduc ance, espec i ely. Ne e heless, because he exis ence o he
shun admi ance does no pa icipa e in PF esul s signi ican ly, hus,
hey a e c owded ou o he o mula ions (he ea e ) [16].
Based on Fig. 1 (a), conside ing ha he IDC-PFC is placed be ween
buses i,j, and k, he cu en lowing om bus-i(Ii) has wo possible pa hs
o low be ween he HVDC cable and HVDC cable u. The e a e wo
swi ching combina ions, one o a posi i e cu en passage pai and he
o he one o a nega i e cu en passage pai . The swi ching combina-
ions ep esen ed in Fig. 1 (a) a e called posi i e cu en pai s in which
he swi ches ({S2,S4,S6}) and he diodes ({D1,D3,D5}) a e u ilized. Fo
he nega i e cu en di ec ions, he swi ches ({S1,S3,S5}) and he diodes
({D2,D4,D6}) a e used, o u he in o ma ion e e o [5]. Fo he
cu en di ec ions shown in Fig. 1 (a), he cu en o HVDC cable (Iij)
passes h ough he swi ches ({D1,EC,S4}and {S2,EC,D3}), while o he
cu en o HVDC cable u(Iik) he swi ches ({S2,EC,D5}and {D1,EC,S6})
a e in ol ed. Conside ing, he du y cycle (D) o he closed s a e o he
swi ches ({S2,S4}) o he cu en (Iij) passage, and he closed s a e o he
swi ches ({S2,S6}) wi h he complemen a y du y cycle (1 −D) o he
cu en (Iik) passage, he a e age cu en passing h ough he IDC-PFC’s
capaci o (IC) mus be ze o. Unde he gi en ci cums ances, he a e age
cu en o he capaci o can be ob ained as:
IC=1
T∫T
0
ICd =1
T(− DIij + (1−D)Iik)(1)
The pa ame e Tand ICa e he ope a ion cycle o he IDC-PFC and
a e age IDC-PFC capaci o ’s cu en , espec i ely. The a e age cu en
needs o be ze o (IC=0) o he s able ope a ion o he IDC-PFC. Then,
one can achie e:
IC=0→D=Iij
Iij +Iik
(2)
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
2
Wi h simple KVL/KCL law and neglec ing he shun admi ance (Y), he
cu en s o HVDC cable (Iij) and HVDC cable u(Iik) can be achie ed as
ollows:
I =Iij = − Iji =1
Z
((UDC,i−UDC,j) + (1−D)EC)(3)
Iu=Iik = − Iki =1
Zu
((UDC,i−UDC,k) − DEC)(4)
The pa ame e s UDC,i,UDC,j,UDC,k,EC,I , and Iua e he ol age o bus-i,
ol age o bus-j, ol age o bus-k, IDC-PFC’s capaci o ’s DC ol age,
cu en o HVDC cable , and cu en o he HVDC cable u, espec i ely.
Now, by subs i u ing equa ions (3) and equa ion (4) in o equa ion (2)
and u he manipula ion, he IDC-PFC’s capaci o ’s DC ol age can be
ob ained:
EC=TiUDC,i+TjUDC,j+TkUDC,k(5)
Ti=DZ − (1−D)Zu
Z D2+Zu(1−D)2(6)
Tj=(1−D)Zu
Z D2+Zu(1−D)2,Tk=−DZ
Z D2+Zu(1−D)2
The e ms Ti,Tj, and Tka e he coe icien s ha ela e he dependency o
he IDC-PFC’s capaci o on he du y cycle (D) and i s in e connec ed bus
ol ages (UDC,i,UDC,j, and UDC,k).
Mo eo e , he powe s o he IDC-PFC in e connec ed HVDC cables/
lines ( and u) a e exp essed, as ollows:
Ps, =Us, I ,Ps,u=Us,uIu;Us, = (1−D)EC,Us,u= − DEC(8)
PExchange
IDC−PFC =Ps, +Ps,u(9)
The symbols Ps, and Ps,ua e he manipula ed powe o he HVDC cable ,
and he manipula ed powe o he HVDC cable u, espec i ely. Addi-
ionally, he pa ame e PExchange
IDC−PFC p esen s he exchanged powe be ween
he HVDC cables and u.
Acco ding o he equa ion (3) and Fig. 1 (b), inc easing ol age in-
jec ion in o HVDC cable (Us, = (1−D)EC)), will inc ease he cu en
lowing h ough he cable. Mo eo e , based on he ac i e powe balance
condi ion, any inc emen in HVDC cable cu en will educe he cu en
o HVDC cable u, and ice e sa. Wi h he gi en desc ip ion and based
on equa ion (9), a posi i e alue o PExchange
IDC−PFC means he powe is
exchanged om he HVDC cable u o he HVDC cable , and ice e sa.
