761
1 INTRODUCTION
The safety and resilience indicators (Kołowrocki &
Soszyńska‐Budny, 2017) for critical infrastructure
defined as a complex system in its operating
environment that significant features are inside‐
system dependencies and outside‐system
dependencies (Lague et al., 2015) are crucial for its
operators. A simple critical infrastructure
safety
modelwithoutconsideringoutsideimpactsproposing
safety indicators SafI1‐SafI8 (Kołowrocki &
Soszyńska‐Budny, 2017, 2018a, 2019) can be
generalizedbylinkingitwiththemodelofthecritical
infrastructure operation process (Kołowrocki &
Soszyńska‐Budny, 2017, 2018b). This way created
jointimpactmodelof
thecriticalinfrastructurerelated
toitsoperationprocesscanoffer,additionallytothe
modified safety indicators SafI1‐SafI8, two resilience
indicators ResI1‐ResI2 which are measures of the
critical infrastructure operation impact on its safety
andresiliencetooperation(Kołowrocki&Soszyńska‐
Budny, 2017, 2018b). The paper is devoted
to
development of this joint model of safety and
operation process of critical infrastructure and its
practical application to safety and resilience
examination of the port oil terminal critical
infrastructure.
2 CRITICALINFRASTRUCTUREIMPACTEDBY
ITSOPERATIONPROCESSSAFETYMODEL
2.1 CriticalInfrastructureoperationprocess
We consider the critical infrastructure
related to the
operation process Z(t),
),,0 t
impacted in a
variousway at its operationstates
,
b
z
.,...,2,1 vb
Weassumethatthechangesoftheoperationstatesof
thecriticalinfrastructureoperationprocessZ(t)have
an influence on the critical infrastructure safety
structure and also on the safety of the critical
infrastructure assets
i
A ,
,,...,2,1 ni
(Kołowrocki
&Soszyńska‐Budny,2011,2017,2018b).
The following critical infrastructure operation
process parameters (OPP) can be identified either
statisticallyusingthemethodsgivenin(Kołowrocki,
2014; Kołowrocki & Soszyńska‐Budny, 2011, 2017,
2018b)orevaluatedapproximatelybyexperts:
thenumberofoperationstates
(OPP1)
v
;
Safety and Resilience Indicators of Critical
Infrastructure Impacted by Operation Application to
Port Oil Terminal Examination
K.Kołowrocki&J.Soszyńska‐Budny
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Modellingofoperationprocessinfluenceonsafetyofacriticalinfrastructureispresented.New
safetyandresilienceindicatorsforacriticalinfrastructurearedefinedandproceduresoftheirdeterminationin
thecaseofthecreatedmodelareproposed.Next,thismodelisappliedtosafety
andresilienceanalysisofthe
portoilterminalcriticalinfrastructureimpactedbyitsoperationprocessandtheresultsarecomparedtothe
indicatorsofthiscriticalinfrastructurewithoutoperationimpacts.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 13
Number 4
December 2019
DOI:10.12716/1001.13.04.08
762
the vector
x1
)]0([
b
p
of the initial probabilities
(OPP2)
),)0(()0(
bb
zZPp
,,...,2,1 vb
of
the critical infrastructure operation process Z(t)
staying at particular operation states
b
z at the
moment
;0t
the matrix
x
][
bl
p
of probabilities of transition
(OPP3)
,
bl
p
,,...,2,1, vlb
of the critical
infrastructure operation process Z(t) between the
operationstates
b
z
and
;
l
z
thematrix
x
][
bl
M
ofmeanvalues ofconditional
sojourn times (OPP4)
],[
blbl
EM
,,...,2,1, vlb
of the critical infrastructure
operation process Z(t) conditional sojourn times
bl
attheoperationstate
b
z
whenthenextstateis
.
