Ci a ion: V al, M.; Fujdiak, R.;
Benedik , J.; P aks, P.; B is, R.; P acek,
M.; Toman, P. Time-Dependen
Una ailabili y Explo a ion o
In e connec ed U ban Powe G id
and Communica ion Ne wo k.
Algo i hms 2023,16, 561. h ps://
doi.o g/10.3390/a16120561
Academic Edi o s: Shuai Li and
Dunhui Xiao
Recei ed: 15 No embe 2023
Re ised: 5 Decembe 2023
Accep ed: 8 Decembe 2023
Published: 10 Decembe 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
algo i hms
A icle
Time-Dependen Una ailabili y Explo a ion o In e connec ed
U ban Powe G id and Communica ion Ne wo k
Ma ej V al 1, Radek Fujdiak 1,* , Jan Benedik 1, Pa el P aks 2, Radim B is 3, Michal P acek 1
and Pe Toman 1
1Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, Technicka 3058/10,
616 00 B no, Czech Republic; [email p o ec ed] (M.V.); [email p o ec ed] (M.P.)
2IT4Inno a ions Na ional Supe compu ing Cen e , VSB—Technical Uni e si y o Os a a,
708 00 Os a a, Czech Republic; [email p o ec ed]
3Facul y o Elec ical Enginee ing and Compu e Science, VSB—Technical Uni e si y o Os a a,
708 33 Os a a, Czech Republic
*Co espondence: [email p o ec ed]
Abs ac :
This pape p esen s a ime-dependen eliabili y analysis c ea ed o a c i ical ene gy
in as uc u e use case, which consis s o an in e connec ed u ban powe g id and a communica ion
ne wo k. By u ilizing expe knowledge om he ene gy and communica ion sec o s and in eg a ing
he enewal heo y o mul i-componen sys ems, a ep esen a i e eliabili y model o his in e con-
nec ed ene gy in as uc u e, based on eal ne wo k loca ed in he Czech Republic, is es ablished.
This model assumes epa able and non- epa able componen s and cap u es he opology o he in e -
connec ed in as uc u e and eliabili y cha ac e is ics o bo h he powe g id and he communica ion
ne wo k. Mo eo e , a ime-dependen eliabili y assessmen o he in e connec ed sys em is p o ided.
One o he signi ican ou pu s o his esea ch is he iden i ica ion o he c i ical componen s o he
in e connec ed ne wo k and hei in e dependencies by he di ec ed acyclic g aph. Nume ical esul s
indica e ha he o iginal design has an unaccep able la ge una ailabili y. Thus, o imp o e he
eliabili y o he in e connec ed sys em, a sligh ly modi ied design, in which only a limi ed numbe o
componen s in he sys em a e modi ied o keep he addi ional cos s o he imp o ed design limi ed,
is p oposed. Consequen ly, nume ical esul s indica e educing he una ailabili y o he imp o ed
in e connec ed sys em in compa ison wi h he ini ial eliabili y design. The p oposed una ailabili y
explo a ion s a egy is gene al and can b ing a aluable eliabili y imp o emen in he powe and
communica ion sec o s.
Keywo ds:
sma g id; powe g id; dis ibu ion ne wo k; communica ion ne wo k; in e connec ion;
eliabili y pa ame e s; inciden pa ame e s; una ailabili y quan i ica ion; acyclic g aph
1. In oduc ion
In he ongoing ans o ma ion o he ene gy and communica ion in as uc u e,
a ma ked ansi ion om cen alized o decen alized sys ems can be obse ed. Con-
cu en ly, he e is a subs an ial deploymen o sma de ices designed o ne wo k man-
agemen , con ol, and moni o ing, u he enhancing he in e connec ion be ween he
ene gy and communica ion ne wo ks. This enhanced in eg a ion is c ucial, especially in
he con ex o dis ibu ion ne wo ks. These ne wo ks a e essen ial in ensu ing a s eady
supply o elec ici y o households, businesses, and indus ies. I is essen ial o men ion
ha any dis up ion in his supply can esul in signi ican inancial implica ions. No only
do Dis ibu ion Sys em Ope a o s (DSO) lose p o i s om such in e up ions, bu hey also
ace po en ial penal ies imposed by egula o s. The e iciency and eliabili y o dis ibu ion
ne wo ks is he e o e o g ea impo ance o bo h consume s and DSOs.
The eliabili y and consis ency o elec ici y dis ibu ion a e key cha ac e is ics, and
hei e ec i eness is measu ed using speci ic me ics, namely he Sys em A e age In e up-
Algo i hms 2023,16, 561. h ps://doi.o g/10.3390/a16120561 h ps://www.mdpi.com/jou nal/algo i hms
Algo i hms 2023,16, 561 2 o 21
ion F equency Index (SAIFI) and Sys em A e age In e up ion Du a ion Index (SAIDI).
These indices p o ide a quan i iable measu e o he elec ici y’s deli e y quali y o he end
use s. As he ene gy sec o ansi ions o decen alized sys ems and inco po a es in elligen
ne wo k managemen de ices, i is e iden ha communica ion de ices play a subs an ial
ole in in luencing hese me ics. Speci ically, de ices such as Remo e Te minal Uni (RTU)
and Ad anced Me e ing Moni o (AMM) a e in oduced. These componen s no only
se e as essen ial in e aces be ween he dis ibu ion ne wo k and he con ol cen e , bu
also ensu e eal- ime moni o ing and con ol o e he en i e g id, encompassing powe
elemen s like ans o me s, lines, and ci cui b eake s.
To enhance he eliabili y o he dis ibu ion sys em, one e ec i e s a egy is op i-
mizing i s main enance wi h ega d o epai cos s, e isions, and he equency o hese
asks. This op imiza ion ask, howe e , becomes conside ably mo e in ica e when wo
in e connec ed in as uc u es, namely he powe and communica ion ne wo ks, a e aken
in o accoun . This pape in oduces a no el me hodology o calcula e he una ailabili y
o he en i e sys em which is c ucial o compu ing SAIFI and SAIDI pa ame e s. Fu -
he mo e, we del e in o he possibili ies o op imizing main enance on a es ne wo k
o hese in e connec ed in as uc u es, laying emphasis on po en ial dependencies and
in e ac ions du ing hei ope a ion. On he basis o p io esea ch, a ma hema ical model
employing he Weibull dis ibu ion is adop ed o e alua e sys em con ingencies wi hin his
combined in as uc u e.
