Tensor Model for Representing Critical Infrastructure

109
Ia osla Do ohyi1,2,3,†, Vladysla K a chuk1,† and Vasyl Tsu kan2,4,*,†
1 Done sk Na ional Technical Uni e si y, 76, Sambi ska S ., D ohobych, L i egion, 82111, Uk aine
2 Na ional Technical Uni e si y o Uk aine “Igo Siko sky Kyi Poly echnic Ins i u e”, 37, P ospec Be es eiskyi ( o me
Pe emohy), Kyi , Uk aine, 03056, Uk aine
3 Ta as She chenko Na ional Uni e si y o Kyi , 60 Volodymy ska S ee , Kyi , 01033, Uk aine
4 G. E. Pukho Ins i u e o Modeling in Ene gy Enginee ing o Na ional Academy o Sciences o Uk aine, 15, Oleha Mud aka
S ., Kyi , 03164, Uk aine
Abs ac
C i ical in as uc u e ensu es he s abili y o socie y, economic g ow h, and na ional secu i y. This pape
p esen s a enso -based model o assessing in as uc u e esilience, conside ing echnical, economic,
en i onmen al, social, and manage ial aspec s.
The p oposed model ep esen s in as uc u e subsys ems – gene a ion, anspo a ion, and esou ce
consump ion–ac oss di e en ope a ional phases: p e-disas e , c isis, and eco e y. Tenso analysis enables
a comp ehensi e e alua ion o in e ac ions be ween sys em componen s, mul iple h ea impac s, and
esilience c i e ia such as unc ionali y, eco e y ime, h ea esis ance, adap abili y, and economic
e iciency.
By de ining h ea ec o s o a ious dis up ions, including cybe a acks, na u al disas e s, and
echnological ailu es, he model iden i ies ulne abili ies and p o ides insigh s o s eng hening
in as uc u e esilience. The indings suppo s a egic managemen , isk mi iga ion, and policy
de elopmen in na ional secu i y and enginee ing planning.
Keywo ds
esilience o c i ical in as uc u es, enso analysis, esilience c i e ia, h ea modeling, isk managemen ,
s a egic planning, mul idimensional analysis, sys em adap abili y, cybe a acks, en i onmen al
sus ainabili y. 1
1. In oduc ion
C i ical in as uc u es, including ene gy, anspo a ion, communica ions, wa e supply, and
heal hca e, a e essen ial o na ional secu i y and economic s abili y. Howe e , hese sys ems ace
inc easing h ea s om na u al disas e s, cybe a acks, echnological ailu es, and geopoli ical
con lic s. Thei in e connec ed na u e makes esilience a key p io i y.
Resilience de ines a sys em’s abili y o main ain unc ionali y, esis ex e nal impac s, and eco e
a e c ises. I in ol es echnical, o ganiza ional, social, and economic ac o s ha ensu e adap abili y
o dynamic h ea s. A s uc u ed app oach o assessing esilience equi es clea c i e ia,
ma hema ical modeling, and analy ical me hods.
This pape explo es enso analysis as a ool o modeling esilien c i ical in as uc u e. Tenso
models enable he e alua ion o complex in e ac ions be ween sys em componen s, h ea scena ios,
and adap i e s a egies. The p oposed app oach in eg a es echnical, economic, and manage ial
aspec s o enhance in as uc u e secu i y and ope a ional s abili y.
ITS-2024: In o ma ion Technologies and Secu i y, Decembe 19, 2024, Kyi , Uk aine
∗ Co esponding au ho .
† These au ho s con ibu ed equally.
[email p o ec ed] (I. Do ohyi); [email p o ec ed] (V. K a chuk); [email p o ec ed] (V.
Tsu kan)
0000-0003-3848-9852 (I. Do ohyi); 0009-0000-7929-6796 (V. K a chuk); 0000-0003-1352-042X (V. Tsu kan)
© 2024 Copy igh o his pape by i s au ho s. Use pe mi ed unde C ea i e Commons License A ibu ion 4.0 In e na ional (CC BY 4.0).
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2. Li e a u e Re iew
The a icle [1] p esen s an inno a i e app oach o assessing he esilience o c i ical in as uc u e
unde condi ions o mul i-le el h ea s, pa icula ly o anspo a ion acili ies as key componen s
o c i ical in as uc u e. The au ho s ha e de eloped an adap i e me hodology ha in eg a es
a ious isk pa ame e s, p o iding a obus ounda ion o decision-making du ing c ises. This
app oach se es as a basis o u he esea ch in in as uc u e esilience and isk managemen .