l
z
The main critical infrastructure operation process
characteristic (OPC) that can be either calculated
analytically using the above parameters of the
operation process or evaluated approximately by
experts(Kołowrocki&Soszyńska‐Budny,2011,2017,
2018b)isthevector
],...,,[][
211
pppp
b
x
, (1)
oflimitvaluesoftransientprobabilities(OPC1)
)(tp
b
=P(Z(t)=
b
z ),
),,0
t
,,...,2,1 vb
(2)
of the critical infrastructure operation process
)(tZ
attheparticularoperationstates
,
b
z
.,...,2,1 vb
2.2 CriticalInfrastructuresafetyandresilienceindicators
We denote the critical infrastructure conditional
lifetime in the safety state subset
},,...,1,{ zuu
,,...,2,1 zu
while its operation process Z(t),
),,0 t
is at the operation state
,
b
z
,,...,2,1 vb
by
)(1
)]([
b
uT
,
,,...,2,1 zu
and
the conditional safety function of the critical
infrastructure related to the operation process Z(t),
),,0 t
bythevector(Kołowrocki&Soszyńska‐
Budny,2017,2018b)
)(1
)],([
b
t S
=[1,
)(1
)]1,([
b
tS
,...,
)(1
)],([
b
ztS
], (3)
withthecoordinatesdefinedby
)(1
)],([
b
utS
))()](([
)(1
b
b
ztZtuTP
(4)
for
),,0 t
,,...,2,1 zu .,...,2,1
b

Thesafetyfunction
)(1
)],([
b
utS
,
,,...,2,1 zu
is
the conditional probability that the critical
infrastructure related to the operation process Z(t),
),,0 t
lifetime
)(1
)]([
b
uT
,
,,...,2,1 zu
in the
safety state subset
},...,1,{ zuu
,
,,...,2,1 zu
is
greater than t, while the critical infrastructure
operationprocessZ(t)isattheoperationstate
.
b
z
Next,we denotethe critical infrastructure related
to the operation process Z(t),
),,0
t
unconditional lifetime in the safety state subset
},,...,1,{ zuu
,,...,2,1 zu
by
),(
1
uT
,,...,2,1 zu
and the unconditional safety
function,thefirstsafetyindicatorSafI1(Kołowrocki
& Soszyńska‐Budny, 2017, 2018b) of the critical
infrastructure related to the operation process Z(t),
),,0
t
bythevector
),(
1
tS
=[1,
)1,(
1
tS
...,
),(
1
ztS
], (5)
withthecoordinatesdefinedby
),(
1
utS ))((
1
tuTP
(6)
for
),,0
t
.,...,2,1 zu

Inthecasewhenthesystemoperationtime
is
large enough, the coordinates of the unconditional
safetyfunctionofthecriticalinfrastructurerelatedto
theoperationprocessZ(t),
),,0 t
definedby(6),
aregiven by(Kołowrocki &Soszyńska‐Budny, 2017,
2018b)
),(
1
utS
)(
1
1
]),([
b
v
b
b
utp
S
for
0t
,
,,...,2,1 zu
(7)
where
)(1
)],([
b
utS
,
,,...,2,1 zu
,,...,2,1
b
are
thecoordinatesofthecriticalinfrastructurerelatedto
the operation process Z(t),
),,0 t
conditional
safety functions defined by (3)‐(4) and
b
p
,
,,...,2,1
b
arethecriticalinfrastructureoperation
processZ(t),
),,0
t
limittransientprobabilities
atoperationstates
b
z
,
,,...,2,1
b
definedby(1).
Other safety indicators corresponding to SafI2‐
SafI8, defined in (Kołowrocki & Soszyńska‐Budny,
2017,2018b)areasfollows:
theriskfunction(SafI2)
r
1
(t)=
1
1
(, )trS ,
0, ),t

{1, 2,..., },rz
(8)
whereristhecriticalsafetystate;
thegraphofthecriticalinfrastructureriskfunction
r
1
(t),
),,0
t
calledthefragilitycurve(SafI3)
(Ben,Gouldby,Shultz,Simm&Wibowo,2010);
the mean value of the critical infrastructure
unconditional lifetime
)(
1
rT
up to exceeding
criticalsafetystate
r
(SafI4)givenby
11
0
() [ (,)]rtrdt
μ S
,)]([
1
)(1
b
b
b
rp
(9)
where
)(1
)]([
b
r
are the mean values of the
critical inf r astructure conditional lifetimes
)(1
)]([
b
rT
in the safety state subset
},...,1,{ zrr
at the operation state ,
b
z
,,...,2,1
b
givenby
0
)(1)(1
,)],([)]([ dtrtr
bb
S
,,...,2,1
b
(10)
and
)(1
)],([
b
rtS
,
,,...,2,1
b
aredefined by
(3)‐(4)and
b
p
aregivenby(1);
763
the standard deviation
),(
1
r
of the critical
infrastructurelifetimeinthesafetystatenotworse
thanthecriticalstater(SafI5);
themoment
1
ofexceedingacceptablevalueof
criticalinfrastructureriskfunctionlevel
(SafI6);
the intensities of degradation of the critical
infrastructure / the intensities of critical
infrastructure departure from the safety state
subset
},...,1,{ zuu
(SafI7)
,
),(
),(
),(
1
1
1
ut
dt
utd
ut
S
S
λ
,0t ;,...,2,1 zu
(11)
themeanlifetimes
),1()()(
111
uuu μμμ
,1,...,2,1 zu
).()(
11
zz μμ
(12)
of the critical infra structure in the particular
safety states (SafI8) where
),(
1
uμ
,,...,2,1 zu
maybedeterminedfrom(10)bysubstitutingr=u.
