=Paper=
{{Paper
|id=Vol-2805/paper3
|storemode=property
|title=Computational Models and Methods for Automated Risks Assessments in Deterministic Stationary Systems
|pdfUrl=https://ceur-ws.org/Vol-2805/paper3.pdf
|volume=Vol-2805
|authors=Ivan Izonin,Igor Nevliudov,Yurii Romashov
|dblpUrl=https://dblp.org/rec/conf/citrisk/IzoninNR20
}}
==Computational Models and Methods for Automated Risks Assessments in Deterministic Stationary Systems==
https://ceur-ws.org/Vol-2805/paper3.pdf
Computational Models and Methods for Automated
Risks Assessments in Deterministic Stationary Systems
Ivan Izonin 1[0000-0002-9761-0096], Igor Nevliudov 2[0000-0002-9837-2309] and Yurii
Romashov2,3[0000-0001-8376-3510]
1 Lviv Polytechnic National University, 12, Bandera str., Lviv, Ukraine, 79013, Ukraine
2 Kharkiv National University of Radio Electronics, Nauky Ave. 14, Kharkiv, 61166, Ukraine
3 V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine
yurii.romashov@nure.ua
Abstract. It is developed the generalized computational models and methods for
automated assessments the risks due to uncertainties of influencing factors in the
complicated deterministic stationary systems. The principal idea of these models
and method is based on computational solving the finite set of boundary value
problems modelling the considered systems to represent the deterministic prop-
erties of researched possible risks in the general case. It is shown how further
computational handling of these established deterministic properties by using the
results of the probability theory will allow to have assessments of the researched
risks in the form of probabilities of the dangerous events associated with these
risks. These proposed generalized computational models and methods can be
used for assessments the risks in the deterministic stationary systems with the
different nature and they are can be useful for different purposes including for
manufacturers to substantiate the possible warranty life as well as for insurance
companies to estimate the possible hazards. The example of using the proposed
models and methods deals with the risks assessment of melting the ceramic nu-
clear fuel pellets during operating in nuclear reactors. Using this example, it is
quantitatively shown that the deterministic properties have the significant influ-
encing on the risks assessments. Besides, it is shown that the deterministic prop-
erties can almost nullify the uncertainties in the influencing factors, and these
properties can provide the low risks of dangerous states during the system oper-
ation.
Keywords: Risks, Computational Assessment, Deterministic system.
1 Introduction
Considering risks for different kinds of systems is one of in current interest scientific
problem in present necessary to predict possible states and to substantiate required
states of the systems under permanently existed uncertainties in their influencing fac-
tors for optimal management.
Copyright © 2020 for this paper by its authors. This volume and its papers are published under
the Creative Commons License Attribution 4.0 International (CC BY 4.0).
2
The state of deterministic systems is fully unequivocally predefined by the influenc-
ing factors and this is widely used for the different industrial purposes. The stability of
operating the most industrial systems is based on supporting the stability of their state
and in some cases it is suitable to neglect of changing the state of such systems during
the time and to consider such systems as the stationary. Thus the deterministic station-
ary systems can represent the wide kinds of the industrial used systems. Although, the
state of the deterministic systems is predefined by the influencing factors, but the prob-
able deviations of the state are had presenting in such systems during their operation as
shown by the experience.
2 Related works
The problems about risks are considered for different purposes including to predict
damages in structures [1-3], to estimate the possible cost of new products and services
[4-6], to substantiate possibilities of complicated power generation systems [6-8]. Alt-
hough, listed above [1-8] and the most other researches about risks are dealt with the
particular problems only it is obviously that the different particular approaches for risks
assessments have some uniform items: the probabilistic nature of the risk as well as the
sensitivity conceptions are used explicitly or implicitly. Opportunities of significant
improving the management (control) of complicated systems in agriculture, transport,
technical, economic, environment and human areas on the basis of risks assessments
lead to design the special decision support systems ground on using of modern infor-
mational technologies [9-11]. Industrial designing of such decision support systems is
based on risk assessments requires to develop of the serious methodological support
grounded on the general representations about different aspects of the risk assessments,
and such researches are known [12, 13] at present. At the same time, the automation of
risk assessments required to efficient decision support systems is not fully developed
at present although, some researches [10, 14] are dealt with this.