In his esea ch, a key objec i e is o s anda dize he e minology associa ed wi h
eliabili y pa ame e s ele an o bo h powe g id and communica ion ne wo k sys ems.
I has been obse ed in a ious publica ions ha hese e ms a e o en misunde s ood o
inaccu a ely labeled. When mul iple in e connec ed sys ems a e aken in o accoun ins ead
o a singula sys em, unde s anding hei in e ela ionships and dependencies becomes
essen ial. To add ess his, he pape aims o cla i y hese ela ionships using Acyclic G aph
(AG), a g aphical ep esen a ion me hod ha has been success ully u ilized by he au ho s
in pas esea ch, as e idenced by [1].
The es o he pape is o ganized as ollows. I begins wi h a discussion o he s a e
o he a in Sec ion 2, whe e an o e iew o ele an li e a u e and he esul s in he
ield a e p esen ed. In Sec ion 3.1, he in icacies o he in e connec ed in as uc u es’ es
ne wo k a e explained. This sec ion p o ides a de ailed desc ip ion o he de ices in hese
in as uc u es and pains akingly delinea es hei mu ual dependencies. As i p og esses
o Sec ion 3.2, he a icle is ocused on he de ini ion o he sys em’s op imal unc ion.
He e, he main enance op imiza ion challenges o such in e connec ed in as uc u es
ake he limeligh , and discussion wi h an in e p e a ion h ough AG is supplemen ed.
In Sec ion 3.3, he me hodology and heo y behind he analysis o inciden - ela ed Key
Pe o mance Indica o s (KPI) a e summa ized. This is pe o med wi h he aim o achie e
a uni ied unde s anding o pa ame e s in cong uence wi h his pape o e a ching esea ch
goals. Sec ion 3.4 ocuses on he me hods used o asce ain he una ailabili y o he
in e connec ed in as uc u e. Pi o al esul s a e p esen ed in Sec ion 4. Sec ion 5concludes
he pape wi h a discussion o he model’s limi a ions and maps he u u e esea ch, which
includes main enance op imiza ion.
2. S a e o he A
The ex ensi e body o li e a u e on he opic unde sco es he key ole o eliabili y and
isk modeling in in e connec ed in as uc u es. This e iew commences wi h a ho ough
assessmen o bo h ounda ional ex s and he mos ele an ecen s udies on he subjec .
In [
2
], he au ho s unde ake an exhaus i e su ey o mo e han 150 pape s pe aining
o Faul T ee Analysis (FTA), he eby ende ing an in-dep h insigh in o he s a e-o - he-
a me hodologies o FTA. This explo a ion no only co e s he adi ional aspec s o he
Faul T ee (FT), bu also discusses i s se e al ex ensions, including Dynamic Faul T ee
(DFT), Repai able Faul T ee (RFT), and Ex ended Faul T ee (EFT). Re e ence [
3
] discusses
he ma hema ical amewo k behind he op imal alloca ion and planning o main enance
Algo i hms 2023,16, 561 3 o 21
pe sonnel, cas ing main enance op imiza ion as a mul i- ace ed op imiza ion p oblem. In
ano he signi ican con ibu ion [
4
], a b oad spec um o eliabili y and isk modeling
echniques is in oduced. These ange om he Con en ional Faul T ee (CFT), E en T ee
(ET), and Bina y Decision Diag am (BDD) o mo e nuanced me hods like Pe i Ne s (PN),
Ma ko Modeling (MM), and A ack T ee (AT), designed o iden i y and mi iga e he la en
isks which a e cha ac e is ic o Cybe -Physical Sys ems (CPS).
In line wi h DFT, a chap e in [
5
] analyzes he me hodology, e iewing popula ech-
niques including Ma ko chains and Bayesian ne wo ks. Sys ema ic e iew [
6
] co e s
Reliabili y, A ailabili y, Main ainabili y, and Sa e y/Secu i y (RAMS) analysis o C i ical
In as uc u e (CI) pape s. Al oge he , 1500 pape s which co e he RAMS opic published
be ween 2011 and 2020 a e analyzed. The a ge applica ions include g id s a ions, cybe -
physical sys ems, cloud compu ing, so wa e-de ined ne wo ks, indus ial con ol sys ems,
and Supe iso y Con ol And Da a Acquisi ion (SCADA) sys ems.
In [
7
], AT a e ca ego ized in o wo dis inc dimensions: (i) p ope ees e sus di ec ed
AG, and (ii) s a ic ga es e sus dynamic ga es. On ano he on , in [
8
], a comp ehensi e
o e iew o eliabili y modeling in CPS is deli e ed. This includes highligh ing he in ica-
cies associa ed wi h eliabili y and aul modeling ac oss he h ee co e componen s o CPS:
ha dwa e, so wa e, and humans. Fu he mo e, he au ho s analyze challenges a ising om
he in eg a ion o hese componen s o o e a holis ic app oach o CPS eliabili y modeling.
Inno a i e pape [
9
] p esen s a me hodology ha ocuses on de i ing epai able mul i-s a e
FT om ime se ies aul da a. This me hod is adep a analyzing non-exponen ial dis ibu-
ions o bo h eliabili y and main ainabili y, and i p o es ins umen al in p edic ing he
sys em’s u u e eliabili y along wi h de ailing he FT s uc u e. Las ly, in [
10
], a en ion
is cen e ed on he gene a ion o DFT o sys ems ha inco po a e edundancies. These
edundancies, commonplace in sa e y-c i ical sys ems, se e he p ima y pu pose o en-
hancing sys em eliabili y. The au ho s’ objec i e in his pape is wo old: i s , o in oduce
a edundancy p o ile, and second, o pa e he way o he au oma ic gene a ion o DFT
based on sys em models.