2.2. Necessi y o In eg a ing IDC-PFC cha ac e is ics in o he adap i e
d oop gains o he MMCs/VSCs
The IDC-PFC ope a ion has some es ic ions. Rega ding he ac ha
he IDC-PFC only ope a es by one deg ee o eedom (du y cycle), i
migh ace some limi a ions including in e connec ed DC ol ages o i s
in e connec ed buses, IDC-PFC capaci o ’s DC ol age iola ion, and
cu en limi a ions. Hence, he du y cycle o he IDC-PFC would no be
allowed o swing ho oughly om 0 o 1 o a leas mo e ex ensi ely.
The e o e, o a oid and escape hese es ic ions as much as possible,
adap i e d oop con ol o he IDC-PFC’s in e connec ed con e e s
could be a p ope solu ion. The adap i e d oop could mi iga e his issue
a some le el. The e o e, in he p esen pape , a new s a egy o adap i e
d oop con ol wi h he conside a ion o he IDC-PFCs is p oposed and
he impo an cha ac e is ics o he IDC-PFC wi h a iable d oop a e
analyzed.
2.3. D oop Concep
Conside ing he imbalance o he in low powe and ou low powe ,
he gene alized d oop con ol equa ion o he con e e wi h he d oop
(KD oop) is p esen ed below:
Fig. 1. (a) IDC-PFC ins alled be ween HVDC cables and uin a meshed HVDC g id, and (b) IDC-PFC modeled by dependen ol age sou ces h ough i s in e -
connec ed HVDC cables and u.
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
3
(P*−P) + KD oop(U*
DC −UDC) = 0 (10)
In (10), he pa ame e s UDC,U*
DC,Pand P*a e he DC ol age, DC ol age
e e ence, powe , and powe e e ence, espec i ely. Mo eo e , he
symbol KD oop ep esen s he d oop gain o an MMC. Acco ding o (10),
an MMC-HVDC e minal migh ope a e unde one o he h ee modes:
cons an powe , cons an ol age, and ol age-powe d oop-con olled
as shown in Fig. 2. The cons an powe mode main ains he powe o he
con e e a a cons an e e ence (KD oop =0) ega dless o he DC
ol age swing. Fo he cons an DC ol age mode, he con e e keeps
he ol age a a ixed amoun (KD oop =∞) ega dless o powe change.
Finally, he d oop-con olled mode is a combina ion o he wo p e ious
con ol modes in which he DC ol age o he con e e changes ac-
co ding o a d oop alue (slope, m) o changing a speci ic powe o he
con e e [11].
2.4. In eg a ing IDC-PFC in o he o mula ions o adap i e d oop o he
MMCs/VSCs
Gene ally, he e a e some conside a ions o adap i e d oop con ol
o MMCs/VSCs. Fo example, e en hough he ixed d oop gains o he
con e e s a e chosen in acco dance wi h hei a ings, his does no
imply ha hey a e ope a ing a hei ull capaci y. In o he wo ds,
usually, he e would be some a ailable head oom o sha ing he addi-
ional powe imbalance. As such, one applica ion o adap i e d oop
con ol is ela ed o achie ing easonable powe sha ing among con-
e e s p e en ing con e e s’powe limi iola ions. Mo ing o wa d,
he second applica ion is he ol age de ia ion o con e e s which is
a ained by p ope d oop swinging, which won’ le he ol age iola ion
happen. Hence, based on he gi en concep s, he below coe icien s a e
conside ed o MMC’s/VSC’s d oop gain ha is no in connec ion wi h
any IDC-PFC:
KAdap i e
D oop,i=λKD oop
Pmax − |Pi|
⏞⏟⏟⏞
A
σ
− |ΔUDC,i|
⏟⏞⏞⏟B
;ΔUDC,i=1− |UDC,i|(11)
The e m Aindica es he a ailable head oom o he i- h con e e , while
he e m Bdeno es he DC ol age de ia ion o he i- h con e e om i s
e e ence alue and i is only conside ed o ha e a
σ
=5 % de ia ion. The
symbol λis a use -de ined ac o o ha e u he con ol o e he d oop
alue. In his equa ion, when he con e e ope a es nea i s ull ca-
paci y, he amoun o Agoes o ze o and causes he con e e o ope a e
a cons an powe mode (KAdap i e
D oop =0). Fu he mo e, when he con-
e e ope a es close o i s DC ol age limi , he e m Bdec eases and
his causes he con e e o ope a e a cons an ol age mode (KAdap i e
D oop =
∞).
On he o he hand, whene e he IDC-PFC is connec ed o a d oop-
con olled con e e , he p oblem o limi a ions o he IDC-PFC’s ope -
a ion could be a oided by assigning op imum adap i e d oop con ol.