The purpose of this research is to develop the computational models and methods
for automated risks assessments in deterministic stationary systems suitable for using
in decision support information systems. To realize this purpose the follows objectives
will be accomplish:
• On the base of the theory of probability it will be proposed the formal definition of
the quantitative measure of the risk and it will be shown that risks in deterministic
systems can be represented as the results of uncertainties in influencing factors.
• The generalized computational models and methods suitable for automated risks as-
sessments in deterministic stationary systems will be developed on the basis of pro-
cessing the especial sampling sets built numerically using the mathematical models
representing the deterministic properties of researched systems by means differential
equations with boundary conditions.
• The conception of automation the risks assessments in the deterministic stationary
systems on the basis of computer information technologies will be presented and
discussed. The different kinds of tasks required for risks assessments will be shown
and using of the programming languages suitable for these tasks will be discussed.
3
• It will be considered the example of using the developed approaches for risk assess-
ments in deterministic stationary systems and this example will be dealt with the
risks assessment of melting the ceramic nuclear fuel pellets during operation in nu-
clear reactors. Using this example, it will be quantitatively shown that the determin-
istic properties have the significant influencing of the risks assessments. Besides, it
will be shown that the deterministic properties can almost nullify the uncertainties
in the influencing factors, and these properties can provide the low risks of danger-
ous states during the system operation despite the uncertainties naturally existing in
the influencing factors.
The presented here results about the computational models and methods for automation
risks assessments in deterministic stationary systems can be used to research the sys-
tems with different natures including the various engineering, economic, human and
environment systems unlike the methods proposed for the each particular problems in
the most of existed researches [1, 4, 7-9].
3 Models and methods for automated computational risks
assessments in deterministic stationary systems
The main principle of using the computational models and methods for risks assess-
ments is to define the quantitative measures of the researched risks, to construct the
necessary mathematical models defining these measures, to design the corresponding
computational models and methods as well as to make the required software and exe-
cuting the computer simulations to calculate these measures.
3.1 Quantitative measures of the risks
The "risk" notion used in relation to some system can be imagined as the likelihood of
some dangerous states of this system and such likelihood can be defined thru the notion
about randomness which is the fundamental notion in the theory of probability. Let
denote as y the real number representing the value of researched parameter of the
considered system. We will assume that the normal states of the considered systems are
corresponded to the parameter y with the values (see Fig. 1)
ymin ≤ y ≤ ymax , (1)
where ymin and ymax are the given minimum and maximum values permissible for the
normal states of the considered system.
Using definition (1), the risks in the considered system can be imagined as any vio-
lations of the double inequality (1). Taking into account this circumstance, the quanti-
tative measure of the risk in the considered system can be defined as follows:
Q =1 − P { ymin ≤ y ≤ ymax } , (2)
4
where Q is the measure of the risk in the considered system and P {} is the proba-
bility of the event corresponding to the satisfied conditions placed inside the brackets.
Fig. 1. Origin of the risks in the deterministic system
Let denote as x the real number representing the value characterized the influencing
factor which can govern the y value. Due to the assumption that the considered system
is deterministic, it is existed the strongly definite functional relation between x and y
values (see Fig. 1):
y = y ( x) . (3)
The character of the functional relation (3) is the consequence of the properties of the
considered system and due to this circumstance the relation (3) represents the determin-
istic properties of this considered system. Thus, all risks with measure (2) in this con-
sidered deterministic system (3) are possible due to the uncertainties of the influencing
factors, and these uncertainties can be imagined as the random value of the x parame-
ter. The probabilistic characteristics of the y random value are predefined by the prob-
abilistic characteristics of the x random value as well as by the deterministic properties
(3) of the considered system (see Fig. 1):
f ( y ) = g ( x ( y ))
dx
, (4)
dy
where f ( y ) is the probability density of the y value; g ( x ) is the probability density
of the x value; x ( y ) is the inverse function to the function (3).