In ecen yea s, a ocus has been placed on so wa e ools ailo ed o he eliabili y
analysis o sys ems. The 2017 epo on open-sou ce FTA ools, as men ioned in [
11
],
highligh ed se e al ools. Thei indings, upda ed o e lec he cu en s a us, lis he
ollowing ools:
•OpenFTA
(Open-sou ce): OpenFTA, an FTA ool, aids in comp ehending and apply-
ing FTA, a me hod in sa e y enginee ing o quali a i e and quan i a i e e alua ion o
CI sys em eliabili y and sa e y. Howe e , ha ing no been upda ed o o e a decade,
i s ele ance in con empo a y applica ions migh be ques ionable.
•OpenAl aRica
(Res ic ed ee access): OpenAl aRica ocuses on isk analysis o
in ica e sys ems using he Al aRica language, a high- ie language c a ed o he
RAMS analysis o sys ems. Ca e ing o as models, i encompasses bo h quali a i e
and quan i a i e examina ion ins umen s.
•Faul T ee Analyse
(Demo e sion a ailable): A segmen o ALD’s sui e ailo ed
o eliabili y enginee ing and isk e alua ion, his ool o e s a isual in e ace o
cons uc ing and sc u inizing FT. I is enginee ed o deduce he likelihood o a p incipal
e en om he p obabili ies o ounda ional e en s.
•Isog aph Faul T ee+
(7-day ial): C a ed by Isog aph, Faul T ee+ is a leading appli-
ca ion o c ea ing FT and execu ing quali a i e and quan i a i e FTA. I s use - iendly
in e ace is equipped o manage a ange o logic ga es and inciden s. Indus ies such
as de ense, ae ospace, nuclea , and ail ha e in eg a ed Isog aph’s so wa e sui e in o
hei ope a ions.
•I em Toolki
(30-day ial): This sui e o e s ools essen ial o eliabili y p edic ions—
Failu e Modes and E ec s Analysis (FMEA), FTA, and o he eliabili y enginee -
ing unde akings. Designed o analyzing bo h udimen a y and ad anced sys-
ems, i aids enginee s ac oss domains, om elec onics o mechanics, o gauge hei
designs’ eliabili y.
Algo i hms 2023,16, 561 4 o 21
•DFTCalc
(Open-sou ce): Essen ially a “DFT Calcula o ”, DFTCalc specializes in DFT
analysis. Di e ing om con en ional FT ha employs jus AND and OR ga es,
DFT encapsula es e en sequences and in ica e in e dependencies. Sc ip ed in C++,
DFTCalc yields me ics o eliabili y and a ailabili y o such ees.
Despi e a weal h o esea ch, signi ican disc epancies in he in e p e a ion o KPI
pe sis ac oss se e al publica ions [
12
–
25
]. Wi h an in en o add ess and ec i y hese incon-
sis encies, in pape [
26
], he au ho s p opose a ime-dependen eliabili y analysis ailo ed
o a eal c i ical ene gy in as uc u e use case, which consis s o in e connec ed u ban
elec ical and communica ion ne wo k eliabili y assessmen o highly eliable elemen s,
which le e ages exac eliabili y quan i ica ion o highly eliable sys ems. The so wa e
p esen ed in his pape can quan i y he eliabili y o e y eliable sys ems up o 10
−45
,
which was demons a ed on a Highly Reliable Ma ko ian Sys em (HRMS) benchma k
and also success ully compa ed wi h MOCA-RP so wa e [
26
]. Di e en ia ing om ea lie
s udies, he cu en in es iga ion o his pape ’s au ho s emphasizes a obus eliabili y
quan i ica ion, which is applied o he eal in e connec ed ene gy in as uc u e. The in-
e connec ed in as uc u e is based on he eal sys em loca ed in he Czech Republic and
co e s wo dis inc in as uc u es: he powe g id and he communica ion ne wo k. Fo
illus a i e pu poses, he pape in oduces a no el e sion o an in e connec ed in as-
uc u e es ne wo k p oposed in [
27
]. Fu he , in he ealm o ime-dependen eliabili y
analysis using AG, pe inen algo i hms o disc e e main enance op imiza ion o in ica e
mul i-componen sys ems ha e been p esen ed in [
26
]. Gi en he cu en discou se in he
s a e o he a , he e is an unequi ocal demand o specialized so wa e ools ha can han-
dle he ime-dependen eliabili y o highly eliable in e connec ed sys ems. Pu suing his
need, he main aim o his pape is o inco po a e he compu a ions ela ed o una ailabili y
in o a p e-exis ing simula o , which has been elabo a ely discussed in [28].
3. Me hodology
This sec ion is de o ed o he in oduc ion o he model o in e connec ed in as uc-
u es, he me hodology and heo y used in de e mining KPIs, and he issue o AGs.
3.1. Desc ip ion o Rep esen a i e In as uc u e
The opology o he chosen es ne wo k, which encompasses in e connec ed in as uc-
u es, is illus a ed in Figu e 1. The ne wo k is made up o wo au onomously unc ioning
in as uc u es: he powe dis ibu ion ne wo k and he communica ion ne wo k. Comp e-
hensi e de ails o each ne wo k a e elabo a ed upon in he ollowing subsec ions. I is im-
pe a i e o no e ha he e a e wo unique dependencies be ween hese
wo in as uc u es.
•
The eliance o he communica ion in as uc u e on he consis en pe o mance o
he dis ibu ion ne wo k. This is due o he wi eless ansmi e s ha sou ce hei
powe om he Low Vol age (LV) le el o he dis ibu ion sys em. Speci ically, hey
de i e powe om LV busba s o he Dis ibu ion T ans o me (DT) labeled DT1, DT2,
and DT3.