The e o e, he ollowing e ms a e also de ined o b ing he limi a ions
o he IDC-PFC in o he adap i e d oop gain o mula ions whe e he IDC-
PFC is ins alled:
KAdap i e
D oop,i=λiKD oop,i×
(Pmax − |Pi|
⏞⏟⏟⏞
Ai
)×((Imax, − |I |) + (Imax,u− |Iu|))
⏞⏟⏟⏞
Ci
(
σ
− |ΔUDC,i|
⏟⏞⏞⏟
Bi
)×(EC,n− |EC|)
⏟⏞⏞⏟
d
(12)
KAdap i e
D oop,j=λjKD oop,j×(Pmax − |Pj|
⏞⏟⏟⏞
Aj
)×(Imax, − |I |)
⏞⏟⏟⏞
Cj
(
σ
− |ΔUDC,j|
⏟⏞⏞⏟Bj
)×(EC,n− |EC|)
⏟⏞⏞⏟d
(13)
KAdap i e
D oop,k=λkKD oop,k×(Pmax − |Pk|
⏞⏟⏟⏞
Ak
)×(Imax,u− |Iu|)
⏞⏟⏟⏞
Ck
(
σ
− |ΔUDC,k|
⏟⏞⏞⏟Bk
)×(EC,n− |EC|)
⏟⏞⏞⏟d
(14)
The pa ame e s EC,n,I ,max, and Iu,max a e he nominal alue o he IDC-
PFC capaci o ’s DC ol age, he maximum cu en o HVDC cable ,
and he maximum cu en o HVDC cable u, espec i ely. The abo e
equa ions (12–14) ep esen he adap i e d oop gains o he buses (i- h,
j- h, and k- h) which a e in connec ion wi h an IDC-PFC. The e ms Ai,
Aj, and Aka e esponsible o easonable powe -sha ing o he con-
e e s i- h, j- h, and k- h, espec i ely. Mo eo e , he e ms Bi,Bj, and
Bk o he con e e s i- h, j- h, and k- h, o de ly, a e conside ed o
p e en ing DC ol age iola ions. Also, he symbols λi,λj, and λka e
use -de ined ac o s.
While he e ms Ai,Aj, and Aka e associa ed wi h he powe s o
con e e s, he e ms Ci,Cj, and Cka e de ined o ela e he d oop gains
o he con e e s o he cu en s o he IDC-PFC’s in e connec ed cables.
Fo he adap i e d oop gain o i- h con e e (KAdap i e
D oop,i), he e m Ci
embeds he cu en limi a ions o he HVDC cables and u(I ,max and
Iu,max) in o he adap i e d oop gain o i- h con e e which is in
connec ion wi h HVDC cables and u. The e m Ci educes he d oop
gain (KAdap i e
D oop,i( o p e en he i- h con e e om ha ing a majo
con ibu ion o powe -sha ing when he cu en s o he IDC-PFC’s
in e connec ed HVDC cables a e bo lenecked. In o he wo ds, when he
HVDC cables and ua e o e loaded, he e m Ci ends o all o cing he
i- h con e e o ope a e as a cons an powe bus (KAdap i e
D oop,i=0) no o
abso b/injec powe /cu en anymo e. This p ocedu e also happens o
he adap i e d oop gains (KAdap i e
D oop,jand KAdap i e
D oop,k) by he e ms Cjand Ck o
he j- h, and k- h con e e s, espec i ely.
In addi ion, since he IDC-PFC capaci o ’s DC ol age (5) is depen-
den on he DC ol age o he buses i- h, j- h, and k- h, he e m dis he
same o he adap i e d oop gains o all IDC-PFC’s in e connec ed
con e e s. I he alue o ECis close o i s limi (EC,n), he d oop gains
ise (KAdap i e
D oop,i=KAdap i e
D oop,j=KAdap i e
D oop,k=∞) e using he DC ol age limi o
IDC-PFC’s capaci o (EC) o be iola ed.
A his ime, conside ing he d oop alue o bus-iin (12), he e ec o
he p oposed nomina o (Ai×Ci) and he denomina o (Bi×d) on he
Fig. 2. Th ee possible ope a ion modes o MMCs: (a) Cons an ol age mode, (b) Cons an powe mode, and (c) D oop-con olled mode.
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
4
o e all d oop alue can be in es iga ed. Fig. 3 shows ha when he
nomina o (Ai×Ci) is low (close o ze o), he con e e is wo king close
o i s nominal powe a ing ( he e o e d oop is low) and should no
ecei e/injec mo e powe , no ma e wha he denomina o is. Based on
Fig. 3, he less he e m (Ai×Ci) is, he less he d oop gain will be, which
means he loading o he bus-iand i s in e connec ed cables/lines will be
less. This will help he IDC-PFC no o be apped in HVDC lines/cables
and con e e limi s. On he o he hand, when he denomina o (Bi×d)
is low (close o ze o), he d oop alue ends o ise apidly o a oid
ol age de ia ion o mo e han he speci ied amoun , as seen in Fig. 3.
This will p e en he IDC-PFC’s capaci o ’s ol age om iola ing i s
limi which e en ually will lead o u he ope a ional a ea o he IDC-
PFC. A las , hese ad an ages will enla ge he ope a ional a ea o he
IDC-PFC. The es o he d oop alues o IDC-PFC’s in e connec ed buses
(KAdap i e
D oop,jand KAdap i e
D oop,k) expose he same beha io .