Toexpress the scaleof influenceof theoperation
process on the critical infrastructure safety, the
followingresilienceindicatorsaredefined:
thecoefficientsofoperationprocessimpactonthe
critical infrastructure intensities of degradation
(the coefficients of operation process
impact on
criticalinfrastructureintensitiesofdeparturefrom
thesafetystatesubset
},...,1,{ zuu
)(ResI1),i.e.
thecoordinatesofthevector
),(
1
tρ
=[0,
)1,(
1
tρ
,…,
),(
1
ztρ
],
,0t
(13)
where
1
(, )tu
=
1
(, ) (, ),tu tu

0
,0t 1, 2,..., ,uz
(14)
i.e.
,
),(
),(
),(
0
1
1
ut
ut
ut
λ
λ
ρ

,0t
1, 2,..., ,uz
(15)
and
),,( ut
0
λ
are the intensities of
degradation of the critical infrastructure without
operation process impact and
),,( ut
1
λ
are the
intensities of degradation of the critical
infrastructurewithoperationprocessimpact,
theindicatorofcriticalinfrastructureresilienceto
operationprocessimpact(ResI2)definedby
,
),(
1
),(
1
1
rt
rt
RI
,0t
(16)
where
),,(
1
rtρ
),,0 t
is the
coefficients of operation process impact on the
critical infrastructure intensities of degradation
givenby(15)for
.
r
u
3 APPLICATION
We consider the portoil terminal critical
infrastructure impacted by its operation process
placed at the Baltic seaside that is designated for
receiving oil products from ships, storage and
sendingthembycarriagesor trucks.Theterminalis
described in details in (Kołowrocki & Soszyńska‐
Budny,
2019).
3.1 Portoilterminalcriticalinfrastructureassets
TheconsideredterminaliscomposedofthreepartsA,
Band C,linked by the pipingtransportation system
with the pier. The area in the neighborhood of the
portoilpiping transportation systemispresented in
Figs.9‐10in(Kołowrocki
&Soszyńska‐Budny,2019).
The main technical assets of the port oil terminal
criticalinfrastructureare:
A
1‐portoilpipingtransportationsystem, 
A
2‐internalpipelinetechnologicalsystem,
A
3‐supportingpumpstation,
A
4‐internalpumpsystem,
A
5‐portoiltankershipmentterminal,
A
6‐loadingrailwaycarriagestation,
A
7‐loadingroadcarriagestation,
A
8‐unloadingrailwaycarriagestation,
A
9‐oilstoragereservoirsystem.
The scheme of the asset A
1, the port oil piping
transportation system is prezented in Figure 11 in
(Kołowrocki & Soszyńska‐Budny, 2018a, 2019). The
port oil transportation system is a series system
composed of two series‐parallel subsystems S
1, S2,
eachcontainingtwopipelinesandoneseries‐“2outof
3” subsystem S
3 containing 3 pipelines. The
subsystemsS
1,S2,S3areformingageneralseriesport
oil piping transportation system safety structure
presentedinFig.1.