•
The second dependency eme ges om he in eg a ion o speci ic RTU and AMM
de ices wi hin he dis ibu ion ne wo k. When hese de ices mal unc ion, hey can
dis up he dis ibu ion sys em’s ope a ions in wo po en ial ways: (i) di ec ly (by
hinde ing he abili y o con ol swi ching de ices), and (ii) indi ec ly ( h ough ailu es
o me e ing de ices). I is c ucial o no e ha he in luence o AMM de ices on he
dis ibu ion ne wo k’s ope a ion is no aken in o accoun o he pu poses o his
pape because hey a e exclusi ely u ilized o me e ing objec i es (non-di ec impac ).
An RTU is a de ice wi hin indus ial con ol sys ems ha acili a es he emo e moni o -
ing and con ol o a ious p ocesses and equipmen . I s a chi ec u e ypically encompasses
Inpu /Ou pu (I/O) modules, a cen al p ocesso , onboa d memo y, and communica ion
in e aces ailo ed o seamless connec i i y o an a ay o de ices, including bu no lim-
i ed o senso s, eclose s, and load b eak swi ches (LBS). To enhance hei esilience, RTU
Algo i hms 2023,16, 561 5 o 21
o en inco po a es ea u es such as backup powe supplies and edundan communica ion
channels. Ne e heless, hey a e no immune o aul s, and ha e hus been neglec ed o
he pu poses o his pape . These aul s, which can signi ican ly dis u b he ope a ion o
he dis ibu ion sys em, may be based on di e se sou ces including ad e se en i onmen al
condi ions, inhe en ha dwa e and so wa e de ec s, o lapses in communica ion. A com-
p ehensi e unde s anding o hese ac o s, coupled wi h insigh s in o hei po en ial impac
on he p obabili y o RTU ailu es, is impe a i e. Such knowledge no only bols e s sys em
eliabili y, bu also aids in minimizing he isk o powe dis up ions and s eamlining
main enance s a egies. In he con ex o he a o emen ioned dependency, he es ne wo k
accoun s o he ollowing ou dis inc RTU ypes:
•
RTU ins alled in Medium Vol age (MV) swi chboa ds a Dis ibu ion T ans o me
S a ions (DTS), e e enced as RTU1.1–4 in Figu e 1.
•
RTU se ing as he con ol mechanism o eclose s on MV lines, deno ed as RTU2.1–4.
• RTU posi ioned wi hin he High Vol age (HV)/MV subs a ions, labeled as RTU3.1.
•
RTU unc ioning as he moni o ing and command uni o sec ion load b eak swi ches,
iden i ied as RTU2.5.
SCADA
Da a cen e
in as uc u e
SCADA ope a ions cen e
RTU2.1
AMM2
COS 1
RTU3.1
RTU2.2
RTU2.5
RTU1.3
RTU1.4
RTU1.1
RTU1.2
AMM1
T1
DT1
DT2
DT3
DT4
R1
R2
R3
LBS1
22 kV
0.4 kV
0.4 kV 22 kV
0.4 kV
0.4 kV
22 kV
22 kV
RTU2.3
22 kV
110 kV
O e head powe lines
Unde g ound powe cables
Wi eless da a connec ion
Fibe op ic da a cabling
R4
RTU2.4
ER 1
Edge Rou e
COS 2
ER 2
Op ical Line Te minal
Passi e Op ical
Spli e
WAN
Figu e 1.
Conside ed ep esen a i e in e connec ed in as uc u es o powe and communica ion
ne wo ks in a Sma G id domain.
In DTS, he RTU is esponsible o moni o ing digi al s a es such as swi ch posi ions
and doo con ac s. I enables emo e con ol o eede swi ches, o e s di ec measu emen
o eede s, and can de ec aul s wi hin he dis ibu ion ne wo k. Mo eo e , he RTU
assesses powe quali y and accumula es da a om o he elec onic ins umen s p esen in
he ins alla ion. In he con ex o eclose s, he RTU showcases s a us indica o s like eclose
s a us and doo con ac . I allows o bo h emo e and local con ol, u nishes 3-phase
ol age and cu en measu emen s, and is equipped wi h he capabili y o de ec aul s
on powe lines. Beyond hese unc ionali ies, he RTU can handle au oma ic ope a ions
inclusi e o p o ec i e elays, eclose s, and he managemen o blocking condi ions. Fo
HV/MV subs a ions, RTU plays a c ucial ole in b idging communica ion wi h he SCADA
sys em. This communica ion is ypically acili a ed ia E he ne LAN o op ical links, bu
he e is an op ion o a cellula modem backup when needed. The RTU is adep a e ie ing
da a om a ious subs a ion de ices, no ably p o ec i e elays and powe quali y me e s.
When i comes o LBS, he RTU ensu es emo e and local con ol o he swi ch, p o ides
3-phase ol age and cu en eadings, and de ec s aul s on powe lines. An ad anced

Algo i hms 2023,16, 561 6 o 21
ea u e includes ini ia ing au oma ic unc ions, no ably disconnec ing a e iden i ying
a sho ci cui du ing a ol age- ee pause, as well as he egula ion o blocking condi ions.
3.1.1. Powe G id Topology
The es ne wo k’s powe g id pa is a ep esen a ion o a sec ion o he ac ual
dis ibu ion ne wo k in he Czech Republic, s uc u ed wi h a ing opology, which can be
iewed in Figu e 2.
110 kV 22 kV 22 kV 0.4 kV
22 kV
22 kV
22 kV
0.4 kV
C3
C2
R2
R1
R4
LBS1
R3
L2
L1
L3
L4
T1
DT1 DT3
DT4
DT2
L5
0.4 kV
0.4 kV
C1
Figu e 2. Conside ed ep esen a i e opology o he powe dis ibu ion ne wo k.