Mo eo e , he IDC-PFC’s capaci y o exchange powe in his s udy is
8 MW (EC×IC). Since bo h he IDC-PFC capaci o ’s DC ol age and i s
cu en a e conside ed in he p oposed d oop gains, iola ing he IDC-
PFC capaci y limi a ion o exchange powe is p e en ed.
In p e ious DC powe low s udies wi h he p esence o IDC-PFC
[5,7,14,15,17], only he magni ude o he IDC-PFC capaci o ’s DC
ol age (EC) was con olled by swinging he du y cycle (D) wi hin a
limi ed ope a ional a ea. Mo eo e , he d oop-con ol powe -sha ing
s a egies [9–13] ha e no conside ed he p esence o any kind o IDC-
PFC o s udy he bene i s. Howe e , in his pape , o a oid es ic ions
on he IDC-PFC’s ope a ion, adap i e d oop is exe ed as a new deg ee o
eedom.
3. IDC-PFC ope a ion analysis unde du y cycle and a iable
d oop
3.1. Th ee- e minal HVDC es G id: A compa ison s udy o [5]
The h ee- e minal HVDC es g id o CIGRE is s udied in his sec ion
speci ically o compa e he esul s o he p oposed d oop s a egy (as a
new deg ee o eedom o con ibu e o he IDC-PFC’s ope a ion) o he
esul s o [5] (which only had conside ed du y cycle o he IDC-PFC
ope a ion), see Fig. 4. The g id in o ma ion is de i ed om [5]. The
HVDC bus-3is a cons an powe bus (connec ed o an o sho e wind
a m) wi h a powe o 800 MW, while he bus-1is a slack bus. The HVDC
bus-2is connec ed o an onsho e-side con e e . The aims o his s udy
a e lis ed below:
•Fi s ly, since he IDC-PFC’s mas e ( ) and sla e (u) HVDC cables a e
connec ed o he bus-2(which a ec s bo h cables’cu en s), he
d oop coe icien o he bus-2is conside ed o be a iable (which was
cons an powe bus in [5] wi h 400 MW powe ) o s udy he e ec o
a iable d oop on he beha io o he IDC-PFC. Fo his pu pose, all
he possible ope a ion o he sys em is swep up o a ious easible
d oop gains. The e o e, he main cha ac e is ics o he IDC-PFC a e
ex ac ed wi h wo deg ees o eedom (du y cycle and d oop gain).
•Secondly, he ad an ages o he p oposed s a egy a e compa ed o
he esul s o [5], and he b oadened ope a ional ou es a e analyzed.
In o he wo ds, i is shown ha he sys em can ope a e in an
ex ensi e ou e i he d oop gain is chosen app op ia ely.
•Finally, a igu e ha seg ega es he widened ope a ional a ea caused
by IDC-PFC and he combina ion o IDC-PFC and a iable d oop is
p o ided. This igu e illus a es he widened ope a ional a ea in
de ail.
3.2. DC powe Flow o Th ee-Te minal es HVDC g id in he p esence o
IDC-PFC wi h a iable d oop
In his sec ion, DC Powe Flow (DC-PF) o he h ee- e minal HVDC
es g id in he p esence o a iable d oop gain is p esen ed. Conside ing
ha he Bus-1is a slack-bus, he DC-PF can be s a ed as ollows:
UDC,1−U*
DC,1=0,slack bus (15)
I2−P2
UDC,2
=0,I3−P*
3
UDC,3
=0 (16)
I1−I12 −I13 =0,I2+I12 +I23 =0,I3+I13 −I23 =0 (17)
1
R12
(UDC,1− (1−D)EC−UDC,2) − I12 =0 (18)
1
R13
(UDC,1−UDC,3) − I13 =0,1
R23
(UDC,3−UDC,2−DEC) − I23 =0 (19)
P2= − KD oop,2× (UDC,2−U*
DC,2)(20)
P*
3=−UDC,2UDC,3−UDC,3DEC+U2
DC,3
R23
+−UDC,1UDC,3+U2
DC,3
R13
(21)
Fo he gi en DC-PF equa ions, he pa ame e s U*
DC,1,U*
DC,2,P*
3,R12,R13,
and R23 a e slack-bus e e ence DC ol age, he DC ol age e e ence
alue o adap i e-d oop-con olled bus-2, cons an powe o bus-3
(o sho e wind a m), and esis ances o cable-1,cable-2, and cable-3,
espec i ely. Mo eo e , conside ing he ac i e powe balance o he IDC-
PFC, he ela ion be ween du y cycle (D) and he d oop o he bus-2
(KD oop,2) can be s a ed as ollows:
P2=UDC,2I2= − KD oop,2× (UDC,2−U*
DC,2),I2D−I12 =0 (22)
KD oop,2=UDC,2I12
D(U*
DC,2−UDC,2)(23)
The p esen ed DC-PF is sol ed by he New on-Raphson me hod [14] in
s eady-s a e condi ions.