S
1
S
2
S
3
A
11
A
12
A
21
A
22
A
31
A
32
A
33
Figure1. General scheme of the port oil piping
transportationsystemsafetystructure
3.2 Portoilterminalcriticalinfrastructuresafety
parameters
Afterconsideringthecommentsandopinionscoming
from experts concerned with the port oil terminal
critical infrastructure and its assets without any
outside impacts, using (GMU Critical Infrastructure
SafetyInteractivePlatform,2018)thefollowingsafety
parameters were fixed (Kołowrocki & Soszyńska
‐
Budny,2019):
thenumbersafetystates(excludingsafetystate0)
2
z
;
764
threesafetystates2,1,0;
thecriticalsafetystater=1;
theriskfunctionpermittedlevel
=0.05;
the mean values of the asset A
1, the port oil
terminal critical infrastructure lifetimes in the
safetystatesubsets
},2,1{:}2{
forsafetystatesubset
}2,1{
)1(
0
1
=63years, (17)
forsafetystatesubset
{2}
)2(
0
1
=46years; (18)
themeanvaluesoftheassetsA
2–A9
lifetimesin
the safety state subsets
},2,1{
},2{
evaluated
approximatelybyexperts,areasfollows:
forsafetystatesubset
{1, 2}
)1(
0
i
=80years,
,9,...,3,2i
(19)
forsafetystatesubset
{2}
)2(
0
i
=50years,
.9,...,3,2i
(20)
From (Kołowrocki & Soszyńska‐Budny, 2011), it
follows that the intensities of assets departure from
thesafetystatessubset
},2,1{
are:
forassetA
1
)1(
0
1
=0.015873,
)2(
0
1
=0.021739, (21)
forassetsA
2–A9
)1(
0
i
=0.0125, )2(
0
i
=0.02,
.9,...,3,2i
(22)
3.3 Portoilterminalcriticalinfrastructuresafety
indicators
Assuming that the oil terminal critical infrastructure
wasfree of anyoutside impacts,itsfollowing safety
indicators were determined (Kołowrocki &
Soszyńska‐Budny,2019):
thesafetyfunction
);,(
0
tS
the expected values of the oil terminal critical
infrastructure lifetimes in the safety state subsets
},2,1{:}2{

)1(
0
8.63,
)2(
0
5.50years; (23)
the mean values of the oil terminal critical
infrastructure lifetimes in the particular safety
states:
)1(
0
3.13,
)2(
0
5.50years; (24)
the port oil terminal critical inf r astructure risk
functionr
0
(t);
‐ the moment when the oil terminal critical
infrastructure risk function exceeds a permitted
level
=0.05
44.0
years. (25)
Theoilterminalcriticalinfrastructureintensitiesof
ageing(SI7)are:
)1(
0
0.115873,
)2(
0
0.181739. (26)
3.4 ParametersandcharacteristicsofPortoilterminal
criticalinfrastructureoperationprocess
Operation of the asset A
1, the port oil piping
transportationsystemisthemainactivityoftheport
oil terminal involving the remaining assets A
2 – A9
anddeterminingtheiroperationprocesses.
On the basis of the statistical data and expert
opinions, it is possible to fix and to evaluate the
following unknown basic parameters of the oil
terminalcriticalinfrastructureoperationprocess:
thenumberofoperationprocessstates (OPP1)
=7
andtheoperationprocessstates:
theoperationstate
1
z
transportofonekindof
mediumfromtheterminalpartBtopartCusing
twooutofthreepipelinesofthesubsystem
3
S
of
the asset A
1 illustratedin Figure 2 and assets A2,
A
4,A6,A7,A9;
S
3
A
31
A
32
A
33
Figure2. The scheme of the port oil piping transportation
systemattheoperationstate
1
z
theoperationstate
2
z
transportofonekindof
mediumfromtheterminalpartCtopartBusing
oneoutofthreepipelinesofthesubsystem
3
S
of
theassetA
1illustratedinFigure3.andassets A2,
A
4,A8,A9;
S
3
A
31
A
32
A
33
Figure3. The scheme of the port oil piping transportation
systemattheoperationstate
2
z
theoperationstate
3
z
transportofonekindof
mediumfromtheterminalpartBthroughpartA
765
to pier using one out of two pipelines of the
subsystem
1
S
andoneoutoftwopipelinesofthe
subsystem
2
S
oftheassetA1illustratedinFigure
4andassetsA
2,A4,A5,A9;
S
1
S
2
A
11
A
12
A
21
A
22
Figure4. The scheme of the port oil piping transportation
systemattheoperationstate
3
z
theoperationstate
4
z
transportofonekindof
medium from the pier through parts A and B to
part C using one out of two pipelines of the
subsystem
1
S
, one out of two pipelines in
subsystem
2
S
and two out of three pipelines of
the subsystem
3
S
of the asset A1 illustrated in
Figure5andassetsA
2,A3,A4,A5,A6,A7,A9;
S
1
S
2
S
3
A
11
A
12
A
21
A
22
A
31
A
32
A
33
Figure5. The scheme of the port oil piping transportation
systemattheoperationstate
4
z
theoperationstate
5
z
transportofonekindof
mediumfromthepierthroughpart A toBusing
oneoutoftwopipelinesofthesubsystem
1
S
and
oneoutoftwopipelines ofthesubsystem
2
S
of
theassetA
1illustratedinFigure6.andassetsA2,
A
3,A4,A5,A9;
S
1
S
2
A
11
A
12
A
21
A
22
Figure6. The scheme of port oil piping transportation
systemattheoperationstate
5
z
theoperationstate
6
z
transportofonekindof
mediumfrom the terminalpartBto C using two
out of three pipelines of the subsystem
3
S
, and
simultaneously transport one kind of medium
fromthepierthroughpartA toBusingoneoutof
twopipelinesofthesubsystem
1
S
andoneoutof
twopipelinesofthesubsystem
2
S
oftheassetA1
illustratedinFig.7andassetsA
2,A3,A4,A5,A6,A7,
A
9;
S
1
S
2
S
3
A
11
A
12
A
21
A
22
A
31
A
32
A
33
Figure7. The scheme of the port oil piping transportation
systemattheoperationstate
6
z
the operation state
7
z
transport of one kind of
mediumfromthe terminalpart Bto C usingone
outof threepipelines ofthesubsystem
3
S
,and
simultaneouslytransport secondkind ofmedium
from the terminal part C to B using one out of
threepipelinesofthesubsystem
3
S
oftheasset
A
1illustratedinFigure8andassetsA2,A 4,A6,A7,
A
8,A9.