The ollowing s udy [2] ocuses on a no el amewo k o e alua ing he esilience o mul i-
componen c i ical in as uc u e. Speci ically, i demons a es how mode n app oaches in
enginee ing sys ems managemen can enhance esilience o complex h ea s. To achie e his, he
au ho s pay signi ican a en ion o ma hema ical models o analyzing in e -sys em dependencies,
which a e c i ical o ensu ing he unin e up ed unc ioning o in as uc u e du ing eme gencies.
In a icle [3], a quan i a i e me hod o e alua ing he esilience o in e dependen in as uc u es
is p oposed. The ma hema ical model de eloped by he au ho s enables he assessmen o
unc ionali y losses and eco e y a es a e eme gency e en s. This app oach is pa icula ly ele an
as i conside s in e connec ions be ween in as uc u e componen s, making i aluable o p ac ical
isk managemen solu ions.
The analysis o u ban c i ical in as uc u e esilience, exempli ied by Ah az, I an, is p esen ed
in s udy [4]. The au ho s employ an indica i e app oach o assess ulne abili ies in u ban ne wo ks
such as wa e supply, elec ici y, and anspo a ion. This me hod emphasizes he in eg a ion o
en i onmen al, social, and echnical ac o s, he eby enhancing he o e all esilience o u ban
sys ems o c ises.
The wo k [5] p oposes me ics and amewo ks o analyzing he esilience o enginee ing and
in as uc u e sys ems. The esea ch ocuses on me hods o e alua ing sys ems' abili y o wi hs and
ex e nal impac s, quickly adap , and es o e unc ionali y. This s udy p o ides a heo e ical
ounda ion o u he wo ks in esilience and isk managemen o in as uc u e.
Resea ch [6] emphasizes he impac o dynamic cos changes on assessing in as uc u e
esilience. I highligh s he impo ance o inco po a ing ime ac o s o imp o e he accu acy o
esilience o ecas ing, de eloping me hodologies o adap i e isk assessmen and in as uc u e
managemen .
The de elopmen o me ics and me hods o quan i a i e assessmen o he esilience o powe
sys ems is p esen ed in s udy [7]. The au ho s p opose an in eg a i e app oach o analyzing
ope a ional and s uc u al esilience. The sugges ed me ics allow he o mula ion o s a egies o
isk minimiza ion and enhancing he eliabili y o ene gy supply sys ems.
A icle [8] p esen s a comp ehensi e amewo k o e alua ing he esilience o c i ical
in as uc u e componen s by inco po a ing echnical, o ganiza ional, and social dimensions. This
mul idimensional app oach aids in designing measu es o enhance esilience ac oss di e en sec o s
o c i ical in as uc u e, o e ing a holis ic iew o how a ious ac o s in e ac .
S udy [9] in oduces a me hodology o assessing he esilience o ne wo ked in as uc u es,
wi h a ocus on he in e dependencies among sys em elemen s and he e ec s o hei po en ial
ailu e. The indings p o ide a ounda ion o c ea ing e ec i e isk managemen s a egies o
main ain and s eng hen c i ical in as uc u e esilience.
In a icle [10], ope a ional models o analyzing in as uc u e esilience a e examined. The s udy
ocuses on iden i ying ulne abili ies in ne wo ks and inding op imal solu ions o ensu ing
con inuous in as uc u e ope a ion du ing c ises, making his wo k a signi ican s ep in de eloping
esilience s a egies.
A icle [11] explo es he esilience o powe sys ems, conside ing app oaches o assessing c i ical
in as uc u e esilience and c i e ia es ablished by go e nmen policies. The wo k p esen s an
in e disciplina y app oach in eg a ing echnical and egula o y aspec s o enhance he eliabili y o
ene gy in as uc u e.
111
Resea ch [12] highligh s he use o expe judgmen o assess he esilience o c i ical
in as uc u es. The au ho s p opose a model ha inco po a es subjec i e expe e alua ions o
quan i a i ely de e mining esilience le els. This s udy is use ul o apid isk analysis in si ua ions
wi h limi ed da a.