Using the probability density (4) and the risk measure (2), we can define the risks in
the considered deterministic system as follows (see Fig. 1):
ymax
Q = 1 − ∫ g ( x ( y ))
dx
dy . (5)
ymin
dy
5
It is necessary to note, that the state of the system can be defined by not one parameter
y , but the several parameters y1 , y2 , y3 ,... and this case can be reduced to considering
the risk measures defined similarly to relation (4) for each of the y1 , y2 , y3 ,... param-
eter. It is necessary to note also, that the influencing factors can be defined by not one
parameter x , but the several parameters x1 , x2 , x3 ,... and deterministic properties (3)
of the system will be the several variable function in this case. The uncertainties of
influencing factors will be reduced to the set of probability densities defining the each
variable. To find the probability density if the case of several variable function
y ( x1 , x2 , x3 ,) is the more complicated than in the relation (4), but this will be the
typical mathematical task not principal for this research and this case will not be dis-
cussed here to avoid the not necessary complications.
3.2 Quantitative defining the uncertainties of the influencing factors
The uncertainties of the influencing factors can be reduced to the random values of the
x parameter defining these influencing factors and it is sufficiently to define the prob-
ability density g ( x ) of the x value to estimate the risks quantitatively as was shown
by the relations (4) and (5). To define the g ( x ) function it is possible to use the prob-
ability densities well-known in the probability theory including uniform and triangle
distributions as well as the Weibull's and others distributions (see Fig. 2). The uniform
and triangle distributions can be defined by the possible minimum xmin and maximum
xmax values of the x parameter (see Fig. 2), but the Weibull’s distribution can be de-
fined by the expected value and the dispersion, which can be defined using the expert
assessments.
Fig. 2. Uniform (a), triangle (b) and Weibull’s (c) probability densities for influencing factors
It is obviously, that the suitable choice of the probability density (see Fig. 2) modelling
the uncertainties of the influencing factors will allow defining the risks (5) precisely
sufficiently. At the same time, it is impossible to give the formal universal recommen-
dations to make such choice, and only the expert assessments are possible.
6
3.3 Modelling the properties of the deterministic stationary system
As noted above, the risks assessments will be reduced to the simplest computing by
using the calculus of definite integrals in the case of the explicitly given relation (3)
and for the given probability density g ( x ) . At the same time, the difficulties of risks
assessments are due to the implicitly given relation (3) such that the y parameter is
defined thru some other state parameters of the considered system as follows:
y = y (u ) , (6)
where u is the vector of the state parameters of the considered system.
Defining the vector u in general case is significantly depends from the nature of the
considered system and can be based on the different mathematical approaches. One of
the most general approaches for defining the vector of state parameters is based on the
differential equations using. The considered system is imagined as the set of the points
r in some suitable space; we will denote as ϒ and υ the full domain including the
boundary and the boundary of the considered system. In the case of the stationary sys-
tem the state is depends on the point r only:
=u u ( r ) ∀r ∈ ϒ . (7)
Depending of the vector u represented as (7) from the influencing factors defined by
the x parameter in the most general case can be represented by using the boundary
value problem
A ( u (=
r ) ; x ) f ( r ; x ) ∀r ∈ рϒυ , (8)
B ( u (=
r ) ; x ) p ( r ; x ) ∀r ∈ υ , (9)
where A is the operator of deferential equations; f is the given vector in some space
agreed with the properties of the A operator; рϒυ is the complementary of the υ
boundary to the domain ϒ ; B is the operator defining the boundary conditions agreed
with the properties of the A operator; p is the given vector in some space agreed with
the properties of the B operator.