S
3
A
31
A
32
A
33
Figure8. The scheme of the port oil piping transportation
systemattheoperationstates
7
z
The port oil terminal critical infrastructure
operationprocessZ(t)characteristicsare(Kołowrocki
&Soszyńska‐Budny,2018b):
thelimitva luesoftransientprobabilities(OPC1)of
the operation process Z(t) at the particular
operationstates
,
b
z
:7,...,2,1b
,395.0
1
p ,060.0
2
p
3
0.003,p
,002.0
4
p
5
0.20,p
6
0.058,p
7
0.282.p
(27)
3.5 Parametersofoperationprocessimpactonport oil
terminalcriticalinfrastructuresafety
Thecoefficientsoftheoperationprocessimpactonthe
port oil terminal critical infrastructure intensities of
ageing at the operation states
,
b
z
,7,...,2,1b
are
asfollows[GMUSafetyinteractivePlatform]:
forassetA
1
)(1
)]1([
b
i
=1.10,
)(1
)]2([
b
i
=1.10,
,7,2,1b
i=1,
)(1
)]1([
b
i
=1.20,
)(1
)]2([
b
i
=1.20,
,5,3b
i=1,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=1
(28)
forassetA
2
766
)(1
)]1([
b
i
=1.10,
)(1
)]2([
b
i
=1.10,
,7,2,1b
i=2,
)(1
)]1([
b
i
=1.20,
)(1
)]2([
b
i
=1.20,
,5,3b
i=2,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=2;(29)
forassetA
3
)(1
)]1([
b
i
=1,
)(1
)]2([
b
i
=1,
,7,3,2,1b
i=3,
)(1
)]1([
b
i
=1.20,
)(1
)]2([
b
i
=1.20,
,5b
i=3,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=3;(30)
forassetA
4
)(1
)]1([
b
i
=1.10,
)(1
)]2([
b
i
=1.10,
,7,2,1b
i=4,
)(1
)]1([
b
i
=1.20,
)(1
)]2([
b
i
=1.20,
,5,3b
i=4,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=4;(31)
forassetA
5
)(1
)]1([
b
i
=1,
)(1
)]2([
b
i
=1,
,7,2,1b
i=5,
)(1
)]1([
b
i
=1.20,
)(1
)]2([
b
i
=1.20,
,5,3b
i=5,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=5;(32)
forassetA
6
)(1
)]1([
b
i
=1,
)(1
)]2([
b
i
=1,
,5,2b
i=6,
)(1
)]1([
b
i
=1.10,
)(1
)]2([
b
i
=1.10,
,7,1b
i=6,
)(1
)]1([
b
i
=1.20,
)(1
)]2([
b
i
=1.20,
,3b
i=6,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=6;
(33)
forassetA
7
)(1
)]1([
b
i
=1,
)(1
)]2([
b
i
=1,
,5,3,2b
i=7,
)(1
)]1([
b
i
=1.10,
)(1
)]2([
b
i
=1.10,
,7,1b
i=7,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=7;(34)
forassetA
8
)(1
)]1([
b
i
=1,
)(1
)]2([
b
i
=1,
,6,5,4,3,1b
i=8,
)(1
)]1([
b
i
=1.10,
)(1
)]2([
b
i
=1.10,
,7,2b
i=8;
(35)
forassetA
9
)(1
)]1([
b
i
=1.10,
)(1
)]2([
b
i
=1.10,
,7,2,1b
i=9,
)(1
)]1([
b
i
=1.20,
)(1
)]2([
b
i
=1.20,
,5,3b
i=9,
)(1
)]1([
b
i
=1.30,
)(1
)]2([
b
i
=1.30,
,6,4b
i=9.(36)
3.6 Parametersofportoilterminalcritical
infrastructuresafety
We assume that the port oil terminal critical
infrastructure assets
,
i
A
,9,...,2,1i
atthe critical
infrastructure operation process Z(t) states
,
b
z
,7,...,2,1
b
conditionalsafetyfunctions
1()
[(,)]
b
i
St
[1,
1()
[(,1)],
b
i
St
1()
[(,2)]
b
i
St
],
,0t
(37)
,7,...,2,1
b ,9,...,2,1
i