In a icle [13], he concep o esilience cu es o in as uc u e is p esen ed. The s udy iden i ies
new pe o mance me ics and da a agg ega ion me hods o e alua e he e ec i eness o
in as uc u e sys ems du ing eme gencies. The p oposed app oach allows o a de ailed analysis o
eco e y dynamics and unc ionali y losses.
S udy [14] in oduces a amewo k o e alua ing he esilience o bo h in as uc u al and
economic sys ems. I o e s an in-dep h analysis o me hods o modeling in e dependencies among
sys em componen s, acili a ing a clea e unde s anding o hei esponses o di e en ypes o
h ea s.
Wo k [15] p esen s a scena io-based me hodology o assessing he esilience o c i ical
in as uc u es, wi h a pa icula ocus on he seismic esilience o seapo s. The au ho s de elop a
mul i-le el app oach ha in eg a es scena io modeling and impac assessmen , enabling p ecise
o ecas ing o po en ial isks.
A icle [16] conduc s a sys ema ic e iew o quan i a i e esilience indica o s o wa e
in as uc u e sys ems. The esea ch ocuses on de eloping me ics ha quan i y he abili y o wa e
sys ems o eco e unc ionali y a e ad e se impac s. This s udy se es as a ounda ion o s a egic
decision-making in wa e esou ce managemen .
S udy [17] analyzes app oaches o measu ing he esilience o anspo a ion in as uc u e. The
wo k ocuses on de eloping indica o s and me hods o esilience assessmen in he con ex o
anspo sys ems, al hough i p o ides limi ed in o ma ion on he speci ics o he in as uc u e.
In a icle [18], he pe o mance o g een in as uc u e is in es iga ed h ough he lens o u ban
esilience. The au ho s p opose an analy ical me hodology o assessing he impac o g een
in as uc u e on he eco e abili y o u ban sys ems, conside ing en i onmen al and socio-
economic aspec s.
Resea ch [19] ocuses on e alua ing and enhancing o ganiza ional esilience in Slo akia's c i ical
in as uc u e. The wo k p esen s a mul i-le el app oach o s eng hening o ganiza ional capaci y
o adap a ion and c isis esponse.
A icle [20] examines p inciples and c i e ia o e alua ing he esilience o ene gy sys ems in
u ban en i onmen s. This e iew s udy highligh s key ac o s ensu ing he eliabili y and con inui y
o ene gy supply unde apidly changing condi ions.
In a icle [21], ime-dependen esilience o u ban in as uc u al sys ems is assessed. The au ho s
p opose a me hodology o esilience e alua ion ha accoun s o dynamic changes in in as uc u e
unc ionali y o e ime, enabling mo e p ecise planning o esilience imp o emen s.
S udy [22] o e s a con inuous and mul idimensional assessmen o esilience based on unc ional
analysis o in e connec ed sys ems. This wo k emphasizes he impo ance o an in eg a ed app oach
o in as uc u e esilience assessmen , whe e each elemen in e ac s wi h o he s, c ea ing a complex
ne wo k o in e dependencies.
A icle [23] explo es me ics and me hods o measu ing esilience in anspo a ion
in as uc u e. The wo k discusses he cu en s a e o de elopmen o c i e ia and me hodologies
o esilience assessmen in anspo sys ems, pa icula ly in he con ex o clima e change and
ex eme e en s.
Resea ch [24] ocuses on he assessmen o in as uc u e esilience, explo ing di e en me hods
and app oaches o measu ing how in as uc u e sys ems adap o changing condi ions.
A icle [25] examines he in eg i y o in as uc u e sys ems h ough a sys emic pe spec i e,
highligh ing he signi icance o in eg a ing mul iple componen s o ensu e hey can ope a e
cohesi ely unde unexpec ed ci cums ances.
112
In a icle [26], he concep o sys em esilience in he con ex o in as uc u e is analyzed using
La ia as a case s udy. The esea ch applies heo e ical app oaches o assessing in as uc u e
esilience in coun ies wi h ansi ional economies.
S udy [27] p oposes a uni ied app oach o assessing he esilience and sus ainabili y o u ban
in as uc u e. This app oach inco po a es a ious pa ame e s o e alua e in as uc u e esilience
unde clima e change and ex eme e en s.