Wi hin his g id, he e a e h ee dis inc ol age le els: he HV segmen (110 kV), he
MV segmen (22 kV), and he LV segmen (0.4 kV). Powe is supplied o his ne wo k om
a subs a ion ha is equipped wi h a 110/22 kV ans o me . Fu he mo e, he ne wo k
consis s o i e indi idual sec ions o o e head lines (L1–L5). Each o hese sec ions is
ou i ed wi h a eclose (R1–R4) o in e up aul cu en s. The o e head line (L5) is
pa i ioned by a sec ion swi ch named LBS1. Unde no mal ( aul - ee) condi ions, his
LBS1 is ypically in an open s a e. Hence, he ne wo k uns in a adial manne o educe
sho -ci cui cu en s. Howe e , i a aul occu s ( o ins ance, in Sec ion L1), sec ion
LBS acili a es he DSO o ini ia e eeding om he o he di ec ion. This LBS is ope a ed
emo ely, and i s communica ion is o ches a ed by an RTU, speci ically labeled as RTU2.5
in Figu e 1. This posi ions RTU2.5 as a pi o al componen o he ope a ion o LBS1. In he
e en o communica ion o i RTU2.5 mal unc ions, immedia e es o a ion o he supply is
impossible due o a aul in he dis ibu ion ne wo k. Such a scena io esul s in an ex ension
o he aul du a ion, subsequen ly impac ing he SAIDI and SAIFI me ics. Mo eo e , he
ne wo k houses ou DTs (22/0.4 kV) ha ans o m MV o LV, ensu ing he demands o
he end consume s a e me ia sub e anean LV cables (C1–3). These cables di e ge a he
LV le el busba and a e in ended o di e en consume g oups. This node highligh s he
dependency o he powe ne wo k on he communica ion ne wo k, as p e iously discussed.
3.1.2. Communica ion Ne wo k Topology
The communica ion ne wo k model, as shown in Figu e 3, in eg a es RTU clien
de ices, segmen ed in o wo ca ego ies based on hei connec ion echniques. The inaugu al
ca ego y u ilizes ibe op ics. De ices in his g oup a e in e linked ia a Passi e Op ical
Spli e (POS), an Op ical Line Te minal (OLT), and ul ima ely an Edge Rou e (ER). These
ou e s in e ace wi h he Wide A ea Ne wo k (WAN), which can ei he be p op ie a y o
he company o a public In e ne ne wo k. In he case o he la e , communica ion be ween
he RTU and he se e is enc yp ed and sa egua ded by a p i a e Vi ual P i a e Ne wo k
(VPN) unnel.
Con e sely, he seconda y connec ion me hod aps in o wi eless cellula modali ies
such as Global Sys em o Mobile Communica ions (GSM), Long-Te m E olu ion (LTE),
Algo i hms 2023,16, 561 7 o 21
o he Fi h-gene a ion b oadband cellula ne wo k (5G) b oadband cellula s anda d.
RTU con igu ed wi h a mobile ne wo k in e ace is dependen on a modem. All ansmis-
sions a e enc yp ed, le e aging VPN unnels o o i ied secu i y, p edominan ly when
ansi ing public mobile ne wo ks. Occasionally, he choice migh lean owa ds a p i a e
mobile ne wo k. This WAN is di ec ly linked o bo h a SCADA ope a ional hub and a da a
eposi o y hos ing he equisi e se e in as uc u e o dialogues wi h he clien RTU.
While VPN unnels ha e he po en ial o culmina e a an ER, high-demand ha dwa e
scena ios migh necessi a e a se e exclusi ely dedica ed o VPN liaisons. The ole o he
SCADA sys em ex ends o supe ising and o ches a ing ope a ions o clien ne wo ks as
well as RTU. Figu e 3 ep esen s he di e se connec i i y modali ies p esen in he sys em.
The wi eless linkages a e ep esen ed by W (
W1
h ough
Wn
). In con as , op ical condui s
a e ca ego ized based on hei spa ial alignmen , ei he abo e-g ound o sub e anean. The
ae ial op ical pa hways a e anno a ed as AO (
AO1
–
AOn
), and he sub e anean op ical
coun e pa s bea he BO (BO1–BOn) no a ion.
SCADA
Da a cen e
in as uc u e
COS 2COS 1
Edge Rou e
ER 2ER 1
Op ical Line Te minal
Base T anscei e S a ion
WAN
Wi eless RTUs
W
AO1AOn
1
AO6
BO1
BO2
AO2
W
RTUs wi h
op ical cable
connec ion
...
...
1
1
n
n
Passi e Op ical
Spli e
SCADA ope a ions cen e
AO3
BO3
BO4
AO7
AO4
AO5
n
Wi eless connec ion
Bu ied ibe op ic cable
Ae ial ibe op ic cable
Figu e 3.
Conside ed ep esen a i e opology o he con ol and communica ion ne wo k pa
including he mos common componen s.
3.2. Desc ip ion o Ne wo k Func ionali y and Ne wo k Con ingency Quan i ica ion
Unin ended powe ou ages a e in a iably unwelcome due o he associa ed cos s a is-
ing om he du a ion o he dis up ion. Componen s o CI a e signi ican ly dependen on
elec ical powe . In he e en o powe dis up ions o hese CI componen s, he consequen
inancial consequences can be signi ican . Mo eo e , ex ended o la ge-scale ou ages can
b ing in no jus inancial de imen s bu also social, cybe , and o he mul i ace ed issues.
The op imiza ion e o o his pape p ima ily aims o minimize hese associa ed losses
and cu ail he h ea isk associa ed wi h he mal unc ion o CI’s c ucial componen s, as
de ailed in [
28
]. To del e deepe in o he isk conce ning he powe ailu e o such pi o al
componen s and consequen ly enhance he o e all sys em’s eliabili y, he pape cla i ies
hese issues using he es ne wo k spo ligh ed in Sec ion 3.1. To commence, i is impe a i e
o de ine he p ecise ope a ional dynamics o he en i e sys em. In his ega d, his pape
de ines a Poin o Deli e y (POD) ha is supplied om he LV bus ia he C4 cable, which
in u n ca e s o he CI ( his could include key da a cen e s o analogous en i ies). The
o e a ching co ec ope a ion o he sys em is cha ac e ized by he seamless elec ici y
ansi ion om he HV s a um igh down o he CI’s load junc ion a he LV echelon.