The relations (8), (9) representing the set of the boundary value problems parame-
terised by the x value will allow defining the set u = u ( r ; x ) of solutions. The solu-
tions u = u ( r ; x ) and the relation (6) will allow representing the deterministic proper-
ties of the considered system as (3). The basic idea of modelling the deterministic prop-
erties of the considered system is to solve the boundary value problem (8), (9) for some
especially chosen values { x1 , x2 ,} of the x parameter and to compute the values
{ y1 , y2 ,} of the y parameter using the solutions {u ( r , x1 ) , u ( r , x2 ) ,} set and the
relations (6). Defining the function (3) required for the risks assessments using the data
7
sets { x1 , x2 ,} and { y1 , y2 ,} is reduced to the well-known problem about identifi-
cation of the system and can be realised by the least square method for example. Thus,
we can build the approximation of the implicit defined function (3). It is necessary to
understand, that solving the boundary-value problem (8), (9) for the given x values
cannot be realised exactly in the most cases, and it is necessary to use the numerical
methods like the finite elements method or the finite differences (the grid) method.
Numerical methods for solving the boundary value problems are well-known and no-
tions about these are not necessary in this research.
3.4 Automation the risks assessments procedure
The decision support systems wide implementation in the modern managing procedures
requires providing the opportunities of quickly risks assessments for different variants
of decisions which can be achieved due to the complex automation of the risks assess-
ments procedures. The algorithm of the automated procedure of the risks assessments
in the deterministic stationary systems based on the proposed approach is presented on
the Fig. 3. It is possible to provide the wide automation and in some cases the full
automatic solving the separate tasks using the different programming technologies, but
it is impossible to provide the fully automatic of the risks assessments procedure glob-
ally due to the required expert estimations of the interim results (see Fig. 3). Thus, the
risks assessments procedure can include the cyclic solving of some interim tasks until
the expert positive qualifying of results will be given, and then solving the next required
task will be started (see Fig. 3). Considering these circumstances, the automated risks
assessments systems can be operated only by the experts fully qualified in the corre-
sponding necessary subject areas, but automation of typical processing required for the
risks assessments procedure (see Fig. 3) will allow increasing the productivity of this
procedure.
Risks assessments procedure is actually the data (information) transformation pro-
cess, and due to this automation of the risks assessments procedure can be realized
using the computer aided technologies. The risks assessment procedure includes a lot
of the typical mathematical computing like the matrix computing required to numeri-
cally solve the boundary value problem (see Fig. 3) representing the mathematical
model of the considered deterministic stationary system as well as for computing the
risk quantitative measures. Besides, the risks assessment procedure must use the expert
systems based on the artificial intelligence technologies like the logical programming
and knowledge bases required for the automated analysing of the influencing factors
uncertainties considering with the known knowledge in the corresponding subject ar-
eas. Also, the data analysis for the system properties identification must be necessarily
used in any automated risk assessments procedure. It is suitable to use the FORTRAN
language for the mathematical computing required to solve the boundary value prob-
lems representing the mathematical model of the considered deterministic stationary
system as well as to compute the risks quantitative measures. The PYTHON language
is suitable to solve the data analysis tasks required for the system identification, but the
8
PROLOG, the LISP and similar programing languages are more suitable to realize the
expert systems required for the automated risks assessments.
4 Computer simulations, results and discussions
We will consider further the problem about assessments the risks of melting the ceramic
nuclear fuel pellets in the core of nuclear reactors to show the example of using the
proposed models and methods for automated computational risks assessments in deter-
ministic stationary systems. The nuclear fuel common for the most of existed nuclear
reactors including the VVER-1000 nuclear reactor widely used in Ukraine and some
countries of Eastern Europe and Asia is the compact product is made as the cylinder
with the central hole (see Fig. 4a) to exclude melting in the central volume surrounding
the longitudinal axis; the typical sizes of the fuel pellets are the height h 20mm and
the radii a 5mm the hole and b 10mm of the pellet. These fuel pellets are placed
inside the cladding of the long fuel rods (see Fig. 4b) with lenght about 5000mm, and
the heat producing due to the nucleus fission reactions inside these pellets 1 is trans-
ferred thru the gaseous gap 2 and thru the cladding 3 to the heat carrier 4 surrounding
the cladding on moving along the cladding's longitudinal axis.