3.3. 3D analysis o he IDC-PFC Cha ac e is ics: Du y cycle and d oop
gain a e au onomous deg ees o eedom
In his sec ion, he con ibu ion o a iable d oop as a new deg ee o
eedom on he IDC-PFC’s beha io is analyzed and he esul s a e
compa ed o hose o he [5], see Fig. 5 and Fig. 6. The ollowing 3D
cu es a e he ou pu s o DC-PF s udies sweeping all he possible du y
cycle and d oop gain o bus-2(in MATLAB 2023b: m. ile coding).
By compa ing he esul s o he IDC-PFC cha ac e is ics in Fig. 5
(wi h only du y cycle as a con ol a iable) and Fig. 6 (wi h du y cycle
and a iable d oop as con ol a iables) in b ie , he ollowing ad an-
ages a e achie ed:
•Compa ing Fig. 5 (a) and Fig. 6 (a), he DC ol age o bus-2is
swingable wi hin he ange 203871 V ≤UDC,2≤205991 V(as he
a iable d oop in e enes as an addi ional deg ee o eedom), while
Fig. 3. 3D esponse o he p oposed adap i e d oop (KAdap i e
D oop,i) conce ning
powe /cu en and ol age de ia ion om nominal alues.
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
5
i was 201854 V ≤UDC,2≤202674 V when only he du y cycle was
in cha ge o he IDC-PFC ope a ion.
•Compa ing Fig. 5 (b) and Fig. 6 (b), he IDC-PFC’s capaci o ’s DC
ol age is ully swingable wi hin i s a ed alue (as he a iable
d oop in e enes as an addi ional deg ee o eedom), while i was
0 V ≤ |EC| ≤ 3155 V when only he du y cycle was in cha ge o he
IDC-PFC ope a ion.
•Compa ing Fig. 5 (c) and Fig. 6 (c), he IDC-PFC’s exchangeable
powe be ween HVDC cable and HVDC cable uis swingable wi hin
he ange −0.36699 MW ≤PExchange
IDC−PFC ≤6.45541 MW (as he a iable
d oop in e enes as an addi ional deg ee o eedom), while i was
−1.23579 MW ≤PExchange
IDC−PFC ≤1.89367 MW when only he du y cycle
was in cha ge o he IDC-PFC ope a ion. Acco ding o he esul , he
ange o exchanged powe has mo ed o he posi i e pa , which
means mos powe is ans e ed om he HVDC cable u o he HVDC
cable .
•Compa ing Fig. 5 (d) and Fig. 6 (d), i is concluded ha he swing-
abili y o he HVDC cable ’s DC cu en , has changed om
−2000 <I <−1000 A (descending) o −2000 <I <2000 A (as
he a iable d oop in e enes as an addi ional deg ee o eedom).
The swing-abili y o he HVDC cable u’s DC cu en has shi ed om
−2000 <Iu<−1000 A (ascending) o −254 ≤Iu≤574 A (as he
a iable d oop in e enes as an addi ional deg ee o eedom),
espec i ely. Based on he esul s, he ope a ion ange o he HVDC
cable uis sligh ly educed wi h a iable d oop and du y cycle (828 A)
which was (1000 A) wi h only du y cycle con ol. Howe e , he
HVDC cable is ope a ing a ull capaci y wi h he a iable d oop and
du y cycle (4000 A) which was (1000 A) wi h only du y cycle
con ol.
•Mo e impo an ly, he du y cycle can be swung wi hin 0.45268 ≤
D≤0.88248 (as he a iable d oop in e enes as an addi ional de-
g ee o eedom), while i was wi hin 0.36854 ≤D≤0.68871 whe e
he only con ol a iable was he du y cycle. Fu he mo e, he
adap i e d oop can be swung wi hin 14.66527 ≤KD oop,2≤
34.98027 conside ing sys em cu en and ol age limi a ions.
Finally, o ob ain an o e all ou look o e he ope a ional a ea o he
h ee- e minal es HVDC g id, he ollowing igu e which seg ega es he
widened ope a ional a eas hanks o he p esence o IDC-PFC and bo h
IDC-PFC wi h d oop con ol s a egy is illus a ed, see Fig. 7. The g ey
squa es a e ep esen a i e o he ini ial ope a ional a ea whe e he e is
no ins alled IDC-PFC. Mo eo e , he whi e squa es a e he cu en lim-
i a ions o he HVDC cables. Since he IDC-PFC edis ibu es ex a cu -
en s o o e loaded HVDC cables/lines o he neighbo hood HVDC
cables/lines, he whi e squa es become ope able. In he nex , he blue
squa es a e indica ing he IDC-PFC capaci o ’s DC ol age limi (EC,n).
Wi hou an app op ia e d oop se ing o he IDC-PFC’s in e connec ed
con e e s, he IDC-PFC capaci o ’s DC ol age migh iola e i s limi .
Howe e , by assigning adap i e d oop gains o he IDC-PFC’s in e -
connec ed con e e s, he blue squa es become ope able. As i was
shown in Fig. 6 (b) he IDC-PFC capaci o ’s DC ol age was comple ely
ope able hanks o he se ing app op ia e d oop gain o he IDC-PFC’s
in e connec ed con e e s, he ope a ional a ea o he sys em expanded
Fig. 4. Th ee- e minal es HVDC g id equipped wi h an IDC-PFC [5].