A icle [28] p oposes a quali a i e me hodology o e alua ing he pe o mance o IT
in as uc u e elemen s, conside ing echnical cha ac e is ics and ope a ing condi ions. The au ho
de eloped a model o assess he eliabili y and pe o mance o IT sys em componen s, p edic
po en ial ailu es, and op imize ope a ions, emphasizing he app oach's e sa ili y o a ious ypes
o in as uc u e.
A icle [29] discusses a new app oach o measu ing he esilience o anspo a ion in as uc u e
ne wo ks. The wo k ocuses on de eloping me hods o esilience assessmen ha conside a ious
ex eme e en scena ios.
Resea ch [30] add esses he assessmen o esilience in in e dependen in as uc u e sys ems,
emphasizing he modeling and analysis o join eco e y p ocesses ollowing damage.
A icle [31] e alua es he esilience o in e dependen in as uc u es by examining di e en
esponse s a egies, wi h a ocus on hei capaci y o eco e du ing majo disas e s o c ises.
3. Resul s
3.1. Key C i e ia o C i ical In as uc u e Resilience
Based on he analysis o he li e a u e, he main c i e ia o he esilience o c i ical in as uc u es
ha e been iden i ied (Table 1):
 In as uc u e Func ionali y – assessmen o he abili y o in as uc u e o pe o m i s co e
unc ions du ing and a e s ess e en s (e.g., na u al disas e s o man-made ca as ophes).
 Reco e y Capabili y – he abili y o in as uc u e o quickly eco e a e damage o
unc ional dis up ions. This c i e ion includes ime, esou ces, and measu es needed o e u n
o no mal condi ions.
 Resis ance o Ex e nal Th ea s – he abili y o in as uc u e o wi hs and ex eme ac o s, such
as na u al disas e s, echnological acciden s, economic and social c ises.
 Flexibili y and Adap abili y – he capaci y o in as uc u e o adap o new condi ions and
changes, such as clima e change, echnological ad ancemen s, o shi s in poli ical and
economic con ex s.
 Economic Cos Assessmen – e alua ion o eco e y cos s a e a disas e , including di ec
cos s o es o a ion and indi ec losses om se ice in e up ions.
 In eg a ion wi h O he Sys ems – assessmen o how in as uc u e sys ems in e ac and
depend on each o he . This c i e ion is impo an o iden i ying how one ailu e may a ec
o he s.
 Ins an Analysis and Fo ecas ing – u iliza ion o da a o eal- ime e alua ion o he cu en
s a e o in as uc u e and p edic ion o po en ial issues.
 Flexibili y o Managemen S uc u es – he abili y o managemen bodies and o ganiza ions
esponsible o in as uc u e o quickly adap o new condi ions, o ganize e ec i e
esponses, and coo dina e eco e y e o s.
 Da a Secu i y and P o ec ion – ensu ing he secu i y o da a and in o ma ion sys ems agains
cybe a acks and o he h ea s ha may dis up in as uc u e ope a ions.
 En i onmen al Sus ainabili y – assessmen o he ex en o which in as uc u e complies
wi h en i onmen al s anda ds and can adap o changes in he su ounding en i onmen .
113
Table 1
Key C i e ia o C i ical In as uc u e Resilience
C i e ion C i e ion Desc ip ion Sou ces (Re e ences)
In as uc u e
Func ionali y Assessmen o he abili y o in as uc u e
o pe o m i s co e unc ions du ing and
a e s ess e en s.
[1], [6], [12], [17],
[19], [29]
Reco e y Capabili y The abili y o in as uc u e o quickly
eco e a e damage o unc ional
dis up ions, including ime, esou ces, and
measu es needed o es o a ion.
[2], [9], [14], [17],
[18], [29]
Resis ance o
Ex e nal Th ea s The abili y o in as uc u e o wi hs and
ex eme ac o s, such as na u al disas e s,
echnological acciden s, economic, and
social c ises.
[3], [7], [15], [20],
[24], [28]
Flexibili y and
Adap abili y The capaci y o in as uc u e o adap o
new condi ions and changes, such as
clima e change, echnological
ad ancemen s, o shi s in poli ical and
economic con ex s.
[4], [12], [14], [17],
[26], [30]
Economic Cos
Assessmen E alua ion o eco e y cos s a e a
disas e , including di ec es o a ion cos s
and indi ec losses om se ice
in e up ions.
[8], [11], [18], [23],
[28]
In eg a ion wi h
O he Sys ems Assessmen o how in as uc u e sys ems
in e ac and depend on each o he ,
iden i ying how one ailu e may impac
o he s.