9
Fig. 3. Risks assessments procedure
10
Fig. 4. The sizes (a) nuclear fuel pellet and position of this pellet inside the fuel rod (b)
The temperature of the nuclear fuel pellet 1 is defined by the intensity of the internal
heat volume sources due to the nuclear fission reaction as well as by the heat transfer
of generated heat thru the gaseous gap 2, thru the cladding 3 and outside the cladding
into the heat carrier 4 (see Fig. 4b). The heat transfer processes are very sensitive to the
mode of moving the heat carrier and it is possible to have the local perturbations of
moving the heat carrier which can lead to the minimum heat transfer from the cladding
to the heat carrier and as the result of this can lead to the extremely high temperatures
and to melting the fuel pellets. It is impossible to fully exclude melting of fuel pellets
during operation the nuclear reactor, but melting the pellets inside of 1 or 2 or even 10
fuel rods from about 50000 fuel rods forming the core is not dangerous. Thus, it is
necessary to substantiate that the risks of melting the nuclear fuel pellets inside the fuel
rods are negligible during operation the nuclear reactor.
4.1 Modelling the fuel pellet temperature state with the uncertainty in the heat
transfer to outside the cladding
The mathematical model of the fuel pellet temperature state can be represented using
the equations and the boundary conditions all-known in the theory of heat conduction.
We will consider the radial heat flows only and we will neglect the axial and the cir-
cumferential heat flows in the fuel pellet. Such simplification will allow considering
the principal heat flows providing the heat exchange during operation the fuel pellets,
and it is useful for initial researches. Considering these circumstances, the mathematical
model of the fuel pellet temperature state can be represented as follows [15]:
dq q
+= H , a < r < b , (10)
dr r
dT
q =−λ (T ) , a≤r ≤b, (11)
dr
11
=q 0,=r a, (12)
k (T TH .C . ) , r =
q =− b, (13)
where q = q ( r ) and T = T ( r ) is the radial heat flow and the temperature inside the
fuel pellet; r , a ≤ r ≤ b is the radial coordinate; H is the intensity of the volume heat
sources due to the fission nuclear reactions in the pellet; λ (T ) is the thermal conduc-
tivity of the fuel pellet depending on the temperature; k is the heat transfer coefficient
defining the heat flow from the fuel pellet to the heat carrier; TH .C . is the temperature
of the heat carrier.
The intensity of the volume heat sources can be approximately defined considering
the axial offset as [15]
1,5W
H= , (14)
nπ ( b 2 − a 2 ) L
where W is the heat power of the nuclear reactor; n is the count of the fuel rods form-
ing the core; L is the length of the fuel cylinder inside the fuel rod.
The heat transfer coefficient from the fuel pellet to the heat carrier is defined by the
widths and the thermal conductivities of the gaseous gap and the wall of the cladding,
as well as by the heat transfer from the cladding to the heat carrier and can be repre-
sented as all-known in the heat transfer theory for the cylindrical wall [15]:
−1
b Rg b R b
k = ln + ln c + , (15)
λ b λc Rg α Rc
g
where λg and Rg is the thermal conductivity and the radius (see Fig. 4b) of the gaseous
gap; λc and Rc is the thermal conductivity and the radius (see Fig. 4b) of the cladding;
α is the heat transfer coefficient defines the heat flow from the cladding to the heat
carrier.