Fig. 5. IDC-PFC cha ac e is ics wi h du y cycle (one au onomous con ol a iable) [5]: (a) DC ol age o bus-2, (b) IDC-PFC’s capaci o ’s DC ol age, (c) Exchange
powe be ween mas e HVDC cable and sla e HVDC cable u, and (d) he cu en s o mas e and sla e HVDC cables and u.
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
6
( he blue squa es became ope able), see Fig. 7. The da kes g ey squa es
a e non-ope a ional a eas whe e he powe limi a ions o he con e e s
will no allow u he ope a ion o he sys em.
4. S a e-Space HVDC sys em Modeling: Con ol s uc u e &
s abili y analysis
4.1. S a e-Space Modelling o Mul i-Te minal HVDC g id wi h con ol
s uc u e
To assess he accu acy, e iciency, and ole o he p oposed s a egy
in escaping IDC-PFC’s limi a ions, a con ol s uc u e needs o be
de ised. Fo he dynamic simula ions (in MATLAB-Simulink), he elec-
omagne ic linea ized ansien model is implemen ed.
Taking he h ee- e minal HVDC g id as an example in Fig. 4, he
s a e-space equa ions a e as ollows:
S a e-Space Equa ions:
To gene a e s a e-space equa ions, a iables a e conside ed o be
ope a ing a hei assumed linea ized poin s:
X≃X0+dX
d (24)
Fig. 6. IDC-PFC cha ac e is ics wi h a iable d oop and du y cycle ( wo au onomous con ol a iables): (a) DC ol age o bus-2, (b) IDC-PFC’s capaci o ’s DC ol age,
(c) Exchange powe be ween mas e HVDC cable and sla e HVDC cable u, and (d) he cu en s o mas e and sla e HVDC cables and u.
Fig. 7. Widened Ope a ional a ea hanks o he combina ion o a iable d oop and du y cycle.
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
7
The pa ame e s X,X0, and dX/d a e a iable, linea ized poin , and
a ia ion o he ela ed a iable, espec i ely. The HVDC ansmission
cables/lines a e simula ed in
π
−model, whe e Rij and Lij a e he esis-
ance and induc ance o he cable/line, espec i ely. Ne e heless, he
exis ence o he shun esis o s and capaci ances (Y) does no con ibu e
signi ican ly o he esul s. Thus, hey a e elimina ed om he space-
s a e equa ions [16]. Finally, he linea ized model o he p esen ed
space-s a e equa ions is depic ed as (25).
dX
d =AX +BU (25)
The di e en ial equa ions desc ibing he modeled h ee- e minal es
VSC-HVDC g id a e illus a ed as ollows:
dUDC,1
d =1
Cʹ
1
(P1
UDC,10
−UDC,1P10
U2
DC,10
−I12 −I13) ≈ 0,slack −bus (26)
dUDC,2
d =1
Cʹ
2
(−KAdap i e
D oop,2
UDC,20
UDC,2+I12 +I23)(27)
dUDC,3
d =1
Cʹ
3
(P3
UDC,30
−UDC,3P30
U2
DC,30
+I13 −I23)(28)
dI12
d =1
L12
(UDC,1−UDC,2−R12I12 − (D0−1)EC−EC0D)(29)
dI13
d =1
L13
(UDC,1−UDC,3−R13I13)(30)
dI23
d =1
L23
(UDC,2−UDC,3−R23I23 −D0EC−EC0D)(31)
dEC
d =1
C((D0−1)I12 +D0I23 + (I120 +I230)D)(32)
whe e Aand Ba e s a e and inpu ec o coe icien s, espec i ely:
A=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−P10
Cʹ
1U2
DC,10
0 0 −1
Cʹ
1
−1
Cʹ
1
0 0
0−KAdap i e
D oop,2
UDC,20Cʹ
2
01
Cʹ
2
01
Cʹ
2
0
0 0 −P30
Cʹ
3U2
DC,30
01
Cʹ
3
−1
Cʹ
3
0
1
L12
−1
L12
0−R12
L12
0 0 1−D0
L12
1
L13
0−1
L13
0−R13
L13
0 0
01
L23
−1
L23
0 0 −R23
L23
−D0
L23
0 0 0 D0−1
C
D0
C0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(33)
B=⎛
⎜
⎜
⎜
⎝
1
Cʹ
1UDC,10
1
UDC,20Cʹ
2
0 0 0 0 0
0 0 0 −EC0
L12
0−EC0
L23
I120 +I230
C
⎞
⎟
⎟
⎟
⎠
T
(34)
The pa ame e Xis he s a e ec o and he pa ame e U(U= {P1,
KAdap i e
D oop,2,D}) is he inpu ec o o he s a e-space equa ions. The pa-
ame e s Cʹ
1,Cʹ
2, and Cʹ
3a e he con e e s’i,j, and kHVDC link ca-
paci o s, espec i ely. Mo eo e , he pa ame e C ep esen s he IDC-
PFCs capaci o . The pa ame e (P1) decides how much powe o be
injec ed in o he h ee- e minal HVDC g id. In addi ion, he du y cycle
(D) and he adap i e d oop gain o he bus-i(KAdap i e
D oop,2) a e he con ol
a iables.