areexponentialwiththecoordinates
],)]([exp[)],([
)(1)(1
tuutS
b
i
b
i
,0t
(38)
,2,1
u
,7,...,2,1
b ,9,...,2,1
i
where
),()]([)]([
0)(1)(1
uuu
i
b
i
b
i
,2,1u
(39)
,7,...,2,1
b ,9,...,2,1
i
and
,)]([
)(1 b
i
u
,2,1
u
,7,...,2,1
b ,9,...,2,1i
arethecoefficientsofoperationprocessimpactonthe
intensities of degradation of the port oil critical
infrastructure assets
,
i
A
,9,...,2,1i
at the
operation states
,
b
z
,7,...,2,1
b
defined by (28)‐
(36)and
),(
0
u
i
1, 2,u
1, 2,...,9,i
aretheintensitiesofdegradationoftheportoilcritical
infrastructure assets without the operation process
impact,definedby(21)‐(22).
Under the assumption (39), considering (28)‐(36)
and (21)‐(22), it follows that the intensities of assets
departure from the safety states subset
},2,1{
},2{
withoperationimpactontheirsafetyare:
forassetA
1
)(1
)]1([
b
i
=0.017460,
)(1
)]2([
b
i
=0.023913,
,7,2,1
b
i=1,
)(1
)]1([
b
i
=0.019048,
)(1
)]2([
b
i
=0.026087,
,5,3
b
i=9,
)(1
)]1([
b
i
=0.020635,
)(1
)]2([
b
i
=0.028261,
,6,4
b
i=9; (40)
forassetA
2
)(1
)]1([
b
i
=0.01375,
)(1
)]2([
b
i
=0.022,
,7,2,1
b
i=2,
)(1
)]1([
b
i
=0.015,
)(1
)]2([
b
i
=0.024,
,5,3b
i=2,
)(1
)]1([
b
i
=0.01625,
)(1
)]2([
b
i
=0.026,
,6,4
b
i=2; (41)
767
forassetA
3
)(1
)]1([
b
i
=0.0125,
)(1
)]2([
b
i
=0.02,
,7,3,2,1b
i=3,
)(1
)]1([
b
i
=0.015,
)(1
)]2([
b
i
=0.024,
,5b
i=3,
)(1
)]1([
b
i
=0.01625,
)(1
)]2([
b
i
=0.026,
,6,4b
i=3; (42)
forassetA
4
)(1
)]1([
b
i
=0.01375,
)(1
)]2([
b
i
=0.022,
,7,2,1b
i=4,
)(1
)]1([
b
i
=0.015,
)(1
)]2([
b
i
=0.024,
,5,3b
i=4,
)(1
)]1([
b
i
=0.01625,
)(1
)]2([
b
i
=0.026,
,6,4
b

i=4; (43)
forassetA
5
)(1
)]1([
b
i
=0.0125,
)(1
)]2([
b
i
=0.02,
,7,2,1b
i=
5,
)(1
)]1([
b
i
=0.015,
)(1
)]2([
b
i
=0.024,
,5,3b
i=5,
)(1
)]1([
b
i
=0.01625,
)(1
)]2([
b
i
=0.026,
,6,4b
i=5; (44)
forassetA
6
)(1
)]1([
b
i
=0.0125,
)(1
)]2([
b
i
=0.02,
,5,2b
i=6,
)(1
)]1([
b
i
=0.01375,
)(1
)]2([
b
i
=0.022,
,7,1
b
i=
6,
)(1
)]1([
b
i
=0.015,
)(1
)]2([
b
i
=0.024,
,3b
i=6,
)(1
)]1([
b
i
=0.01625,
)(1
)]2([
b
i
=0.026,
,6,4b
i=6; (45)
forassetA
7
)(1
)]1([
b
i
=0.0125,
)(1
)]2([
b
i
=0.02,
,5,3,2b
i=7,
)(1
)]1([
b
i
=0.01375,
)(1
)]2([
b
i
=0.022,
,7,1
b
i=
7,
)(1
)]1([
b
i
=0.01625,
)(1
)]2([
b
i
=0.026,
,6,4b
i=7; (46)
forassetA
8
)(1
)]1([
b
i
=0.0125,
)(1
)]2([
b
i
=0.02,
,6,5,4,3,1b
i=8,
)(1
)]1([
b
i
=0.01375,
)(1
)]2([
b
i
=0.022,
,7,2b
i=8; (47)
forassetA
9
)(1
)]1([
b
i
=0.01375,
)(1
)]2([
b
i
=0.022,
,7,2,1
b
i=9,
)(1
)]1([
b
i
=0.015,
)(1
)]2([
b
i
=0.024,
,5,3b
i=9,
)(1
)]1([
b
i
=0.01625,
)(1
)]2([
b
i
=0.026,
,6,4
b
i=9. (48)
3.7 Predictionofsafetyandresiliencecharacteristicsof
portoilterminalcriticalinfrastructure
Considering that the coordinates of the conditional