For numerical simulations of the fuel temperature states we will use the parameters
values included in the (10)–(14) corresponding to the fuel of the VVER-1000 nuclear
reactor and all-known in scientific literature [15]:
= =
a 1,15mm, =
b 3, 765mm, =
W 3000MW, =
n 50856, =
L 3530mm, THC 583K , (16)
R=
g 3,86mm, R=
c 4,55mm, λ=
g 0,3 W ( m ⋅ K ) , λ=
c 20,5 W ( m ⋅ K ) . (17)
The thermal conductivity of the fuel pellet is significantly depended on the temperature
and this is the one of principal circumstance which must be considered. We will use the
follows function defining this temperature dependence:
λ (T )= A ⋅ T B , (18)
12
where A ≅ 793,909118652344 and B ≅ -0,766199052333832 are corresponded to the
approximation of all-known literature data about the thermal conductivity of the UO2
ceramic nuclear fuel; λ (T ) and T is measured in W ( m ⋅ K ) and in K .
In the proposed mathematical model (10)–(18) all uncertainties of the influencing
factors are concentrated in the value α of the heat transfer coefficient. It is known [10]
that the α values corresponded to the normal operation of the nuclear reactor VVER-
1000 must satisfy the follows double inequality [15]:
33 ≤ α ≤ 35 kW ( m 2 ⋅ K ) . (19)
For fuel pellets melting risks assessments we will consider the more wide values of the
α coefficient:
α m ≤ α ≤ 50 kW ( m 2 ⋅ K ) , (20)
where α m is the value of the α coefficient corresponding to the melting temperature
Tm 2820K on the radius r = a (see Fig. 4) of the fuel pellet.
Solving the nonlinear boundary value problem (10)–(15) for the input data (16)–
(18), (20) was realized numerically using the finite differences method and all-known
the Picard’s iteration method. It was used the LU-method to solve the linear algebraic
equations system representing the linear problem in each of iterations. The programs
for automated processing of these was developed using the FORTRAN programming
language considering the notes about automation discussed above. These programs re-
alize the typical all-known numerical methods and they will not be discussed here.
4.2 Results and discussing
Computer simulations of the fuel pellet temperature state using the mathematical model
(10)-(15) for the initial data (16)-(18), (20) allow to find the α m value, corresponded
to melting temperature of the fuel pellet on its internal radius:
α m ≅ 0, 75 . (21)
Besides, such computer simulations allow to build the sampling data required to estab-
lish the deterministic properties of the considered system as the functional depending
of the α coefficient from the temperature = Ta T= ( r a ) on the internal radius of the
fuel pellet (see Table 1). The sampling data (see Table 1) can be approximated as:
α (Ta ) = eC ln T + C ln T + C ,
2
1 a 2 a 3
(22)
13
where numerical parameters C1 ≅ 28,39302401293563 , C2 = −436,299681002904
and C3 ≅ 1673,757077973671 are defined using the all-known list square method for
the α coefficient measured in kW ( m 2 ⋅ K ) and the temperature Ta measured in K .
Representing the deterministic properties of the fuel pellet temperature state using
the approximation (22) and the sampling data (see Table 1) are shown on the Fig. 5a.
Comparing the line and the markers on the Fig. 5a, allows seeing that the proposed
approximation (22) has the good agreement with the sampling data.
Table 1. Sampling data for defining the deterministic properties.
The temperature Ta , K on the internal (
The heat transfer coefficient α , kW m 2 ⋅ K )
radius of the fuel pellet defining the heat flow from the cladding
2840,85696253441 0,75
1458,60125430316 10,0
1420,06137960674 15,00
1400,76319976693 20,00
1389,1751065182 25,00
1381,44584776636 30,00
1375,92304476484 35,00
1371,77989675907 40,00
1368,5568265724 45,00
1365,97797773063 50,00
Fig. 5. Representing the deterministic properties (a) of the temperature state and the probability
density (b) of the researched temperature
The values of the heat transfer coefficient α are inside the interval defining by the
double inequality (20) and the value (21), but the real value is uncertain. Let assume
that the heat transfer coefficient α can have any value from the interval (20), (21) with
14
the equal probability, i.e. the probability density of the α value is uniform as on the
Fig. 2a inside the interval defining by the double inequality (20) and the value (21). The
probability density of the temperature Ta can be defined using the relation (4) and the
approximation (22) and considering the noted above assumption as:
C1 C
f (Ta ) PeC1 ln Ta + C2 ln Ta + C3 2
2
= ln Ta + 2 , (23)
Ta Ta
where P ≅ 0, 020304568527919 is defined by the uniform probability density of the
heat transfer coefficient α value in the interval defined by the inequality (20) and the
value (21).