Fo he bus-i, he con ol s uc u e ep esen ing equa ion (12) is
gi en, see Fig. 8 (a). In he p oposed d oop con ol, he e ms Ai,Bi,Ci,
and d ep esen he limi s o he con e e -ipowe , cu en s o mas e
and sla e HVDC cables whe e he IDC-PFC is ins alled, DC ol age de-
ia ion o he con e e -i, and IDC-PFC capaci o ’s DC ol age, espec-
i ely. Fig. 8 (a) gene a es he equa ion (12) ollowed by a PI con olle
and a limi e . The d oops o he bus-jand bus-ka e se simila ly
ollowing he equa ions (13) and (14), espec i ely. The PI con olle is
uned/op imized by MATLAB 2023b PI- uning oolbox. Also, he limi e
conside s he d oop gain pe missible ope a ion ou e.
Based on he gi en s a e-space model, he ans e unc ion ha e-
la es he du y cycle (D) o he mas e HVDC cable cu en (I12) is
achie ed (G(s) = I12/D). The e o e, based on he ans e unc ion G(s),
he ollowing con ol s uc u e o mas e HVDC cable (I12) cu en is
shown in Fig. 8 (b). The ollowing con olle consis s o ou componen s
including a PI con olle ( uned by MATLAB Simulink Tuning Toolbox),
a dampe ( o high- equency oscilla ion damping), a limi e (main ains
he du y cycle o ope a ing wi hin 0 ≤D≤1) and inally, he ans e
unc ion ex ac ed by s a e-space equa ions. The con ol sys em o Fig. 8
(b) se s he du y cycle o ollow he gi en e e ence o he mas e HVDC
cable (I12).
Mo eo e , o ensu e he p oposed adap i e d oop con ol s a egy
wo ks well, he measu emen signals mus be as e han he ac ion o
he con ol sys em o sepa a e he dynamics o di e en con ol ac ions.
4.2. S abili y analysis
In his sec ion, he s abili y aspec s o he p oposed adap i e d oop
con olle and mas e HVDC cable cu en con olle a e analyzed o
he h ee- e minal HVDC g id. To make su e he con ol s uc u e wi h
a iable d oop and du y cycle will ope a e smoo hly, he closed-loop
ans e unc ion o he con ol s uc u e is assessed.
Closed - loop ans e unc ion =G(s)K(s)
1+G(s)K(s)(35)
In he ollowing, he pole map o he closed-loop ans e unc ion is
illus a ed, see Fig. 9. The pole map o he closed-loop ans e unc ion
depic s wo se s o poles, one neu al and he o he sensi i e o he
change o bus-2d oop gain. Fo he sensi i e poles, i is seen ha by
swinging he d oop alue wi hin he iden i ied pe missible ope a ion
Fig. 8. Con ol S uc u e: (a) d oop con ol s uc u e o he bus-iand (b)
mas e HVDC cable cu en con ol s uc u e.
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
8
ou e (14.66 <KD oop,2<34.98), he poles mo e owa d he s able pa
(mo e nega i e side), which shows ha he p oposed con ol s uc u e is
s able.
One impo an aspec o he p oposed s a egy is he in e ac ion o
he d oop con ol dynamic wi h he du y cycle dynamic. Acco ding o
he con ol s uc u e in Fig. 8 and equa ion (35), he e is only one
ans e unc ion ha ela es he mas e HVDC cable cu en o he du y
cycle. The e o e, he dynamics o he sys em a e dic a ed by he exis ing
ans e unc ion. As he du y cycle imposes i s dynamics on he ou pu
mas e HVDC cable cu en , hese dynamics di ec ly show hemsel es
in he adap i e d oop con ol s a egy. The e o e, since hei dynamics
a e he same, he e a e no in e ac ions be ween he du y cycle and
adap i e d oop con ol s a egy.
5. S eady-S a e Analysis: Op imal-Adap i e ¡D oop-Con olled
powe Flow (OADC-PF)
5.1. Impo ance o applying OADC-PF in he p esence o an IDC-PFC
The p oposed adap i e d oop con ol s uc u e o Fig. 8 ies o adap
he d oop alue o he con e e s o a oid limi a ions o he IDC-PFC.
Howe e , his is no op imal. I is wo hwhile knowing he op imal
alue o he con e e s’d oops. In o he wo ds, he OADC-PF wi h he
op imal esul s means ha he con e e s’d oops a e se op imally,
he e o e, hey con ibu e o he IDC-PFC limi a ion emo al as much as
possible.