Unde his ope a ional de ini ion, speci ic componen s such as DT1, DT4, POS, OLT, and
ele an condui s, which do no play a decisi e ole in he ou lined unc ion, can be a oided
Algo i hms 2023,16, 561 8 o 21
in he holis ic es ne wo k ep esen a ion. The opology o his dis ibu ed ne wo k is
isually ep esen ed in Figu e 4. Gi en he p emise ha eclose s (deno ed as R1–R4) do
no spon aneously in e up he powe supply du ing mal unc ions, hey a e omi ed om
he scheme. None heless, he LBS, pi o al o swi backup powe p o isioning, emains
in eg al and is hus inco po a ed wi hin he schema ic.
Fo accu a e comp ehension o dependencies, i is essen ial o co ec ly in e p e
bo h indi idual sys ems and hei in e connec ions. The comp ehensi e diag am ha
illus a es hese sys ems, e e enced in Figu e 4, can be complex and po en ially con using
o some. Consequen ly, his s udy uses he o ien ed AG modelling, as i b ings clea and
unambiguous ep esen a ion o hese in e connec ed sys ems.
110 kV 22 kV
22 kV
0.4 kV
C3
C2
LBS1
L2
T1
DT3
DT2
L5L3L2
L4L1
BTS WAN
ER1
WAN
COS1
SCADA
COS2
AO4
AO6
RTU
BO3
BO4
BO1W1
AO7
AO5
22 kV 0.4 kV
C4 POD
Figu e 4.
Diag am o in e connec ed in as uc u es simpli ied by omi ing he elemen s wi hou
di ec in luence on he de ined sys em unc ion.
AGs a e g aph s uc u es p e alen in bo h compu e science and ma hema ics. They
consis o nodes and di ec ed edges, ensu ing he e a e no di ec ed cycles. Such a design
makes hem highly e sa ile and in aluable o a mul i ude o applica ions. No only can
hey illus a e complex ela ionships be ween elemen s, bu hey a e also ins umen al in
scheduling asks based on dependencies. In he specialized domain o powe elec ical
enginee ing, AGs play a c ucial ole in a ious unc ions:
•Powe Flow Analysis
: He e, AGs ac as a ep esen a ion o he powe low in a g id.
The nodes wi hin hese g aphs s and o subs a ions, while he edges deno e powe
ansmission lines. By using AGs, enginee s and esea che s can de e mine he mos e -
icien powe low ou es and de ec po en ial bo lenecks wi hin he g id. Fo a deepe
di e in o his applica ion, eade s can e e o [29,30].
•Main enance Op imiza ion
: Main enance wi hin he powe g id o en equi es in-
ica e scheduling o accoun o dependencies and cons ain s. AGs assis in his
endea o by helping o p io i ize asks. Wi h he help o hese g aphs, i becomes
easie o de e mine which asks need immedia e a en ion and ensu e a sys ema ic
and e icien comple ion sequence. Mo e on his can be explo ed in [26,31,32].
•Powe and Da a Ou age Modeling
: AGs also ind hei applica ion in modeling
powe and da a ou ages. In such models, nodes signi y he a ious componen s o
he g id, and edges ep esen he in e - ela ionships be ween hem. Th ough hese
AG-based models, i is possible o swi ly iden i y he p ima y causes o an ou age.
Fu he mo e, hey p o ide a oadmap o an e ec i e esponse s a egy o es o e
ei he powe g ids, as discussed in [33], o da a ne wo ks, as highligh ed in [34].
AG showcases he dependencies p esen wi hin he sys em, allowing o he iden i-
ica ion o pi o al elemen s in eg al o he ull unc ionali y o he sys em. In p inciple,
an o ien ed AG, as he ep esen ed sys em, is s uc u ed h ough nodes and edges, bu
he implica ions o nodes and edges in he con ex o a g aph signi ican ly di e om
gene al in e p e a ions.
Algo i hms 2023,16, 561 9 o 21
In his case, he e is a comp ehensi e b eakdown o he in e p e a ion o hese nodes
and edges in acco dance wi h [26]:
•
Fi s , he g aph is inhe en ly acyclic, ensu ing ha wo di ec ly connec ed nodes sha e
a singula edge.
•
A he op o he AG is a soli a y SS node. This unique node symbolizes he o e all
sys em’s unc ionali y, illus a ing co ec ope a ion agains sys em ailu e.
•
An inhe en di ec ionali y exis s be ween he nodes o he AG, es ablishing he e-
la ionship o subo dina ion be ween hem, delinea ed as a sla e node in ela ion o
a mas e node.
•
An in e nal node, also e e ed o as a non- e minal node, ypi ies he s ochas ic
beha io inhe en wi hin a subsys em. This subsys em is pe cei ed o be in a s a e
o co ec unc ionali y only when a minimum o msubo dina e nodes (which can
ei he be e minal o non- e minal) concu en ly display co ec unc ionali y. This
s ipula ion equi es he in ege m o eside wi hin a speci ic in e al. Speci ically:
–The o al numbe o inpu edges is ma ked n.
–
Fo a si ua ion whe e mequals 1, he in e nal node e ec i ely emula es a logical
OR unc ion.
–
Con e sely, when mma ches n, he in e nal node esona es wi h a logical AND unc ion.
•
The ole o e minal nodes is pi o al as hey symbolize he ope a ional s a us o he
di e se componen s in eg a ed wi hin he sys em. To be p ecise, hese componen s
a e subjec o e en s which can ei he be s ochas ic in na u e o de e minis ic. Fo
hose e en s cha ac e ized by s ochas ici y, hey need o be a icula ed ia dis inc
p obabili y dis ibu ions, speci ically ca e ing o he occu ence o aul s. Addi ionally,
e en s e he ed o main enance, ei he p e en i e o co ec i e, necessi a e clea and
unambiguous speci ica ions.
In he p esen ed case, Node S1 he e o e indica es he s a e whe e his POD in Figu e 4
is supplied. Componen s ha a e essen ial o his s a e o ope a ion include he C4 cable,
he T1 powe ans o me , and he DT2 dis ibu ion ans o me . Fu he mo e, ensu ing
consis en unc ionali y is a supply pa hway, including bo h lines and LBS. An illus a i e
ep esen a ion o he powe g id’s s uc u e is shown in Figu e 5.