[5], [11], [15], [18],
[22], [29]
Ins an Analysis and
Fo ecas ing U iliza ion o da a o eal- ime e alua ion
o he cu en s a e o in as uc u e and
p edic ion o po en ial issues.
[6], [9], [19], [21],
[30]
Flexibili y o
Managemen
S uc u es
The abili y o managemen bodies and
o ganiza ions esponsible o
in as uc u e o quickly adap o new
condi ions, o ganize e ec i e esponses,
and coo dina e eco e y.
[5], [14], [20], [29],
[30]
Da a Secu i y and
P o ec ion Ensu ing he secu i y o da a and
in o ma ion sys ems agains cybe a acks
and o he h ea s ha may dis up
in as uc u e ope a ions.
[13], [15], [17], [28]
En i onmen al
Sus ainabili y Assessmen o he ex en o which
in as uc u e complies wi h
en i onmen al s anda ds and can adap o
changes in he su ounding en i onmen .
[4], [7], [10], [15],
[27]

114
Nex , we will examine he enso model o c i ical in as uc u e esilience based on he iden i ied
esilience c i e ia.
3.2. Tenso Model o Resilien C i ical In as uc u e
To cons uc a enso model o esilien c i ical in as uc u e based on he de ined c i e ia, each
c i e ion can be conside ed as a sepa a e componen in e ac ing wi h o he s h ough speci ic
pa ame e s. A enso model is a mul idimensional ma hema ical objec ha allows o he desc ip ion
o in e connec ions be ween a ious cha ac e is ics o in as uc u e and i s esilience.
Le T ep esen he enso desc ibing he esilien c i ical in as uc u e. Each elemen o he
enso co esponds o a speci ic esilience cha ac e is ic o he in as uc u e, g ouped acco ding o
i s a ious pa ame e s. The model can be ep esen ed as a hi d-o de enso :
𝑇,,
whe e:
𝑖 – index ep esen ing indi idual esilience c i e ia,
𝑗 – index ep esen ing subsys ems o componen s o in as uc u e (e.g., ene gy, anspo a ion,
u ili y sys ems),
𝑘 – index ep esen ing he ime aspec o s ages o eco e y (e.g., p e-disas e , du ing disas e ,
pos -disas e ).
P inciples o he model ope a ion a e as ollows:
 he enso 𝑇,, de ines all he in e ela ionships be ween esilience c i e ia, subsys ems, and
s ages;
 o each s age 𝑘 changes in in as uc u e esilience can be assessed based on speci ic c i e ia,
as well as in e dependencies be ween subsys ems;
 enso pa ame e s can be calcula ed based on expe assessmen s, ma hema ical models, o
s a is ical da a.
The p oposed model can easily be expanded by in oducing addi ional enso s. Fo example, we
can in oduce an addi ional h ea enso 𝑍, which demons a es he impac o speci ic h ea s on he
esilience c i e ia o he sys em.
Le he e be 𝑛 h ea s a ec ing he esilience o he sys em. Then, o each esilience c i e ion,
we ha e he ollowing h ea ec o 𝑍, which e lec s he impac o each h ea on c i e ion 𝑖:
𝑍=(𝑍 … 𝑍).
The esul o he impac o h ea s on he esilience o subsys ems can be ep esen ed by enso
𝑆, composed o he co esponding ma ices 𝑆, which con ain he esul o mul iplying 𝑇 by he
co esponding h ea elemen 𝑍. Thus, o each c i e ion i we will ob ain a ma ix o size 𝑗 ×𝑘,
whe e each elemen o his ma ix is calcula ed as:
𝑆,,
=𝑇,,×𝑍,
whe e:
𝑆,,
 – elemen o he ma ix o c i e ion 𝑖, subsys em 𝑗 and s age 𝑘 o h ea 𝑙.
𝑇,, – elemen o he ma ix o c i e ion 𝑖, subsys em 𝑗 and s age 𝑘.
𝑍 – elemen o he h ea ec o o c i e ion 𝑖 and h ea 𝑙.
The gene al appea ance o he ma ix 𝑆 o each h ea 𝑙 is as ollows:
𝑆=𝑆,,
𝑆,,
𝑆,,

𝑆,,
𝑆,,
𝑆,,

𝑆,,
𝑆,,
𝑆,,
.