Fo sol ing he op imiza ion p oblem, Sequen ial Quad a ic P o-
g amming (SQP) o MATLAB so wa e is u ilized. Mo eo e , Gene ic
Algo i hm (GA) is also used o sol e he op imiza ion p oblem ensu ing
he esul s a e global op imal poin s.
5.2. Powe injec ion model o he IDC-PFC
I is a well-known me hod o gene a e he impac o he IDC-PFC in
DC-PF o mula ions using powe injec ion models (PIMs). The PIM o -
mula ions o he IDC-PFC a e s udied on he basis o he lumped-
π
model o he in e connec ed HVDC lines. The e ec o IDC-PFC in he
DC-PF o mula ions is achie ed by compa ing powe equa ions in he
p esence and absence o IDC-PFC [17], as ollows:
P(i) = − ((1−D)UDC,iEC
R
−DUDC,iEC
Ru
)(36)
P(j) = (1−D)UDC,jEC
R
(37)
P(k) = − DUDC,kEC
Ru
(38)
In he equa ions (36)-(38), P(i),P(j), and P(k)p esen he impac s o an
IDC-PFC ha a e a i icially injec ed in o he buses i,jand k, espec-
i ely. These equa ions ep esen he e ec s o an IDC-PFC on DC powe
low s udies. Fo u he in o ma ion ega ding he modeling p ocess o
an IDC-PFC, e e o [17].
5.3. Mul i-Objec i e unc ion &op imiza ion p oblem
In his pape , wo goals will be ollowed: (i) minimizing he DC
ol age de ia ions o he HVDC g id’s buses and (ii) minimizing he
cu en index o all he HVDC cables and o e head lines. Thus, he DC
ol age de ia ion e m (FV) and he cu en mi iga ion e m (FI) a e
desc ibed as below:
FV=∑
Nbus
i=1
|(UDC,i−1)i−1|;i=1,2,3, ..., Nbus (39)
FI=∑
NLine
=1
|IDC− |
|IDC− ,max|; =1,2,3, ..., NLine (40)
Fo he abo e equa ions, he pa ame e s Nbus and NLine indica e he
numbe o HVDC buses and HVDC cable/o e head lines, espec i ely.
The pa ame e s IDC− and IDC− ,max a e he HVDC cable/ o e head line
cu en and cu en limi a ion, espec i ely.
Fo mo e cla i ica ion, he e m FVdemons a es he HVDC buses’
DC ol age de ia ion om hei e e ence alues (nominal alues),
while he e m FIis o a oiding bo leneck occu ence in HVDC cables/
o e head lines. The mo e de ia ion o he sys ems’ ol ages is om hei
nominal alue, he mo e he sys em has a b oad ope a ional a ea. In
o he wo ds, he sys em mus ha e had a b oade ope a ional a ea o be
able o de ia e sys ems’DC ol ages om hei nominal alues. The e-
o e, because he DC ol ages o he buses a e allowed o swing
(0.95 ≤UDC,i≤1.05), he DC ol age o each bus is sub ac ed o 1 and
hen he accumula ed alue is sub ac ed o 1 again, see equa ion (39).
This will cla i y whe he he DC Powe Flow (DC-PF) is able o ind a
p ope solu ion in a b oade ope a ional a ea.
Mo eo e , he equa ion (40) illus a es he loading o he mul i-
e minal HVDC sys em. The less he e m FIis, he less he loading o
he sys em is. In o he e ms, ha ing a small FImeans he injec ed
powe s om he O sho e Wind Fa m (OWF)s in o he mul i- e minal
HVDC ansmission sys em ha e been dis ibu ed in a easonable
manne . Thus, all he HVDC cables/lines con ibu e o ca ying he
p oduced bulk powe o he onsho e side a he han a small numbe o
HVDC cables/lines ca ying a g ea deal o powe .
5.4. Equali y and inequali y cons ain s
The op imiza ion p oblem is sol ed subjec o se e al equali y and
inequali y cons ain s. The inequali y cons ain s include some ope a-
ional e ms including DC ol age (UDC,i), cu en s (IDC− ), onsho e-side
con e e powe (Pi), du y cycle (D), and IDC-PFC capaci o ’s DC
ol age. On he o he hand, equali y cons ain s a e he powe balance o
he HVDC g id and he IDC-PFC capaci o ’s DC ol age. The inequali y
and equali y cons ain s a e desc ibed as ollows:
Inequali y cons ain s:
Umin <UDC,i<Umax (41)
IDC− ≤IDC− ,max (42)
−Pmax ≤Pi≤Pmax (43)
0≤D≤1 (44)
Emin ≤EC≤Emax (45)
The DC ol age bounda ies a e se o be wi hinUmin =0.95 andUmax =
1.05. In addi ion, he IDC-PFC capaci o ’s DC ol age is allowed o swing
0.05UDC,i(Emin=-0.05UDC,iandEmax =0.05UDC,i). Fu he , he onsho e-
Fig. 9. Poles o closed-loop ans e unc ion o he con ol s uc u e.
M. Pou mi asgha iyan e al. In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 164 (2025) 110430
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