3
1
2 4
2
3
1
2 4
2
S1
T1
u1
u2
u3
L1
L4
L2
L3
L5
LBS1
u4
C4
DT2
S1
T1
u1
u2
u3
L1
L4
L2
L3
L5
LBS1
u4
C4
DT2
Figu e 5. AG o he powe g id pa o he simpli ied es ne wo k.
In he es ne wo k, as depic ed in Figu e 4, a simila p ocedu e is adminis e ed
o he communica ion segmen , which is highligh ed in blue. Taking in o accoun he
in e dependencies be ween he wo in as uc u es in he comp ehensi e es ne wo k, he
sys em’s unc ional co ec ness is delinea ed as acili a ing LBS ope abili y ia he Con ol
Algo i hms 2023,16, 561 16 o 21
0 1 2 3 4 5
x 104
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 x 10−4
Una ailabili y
Time (hou s)
4.2
(a) Powe g id (sys em S1).
0 1 2 3 4 5
x 104
0
1
2
3
4
5
6
7
8x 10−4
Una ailabili y
Time (hou s)
7.5
(b) Communica ion ne wo k (sys em S2).
Figu e 10.
Una ailabili y e olu ion o he powe g id (
a
) and he communica ion ne wo k (
b
) wi hin
he mission ime o 5 yea s.
0 1 2 3 4 5
x 104
0
0.2
0.4
0.6
0.8
1
1.2 x 10−3
Time (hou s)
Una ailabili y
SS
S2
S1
Figu e 11.
Compa ison o una ailabili y e olu ion o sys ems SS, S1 and S2 o he case wi h
o iginal pa ame e s.
I has al eady been men ioned in his pape ha he una ailabili y calcula ion can also
be used o op imiza ion asks. This ac was demons a ed in ano he calcula ion whe e
he S1 sys em was imp o ed by placing a new ans o me , T2 (connec ed in pa allel o
ans o me T1 in Figu e 4).
The calcula ion o he e olu ion o he una ailabili y o such an imp o ed sys em (S1
imp o ed) is shown in Figu e 12, whe e compa ison wi h he o iginal S1 and S2 sys ems
can be seen (a).
Subsequen ly, he una ailabili y calcula ion o he whole sys em (deno ed as SS im-
p o ed) is also upda ed, and compa ison wi h he o iginal SS sys em is made (b). This
compa ison shows an imp o emen in he una ailabili y o he S1 imp o ed sys em as he
o e all una ailabili y d opped o 2.48 ·10−4(o iginally 4.2 ·10−4).
All compu a ions we e nume ically compu ed using he high-pe o mance p og am-
ming language MATLAB on compu ing equipmen wi h he ollowing pa ame e s: In-
el (R) Co e™ i7-3770 CPU @ 3.4 GHz 3.9 GHz, 8.00 GB RAM.

Algo i hms 2023,16, 561 17 o 21
0 1 2 3 4 5
x 104
0
1
2
3
4
5
6
7
8x 10−4
Time (hou s)
Una ailabili y
S2
S1
S1 imp o ed
(a) O iginal pa ame e s o sys em S1.
0 1 2 3 4 5
x 104
0
0.2
0.4
0.6
0.8
1
1.2 x 10−3
Time (hou s)
Una ailabili y
SS
SS imp o ed
(b) Imp o ed pa ame e s o sys em S1.
Figu e 12.
Compa ison o he una ailabili y e olu ion wi hin he mission ime o 5 yea s wi h o iginal
and imp o ed pa ame e s o Sys ems S1 and S2 (a) andSys em SS (b).
5. Conclusions
In his pape , a ime-dependen eliabili y assessmen o he in e connec ed sys em by
he enewal heo y was p o ided. Calcula ions o he 5-yea ime e olu ion o una ailabili y
we e pe o med successi ely, i s o he sepa a e sys ems o he powe g id and he
communica ion ne wo k and hen o he o e all in e connec ed in as uc u e.
The p esen ed nume ical esul s show an un a ou ably high impac o he powe ne -
wo k una ailabili y (S1) on he o e all in e connec ed in as uc u e una ailabili y, despi e
he ac ha S1 una ailabili y is lowe han he S2 communica ion ne wo k una ailabili y
(see Figu e 12).
Nume ical esul s o he ime-dependen eliabili y analysis indica e educ ion in
he una ailabili y o he imp o ed in e connec ed sys em in compa ison wi h he ini ial
eliabili y design. I is e iden ha una ailabili y educ ion is pa icula ly caused by design
changes (pa allel duplica ion o T ans o me T1).
Figu e 12 shows ha he una ailabili y cu e o he imp o ed Powe g id S1 is
signi ican ly lowe han hose o bo h S1 o iginal and S2. As a esul o his, a signi ican
una ailabili y educ ion in he In e connec ed ne wo k SS can be obse ed. Thus, he
p oposed design b ings a aluable eliabili y imp o emen , especially o he powe g id
ne wo k o he analyzed in e connec ed ene gy in as uc u e.
The inno a i e cha ac e o he p oposed solu ion can be b ie ly desc ibed in he
ollowing pa ag aphs.
•
Desc ip ion o he no el eal c i ical ene gy in as uc u e use case, which consis s
o in e connec ed u ban powe g id and communica ion ne wo k, was p esen ed.
Pa ame e s o eliabili y and main enance models o componen s we e es ima ed om
he li e a u e e iew and expe knowledge.
•
The de eloped compu a ional model assumes he ageing o componen s, which is
simula ed by he Weibull dis ibu ion. The use o he Weibull dis ibu ion is consis en
wi h eal ailu e da ase s o powe dis ibu ion componen s. Mo eo e , he in e con-
nec ed model also exploi s he obse a ion ha he ime o epai o hese componen s
can be modeled by an exponen ial dis ibu ion.
•
Time-dependen eliabili y assessmen o he in e connec ed use case was pe o med.