115
Addi ional enso s can also be in oduced in o he model, which, o example, desc ibe
ulne abili ies and p o ec ion mechanisms wi hin he sys em. Addi ionally, u he in e ac ion and
mu ual in luence ules be ween enso s can be de ined.
3.3. Scena ios o model esea ch
The model, buil on mul idimensional enso s o assessing sys em esilience unde he in luence
o h ea s, allows o a se ies o expe imen al s udies o analyze sys ems in di e en scena ios:
1. Analysis o he impac o di e en h ea s on sys em esilience
 Objec i e – o in es iga e how di e en ypes o h ea s a ec di e en esilience c i e ia.
 Expe imen :
 change he alues o enso 𝑍 o simula e a ying h ea in ensi ies;
 analyze he esul ing enso 𝑆 o iden i y esilience c i e ia mos a ec ed by he
h ea s.
 Resul – iden i ica ion o he mos ulne able c i e ia o subsys ems.
2. E alua ion o he e ec i eness o p o ec ion measu es
 Objec i e – o e i y how speci ic p o ec i e measu es educe he impac o h ea s.
 Expe imen :
 add a co ec ion enso o 𝑍 ep esen ing he in luence o p o ec i e measu es;
 ecalcula e 𝑆 and compa e i o he baseline alue.
 Resul – assessmen o he e ec i eness o speci ic measu es.
3. Analysis o disas e scena ios
 Objec i e – o model a ious disas e scena ios and assess sys em esilience a each s age.
 Expe imen :
 o each s age (p e-disas e , du ing disas e , pos -disas e ), modi y he
co esponding alues o enso 𝑇 and analyze he changes in enso 𝑆;
 speci ically, s udy how he sys em eco e s a e a disas e .
 Resul – iden i ica ion o c i ical s ages equi ing he mos a en ion.
4. Compa ison o sys ems wi h di e en esilience cha ac e is ics
 Objec i e – o assess which sys em has highe esilience unde he same h ea condi ions.
 Expe imen :
 use di e en ini ial alues o enso 𝑇 (e.g., o di e en o ganiza ions, egions,
o sys em ypes);
 analyze he esul ing enso s 𝑆 and iden i y he sys ems wi h he bes
pe o mance.
 Resul – anking o sys ems based on esilience le el.
5. De e mina ion o sys em sensi i i y o changes in h ea in ensi y
 Objec i e – o in es iga e how changes in one o mo e elemen s o 𝑍 a ec he esul ing 𝑆.
 Expe imen :
 g adually change he h ea alues (e.g., inc ease o dec ease he impac o a
speci ic h ea on a speci ic c i e ion);
 analyze he dynamic changes in 𝑆.
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 Resul – iden i ica ion o c i ical h ea s wi h he mos signi ican impac on he sys em.
6. De e mina ion o in e ela ionships be ween esilience c i e ia
 Objec i e – o ind ou how he impac o one h ea on a speci ic c i e ion can change o he
c i e ia.
 Expe imen :
 analyze he ma ices in enso 𝑆 co esponding o di e en esilience c i e ia;
 build co ela ions be ween he esul s.
 Resul – iden i ica ion o in e dependencies be ween c i e ia.
7. Sys em pa ame e op imiza ion
 Objec i e – o de e mine op imal pa ame e alues o educe he impac o h ea s.
 Expe imen :
 use op imiza ion algo i hms o ind alues o 𝑇 ha maximize esilience 𝑆 wi h
ixed 𝑍.
 Resul – ecommenda ions o imp o ing he sys em.
8. Modeling long- e m consequences
 Objec i e – o assess how he sys em esponds o p olonged pe iods o h ea impac .
 Expe imen :
 use a a iable enso 𝑍 o model long- e m o pe iodic h ea s;
 analyze changes in 𝑆 o e ime.
 Resul – o ecas ing he long- e m esilience o he sys em.
9. Model alida ion on eal da a
 Objec i e – o compa e he model's esul s wi h eal-wo ld da a.
 Expe imen :
 use empi ical da a o build 𝑇 and 𝑍.
 compa e he esul ing 𝑆 wi h ac ual pe o mance indica o s o sys ems.
 Resul – alida ion o he model and i s po en ial use o eal sys ems.