safety functions(37) forthe port oil terminal critical
infrastructure assets
,
i
A
,9,...2,1i
are of the form
(38) with the intensities of ageing at the operation
states
,
b
z
,7,...,2,1
b
given respectively by (40)‐
(48), as the oil terminal critical infrastructure is a
three‐state (z =2) seriessystem, then by Corollary 1
from(Kołowrocki&Soszyńska‐Budny,2019),theyare
givenby:
)1(1
)],([
tS
=[1,
)1(1
)]1,([ tS
,
)1(1
)]2,([ tS
],
t0,
where
)1(1
)]1,([ tS
=exp[‐0.12371t],
)1(1
)]2,([ tS
=exp[‐0.193913t]; (49)
)2(1
)],([
tS
=[1,
)2(1
)]1,([ tS
,
)2(1
)]2,([ tS
],
t0,
where
)2(1
)]1,([ tS
=exp[‐0.12246t],
)2(1
)]2,([ tS
=exp[‐0.191913t]; (50)
)3(1
)],([
tS
=[1,
)3(1
)]1,([ tS
,
)3(1
)]2,([ tS
],
t0,
where
)3(1
)]1,([ tS
=exp[‐0.131548t],
)3(1
)]2,([ tS
=exp[‐0.206087t]; (51)
1(4)
[(,)]t
S
=[1,
1(4)
[(,1)]tS
,
1(4)
[(,2)]tS
],t0,
where
)4(1
)]1,([ tS
=exp[‐0.146885t],
)4(1
)]2,([ tS
=exp[‐0.230261t]; (52)
)5(1
)],([
tS
=[1,
)5(1
)]1,([ tS
,
)5(1
)]2,([ tS
],
t0,
where
768
)5(1
)]1,([ tS
=exp[‐0.131548t],
)5(1
)]2,([ tS
=exp[‐0.206087t]; (53)
)6(1
)],([ tS
=[1,
)6(1
)]1,([ tS
,
)6(1
)]2,([ tS
],t0,
where
)6(1
)]1,([ tS
=exp[‐0.146885t],
)6(1
)]2,([ tS
=exp[‐0.230261t]; (54)
)7(1
)],([ tS
=[1,
)7(1
)]1,([ tS
,
)7(1
)]2,([ tS
],
t0,
where
)7(1
)]1,([ tS
=exp[‐0.12496t],
)7(1
)]2,([ tS
=exp[‐0.195913t]. (55)
Hence, applying (10), the expected values of the
portoilterminalcriticalinfrastructurelifetimesinthe
safety state subsets
},2,1{},2{
atthe operationstates
,
b
z
,7,...2,1b
respectivelyare:
)1(1
)]1([
8.08,
)1(1
)]2([
5.16years,
)2(1
)]1([
8.17,
)2(1
)]2([
5.21years,
)3(1
)]1([
7.60,
)3(1
)]2([
4.85years,
)4(1
)]1([
6.81,
)4(1
)]2([
4.34years,
)5(1
)]1([
7.60,
)5(1
)]2([
4.85years,
)6(1
)]1([
6.81,
)6(1
)]2([
4.34years,
)7(1
)]1([
8.00,
)7(1
)]2([
5.10years. (56)
From the results (27) and (49)‐(55), a pplying (7),
the port oil terminal critical infrastructure
unconditionalsafetyfunction(SafI1)isgivenby
1
(,)t S
1
[1, ( ,1),t S
1
(,2)],tS t0,
where
)1,(
1
tS
0.395exp[‐0.12371t]+0.060exp[‐0.12246t]
+0.003exp[‐0.131548t]+0.002exp[‐0.146885t]
+0.200exp[‐0.131548t]+0.058exp[‐0.146885t]
+0.282exp[‐0.12496t] (57)
)2,(
1
tS
0.395exp[‐0.193913t]+0.060exp[‐0.191913t]+
0.003exp[‐0.206087t]+0.002exp[‐0.230261t]+
0.200exp[‐0.206087t]+0.058exp[‐0.230261t]+
0.282exp[‐0.195913t] (58)
Considering(27)and(56)andapplying(9)forr=
u,theexpectedvaluesandstandarddeviations(SafI4
‐SafI5) of the port oil terminal critical infrastructure
lifetimes in the safety state subsets
},2,1{},2{
respectivelyare:
)1(
1
μ
7.89years
)2(
1
μ
5.03years, (59)
)1(
1
σ
7.91,
)2(
1
σ
5.05years, (60)
andfurther, by(12),itfollowsthatthemeanvaluesof