The iden i ica ion o he c i ical componen s o he in e connec ed ne wo k and hei
in e dependencies was p o ided by he gene al di ec ed AG. Highly eliable com-
ponen s and in e connec ed ne wo ks we e p ope ly modeled. The so wa e ool
le e aged exac eliabili y quan i ica ion o highly eliable e en s.
•
Resul s indica ed ha he o iginal design has an unaccep able la ge una ailabili y S1.
Algo i hms 2023,16, 561 18 o 21
•
Sligh ly modi ied design o imp o e he eliabili y o he in e connec ed sys em was
p oposed, in which only a limi ed numbe o componen s in he sys em we e modi ied
o keep he addi ional cos s o he imp o ed design limi ed.
•
Nume ical esul s indica ed educ ion in he una ailabili y o he imp o ed in e con-
nec ed sys em in compa ison wi h he ini ial eliabili y design.
•
The p oposed una ailabili y explo a ion s a egy is gene al and can b ing a alu-
able eliabili y imp o emen in in e connec ed sys ems including he ene gy and
communica ion sec o s.
The aim o u he esea ch is o in es iga e he ac o s ha con ibu e o communi-
ca ion equipmen ailu es in elec ici y dis ibu ion sys ems and o quan i y hei ela i e
impo ance. Speci ically, he in luence o en i onmen al condi ions, ha dwa e and so -
wa e ailu es, and communica ion dis up ions ha a ec he eliabili y o he RTU will
be in es iga ed. A ma hema ical model will be de eloped o es ima e he p obabili y o
ailu e o he RTU and assess he impo ance o each ac o . The esul s o his s udy will
p o ide aluable insigh s in o he design and main enance o dis ibu ion sys ems, which
will con ibu e o inc eased eliabili y and educed down ime.
Au ho Con ibu ions:
Concep ualiza ion, M.V. and R.F.; me hodology, M.V., R.F., R.B. and P.P.;
so wa e, R.B.; alida ion, M.V., R.F., R.B. and J.B.; o mal analysis, M.V., R.F., R.B., J.B., P.P., M.P. and
P.T.; in es iga ion, M.V., R.F., R.B., J.B. and P.P.; esou ces, M.V., R.F., R.B. and J.B.; da a cu a ion, M.V.,
R.F., R.B., J.B., P.P. and M.P.; w i ing—o iginal d a p epa a ion, M.V., R.F. and J.B.; w i ing— e iew
and edi ing, M.V., R.F. and R.B.; isualiza ion, M.V. and J.B.; supe ision, R.F.; p ojec adminis a ion,
R.F. and P.T.; unding acquisi ion, R.F. and P.T. All au ho s ha e ead and ag eed o he published
e sion o he manusc ip .
Funding:
This esea ch was unded by he Minis y o he In e io o he Czech Republic (p ojec
No. VK01030109) in g an p og am “Open call in secu i y esea ch 2023–2029”.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : Da a a e con ained wi hin he a icle.
Acknowledgmen s:
This esea ch wo k was ca ied ou in he Cen e o Resea ch and U iliza ion o
Renewable Ene gy (CVVOZE). Au ho s g a e ully acknowledge inancial suppo om he Minis y
o he In e io o he Czech Republic.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
No a ions
Abb e ia ions
5G Fi h-Gene a ion B oadband Cellula Ne wo ks
AG Acyclic G aph
AMM Ad anced Me e ing Moni o
AO Ae ial Op ical Pa hways
API Applica ion P og amming In e ace
AT A ack T ees
BE Basic E en
BDD Bina y Decision Diag am
BO Sub e anean Op ical Coun e pa
BTS Base T anscei e S a ion
CI C i ical In as uc u e
CM Co ec i e Main enance
COS Co e Op ical Swi ch
CPS Cybe -Physical Sys em
CFT Con en ional Faul T ee
DER Dis ibu ed Ene gy Resou ces
Algo i hms 2023,16, 561 19 o 21
DFT Dynamic Faul T ee
DSO Dis ibu ion Sys em Ope a o
DT Dis ibu ion T ans o me
DTS Dis ibu ion T ans o me S a ion
EFT Ex ended Faul T ee
ER Edge Rou e
ET E en T ees
FTA Faul T ee Analysis
FT Faul T ee
GSM Global Sys em o Mobile Communica ions
HV High Vol age
I/O Inpu /Ou pu
KPI Key Pe o mance Indica o
LBS Load B eak Swi ch
LTE Long-Te m E olu ion
LV Low Vol age
MM Ma ko Modeling
MDE Model-D i en Enginee ing
MTBF Mean Time Be ween Failu es
MTTF Mean Time To Failu e
MTTI Mean Time To Inciden
MTTK Mean Time To Known issue
MTTRep Mean Time To Replica e
MTTRes Mean Time To Repai able e en
MTTV Mean Time To Valida e
MV Medium Vol age
OLT Op ical Line Te minal
PDMP Piecewise De e minis ic Ma ko P ocess
PN Pe i Ne s
POD Poin o Deli e y
POS Passi e Op ical Spli e
PDF P obabili y Densi y Func ion
RFT Repai able Faul T ee
RAMS Reliabili y, A ailabili y, Main ainabili y, and Sa e y/Secu i y
RTU Remo e Te minal Uni
SAIDI Sys em A e age In e up ion Du a ion Index
SAIFI Sys em A e age In e up ion F equency Index
SCADA Supe iso y Con ol And Da a Acquisi ion
VPN Vi ual P i a e Ne wo k
W Wi eless Linkage
WAN Wide A ea Ne wo k
Va iables
X ime o ailu e ( he li e ime)
Y epai ( eco e y) ime a e a ailu e occu s
T ep cumula i e ime alloca ed o epai s
T es cumula i e ime alloca ed o es o e
Indices
F( ) dis ibu ion unc ion o a andom a iable X
( ) p obabili y densi y unc ion o a andom a iable X
U( ) ins an aneous ime-dependen una ailabili y unc ion
A( ) = 1−U( ) ins an aneous a ailabili y unc ion
h(x) enewal densi y
Pa ame e s
βshape pa ame e
θscale pa ame e
Algo i hms 2023,16, 561 20 o 21
Re e ences
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