3.4. Example o using he model o he ene gy sec o
To apply he enso model o c i ical in as uc u e esilience o he ene gy sec o , i is necessa y o
de ine how each o he esilience c i e ia a ec s ene gy sys ems and de e mine he alues o each
c i e ion and subsys em (e.g., ene gy ne wo ks, gene a ion, and elec ici y ansmission).
We desc ibe he esilience c i e ia o he model as ollows:
 in as uc u e unc ionali y 𝑖 – he abili y o elec ical ne wo ks and s a ions o pe o m hei
unc ions a e na u al disas e s o man-made acciden s.
 eco e y capabili y 𝑖 – he ime equi ed o es o e powe supply a e an acciden o disas e .
 esilience o ex e nal h ea s 𝑖 – he abili y o ene gy sys ems o wi hs and na u al disas e s
( loods, snows o ms) o echnological acciden s.
 lexibili y and adap abili y 𝑖 – he abili y o adap o changes in elec ici y demand o new
echnologies, such as he in eg a ion o enewable ene gy sou ces.
 economic cos assessmen 𝑖 – he cos o es o ing ene gy in as uc u e a e a disas e .
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 in eg a ion wi h o he sys ems 𝑖 – he impac o ene gy dis up ions on o he c i ical
in as uc u es, such as anspo a ion o u ili ies.
 eal- ime analysis and o ecas ing 𝑖 – he use o da a o assess he cu en s a e o ene gy
sys ems.
 lexibili y o managemen s uc u es 𝑖 – he abili y o managemen bodies o espond quickly
o ene gy c ises.
 da a secu i y and p o ec ion 𝑖 – p o ec ion o ene gy sys ems om cybe a acks.
 en i onmen al esilience 𝑖 – adap a ion o ene gy sys ems o en i onmen al equi emen s,
pa icula ly educing CO2 emissions.
Le us assume he e a e h ee main subsys ems o ene gy in as uc u e:
 𝑗: ene gy gene a ion;
 𝑗: elec ici y ansmission;
 𝑗: elec ici y consump ion (dis ibu ion and consump ion).
We will e alua e he esilience o he ene gy sys em a h ee s ages:
 𝑘: be o e he disas e (no mal s a e);
 𝑘: du ing he disas e (damage);
 𝑘: a e he disas e ( eco e y).
Fo each c i e ion (10 c i e ia), we ha e a 3x3 ma ix, whe e each ma ix ep esen s a subsys em
a di e en s ages. The e o e, he o e all o m o he enso 𝑇 will be as ollows (1):
𝑇=𝑇
…
𝑇, (1)
whe e each 𝑇 is a ma ix ha desc ibes he co esponding c i e ion.
In pa icula , o each 𝑇 (2): 𝑇=󰇯𝑇,, 𝑇,, 𝑇,,
𝑇,, 𝑇,, 𝑇,,
𝑇,, 𝑇,, 𝑇,,󰇰. (2)
We ha e a h ee-dimensional s uc u e whe e each elemen 𝑇 is a ma ix o size 3 ×3 and
ep esen s a subsys em a di e en s ages o each esilience c i e ion,
whe e:
 he i s index 𝑖 ep esen s he esilience c i e ion ( o 𝑖=1,2,…,10),
 he second index 𝑗 (1 – gene a ion, 2 – anspo a ion, 3 – consump ion),
 he hi d index 𝑘 (1 – be o e he disas e , 2 – du ing he disas e , 3 – a e he disas e ).
Le he e be 4 h ea s ha a ec he sys em's esilience. Then, o each esilience c i e ion, we
ha e he ollowing h ea ec o 𝑍, which e lec s he impac o each h ea on c i e ion 𝑖 (3):
𝑍=(𝑍 … 𝑍). (3)
The esul o he impac o h ea s on subsys em esilience can be ep esen ed by a enso 𝑆,
consis ing o he co esponding ma ices 𝑆, which con ain he esul o mul iplying 𝑇 by he
co esponding h ea elemen 𝑍. Thus, o each c i e ion 𝑖, a ma ix o size 3 ×3 will be ob ained,
whe e each elemen o his ma ix is compu ed as (4):
𝑆,,
=𝑇,,×𝑍, (4)
whe e:
𝑆,,
 is he elemen o he ma ix o he 𝑖- h c i e ion, he 𝑗- h subsys em, and he 𝑘- h s age o
he 𝑙- h h ea .
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