theoilterminalcriticalinfrastructurelifetimesinthe
particularsafetystatesare:
)1(
1
μ
2.86,
)2(
1
μ
5.03years. (61)
Asthecriticalsafetystateisr=1,thenby(8)and
(57), the port oil terminal critical i nfrastructure risk
function(SafI2),isgivenby
)(
1
tr
=1–
)1,(
1
tS
fort0. (62)
The graph of the risk function
)(
1
tr
of the oil
terminalcriticalinfrastructureis(SafI3).
From (3.16) in (Kołowrocki & Soszyńska‐Budny,
2017) and (62), the moment when the oil terminal
critical infrastructure risk function exceeds a
permittedlevel
=0.05(SafI6),is
1
=
)()(
11
r
0.404years. (63)
Applying (11), the oil terminal critical
infrastructureintensitiesofageing(SafI7)are:
)1,(
1
tλ
0.126743,
)2,(
2
tλ
0.198807. (64)
Considering (26) and (64) and applying (15), the
coefficientsoftheoperationprocessimpactontheoil
terminal critical infrastructure intensities of ageing
(ResI1),are:
)1,(
1
tρ
1.094,
)2,(
1
tρ
1.094. (65)
Finally, by (16) and (65), the port oil terminal
critical infrastructure resilence indicator (ResI2), i.e.
the coefficient of the port oil terminal critical
infrastructure resilience to the operation process
impact,is
()t
= 1/ ( ,1) 0.914 91%.1/t

1
(66)
IntheTable1givenbelow,theinventoryofbasic
safetyandresilienceindicatorsfortheportoilpiping
transportationsystemispresented.
Fromthis inventoryit canbe seenthat operation
process of the oil terminal critical infra structure has
significantimpactonitssafetyandresilience.
769
Table1. Inventory of basic safety, risk and resilience
indicatorsofoilterminalcriticalinfrastructure
_______________________________________________
Oilterminalcriticalinfrastructuresafety,riskandresilience
indicators
___________________________________________________________
Impacts meanvalueof momentof resilience
lifetimeupto exceeding indicator
exceeding acceptable inrange0‐1
criticalsafety risklevel RI(1)
state1inyears
δ
=0.05in

)1(
years
τ
_______________________________________________
without 8.6300.4401.00
impacts
operation7.8900.4040.914
impact
_______________________________________________
4 CONCLUSIONS
Thesafetyandresilienceindicatorsweredetermined
forportoil terminal.Further research canberelated
with considering other impacts and solving the
problemsofcriticalinfrastructuresafetyoptimization
andfindingofoptimalvaluesofsafetyandresilience
indicators. These results can help to mitigate critical
infrastructure accident
consequences and to enhance
criticalinfrastructureresiliencetooperationandother
impacts. This research can also result in the
backgrounds for business continuity and cost‐
effectivenessanalysisofcriticalinfrastructuresunder
operationandotherimpacts.
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