Although, the probability density of the random value α representing the influenc-
ing factors was assumed as the uniform (see Fig. 2a) the probability density (23) of the
researched temperature Ta inside the fuel pellet is significantly differ from the uniform
distributions (see Fig. 5b) due to the deterministic properties of the temperature state of
the fuel pellet. Thus, this obtained result (see Fig. 5b) is quantitatively shown that the
deterministic properties (see Fig. 5a) of the considered stationary system can have sig-
nificant influencing on the probability function defining the risks assessments.
In this particular example the risk Q of melting the nuclear fuel pellet during oper-
ation in the nuclear reactor can be computed using the relation (5) which can be reduced
to the follows suitable view:
Tm
Q = ∫ f (Ta ) dTa , (24)
S ⋅Tm
where S is the parameter defining the maximal available value of the temperature Ta
in the internal radius of the fuel pellet during the nuclear reactor operation.
The risk quantitative measure defined by the relation (24) actually shows the proba-
bility of the event corresponded to the double inequality
S ⋅ Tm ≤ Ta ≤ Tm . (25)
Results of the risks assessment using the relation (25) and the defined above probability
density (23) are shown that the risks of melting is significantly decreased with increas-
ing the maximal allowed temperature of the fuel pellet (see Fig. 6). Although, these
results are obtained using the significant assumptions and simplifications these results
are in good quantitative agreement with the all-known experience of the nuclear reac-
tors operating.
15
Fig. 6. Estimation of the quantitative measure of the risk of melting the nuclear fuel pellets
5 Conclusions
The generalized computational models and methods for automated risks assessments in
deterministic stationary systems are developed in this research to be used for research-
ing the risks in the systems with different natures including the various engineering,
economic, human and environment systems unlike the particular approaches discussed
in the most of the existed researches. Due to this research the follows results were de-
veloped:
• It is proposed the formal definition of the quantitative measure of some risk in the
form of the probability value of the hardly events corresponded to the researched
risks. Using this formal definition and the all-known results of the theory of proba-
bility, it is shown that risks in deterministic systems can be represented as the results
of uncertainties naturally presenting in the influencing factors.
• It is developed the generalized computational models and methods suitable for au-
tomated risks assessments in deterministic stationary systems on the basis of com-
putational processing the especial sampling sets built numerically using the mathe-
matical models representing the deterministic properties of researched stationary
systems by means differential equations with boundary conditions.
• It is proposed the general conception of automation the risks assessments in deter-
ministic stationary systems on the basis of computer information technologies. It is
shown that the risks assessment procedure includes a lot of the typical mathematical
computing required to numerically solve the boundary value problem representing
the mathematical model of the considered deterministic stationary system as well as
for computing the risk quantitative measures, and the FORTRAN language is suita-
ble more for them. Besides, it is shown that the automated risks assessment must use
the expert systems based on the artificial intelligence technologies like the logical
programming, and the PROLOG, the LISP and similar programing languages are
suitable for them. It is also shown that the risks assessments must include the data
analysis required for the system identification, and the PYTHON programming lan-
guage is suitable more for them.
• It is considered the example about the risks assessments of melting the ceramic nu-
clear fuel pellets during operation in nuclear reactors to show of using the developed
generalised approaches. It is quantitatively shown that the deterministic properties
have the significant influencing on the risks assessments. Besides, it is shown that
16
the deterministic properties can almost nullify the uncertainties in the influencing
factors, and these can provide the low risks of hazard states during the system oper-
ation despite the uncertainties naturally existing in the influencing factors.
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