**4. The analysis of limiting states**

To reach the acceptable protectability of the NPP equipment, implementation of

If on axes *RT*, *RN*, and *RO* to put aside classes from 1 to 7 for accidents and disasters on extent of increment of their severity (1—local, 2—object, 3—district, 4—regional, 5—national, 6—global, and 7—planetary), then the quantitative assessment of extent of the NPP safety and any of its components by criteria of risks is represented possible. Such estimation is given by the radius vector in threedimensional space "*RT*-*RN*-*RO*". The strength and life time improvement on all stages of installation design, making, and service should promote decrease in danger

For an NPP transfer in safe states with the use of risk criteria *RN*, *RT*, and *RO* (**Figure 19**), it is necessary to reduce the possibility (risk *RS*) of uncontrollable emission of potentially dangerous substances *W* and energies *E* and also a loss of

> *R*2 *<sup>W</sup>* <sup>þ</sup> *<sup>R</sup>*<sup>2</sup>

or to reduce the relative risks of accidents and disasters *RN*, *RT*, and *RO* as in

This result can be attained by the creation of monitoring systems for diagnostics and monitoring of risk parameters *RN*, *RT*, *RO*, *RW*, *RE*, and *RI* and guard *Z*(*τ*), and also by the introduction in the analysis of safety *S*(*τ*) scenarios of occurrence and

The state, regional and object control, regulating and providing of safety *S*ð Þ*τ* by system risks criteria *RS*ð Þ*τ* comes to the qualitative both quantitative statistical and determined analysis on the given interval of time Δ*τ* of all service parameters and to implementation of complex activities on decrease of system risks from actual

q

� � <sup>¼</sup> ð Þ� <sup>1</sup>*=nS RSc* <sup>¼</sup> *FR PS*

where *nS* is the safety factor on system risks; *RSc* is the unacceptable (critical)

Safety of the NPP by criteria of risks can be considered ensured if the inequality

The interval of time Δ*τ* for which risks *RS* are defined usually is accepted to equal

According to Eqs. (15) and (16), control and planning with the use of the criteria

To decision making about the level of allowable values [*RS*], [*PS*], and [*US*] with

To scientifically well-founded level of definition of necessary expenditures [*M*] on decreasing risks with sampling and improving of efficiency of these expenditures *mM* Thus, predicting, monitoring, and forestalling of accidents and disasters for an

NPP (including by improving of all parameters of strength, life time and

To the development of scientifically well-founded methods of the analysis of

the necessary acceptable expenditures for decrease of risks; and *mM* is the cost-

� � are the acceptable (permissible) probabilities and losses; [*M*] is

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>E</sup>* <sup>þ</sup> *<sup>R</sup>*<sup>2</sup> *I ,*

� �*; US*

� � � � <sup>¼</sup> *FM mM*�<sup>1</sup> *<sup>M</sup>* � � � � *,*

(16)

(17)

complex steps on the decrease of system risks *RS* is necessary.

*RS* ¼

of these installations.

*RS* ¼ *FR PS; US*

risk; [*PS*] and *US*

*nS* ≥ 1 is attained.

**210**

1 year (Δ*τ* = 1 year).

� �≤ *RS*

effectiveness ratio 1ð Þ ≤ *mM* ≤10 .

an estimation of margin values *ns*

control (disruption of data flows *I*),

*Probability, Combinatorics and Control*

Eq. (15) and *RW*, *RE*, and *RI* as in Eq. (16).

propagation of emergency and catastrophic situations.

unacceptable *RS* to acceptable (admissible) levels [*RS*]:

baseline grounded on risks come to following primal tasks:

risks *RS* and their basic quantities *PS* and *US*

In nuclear energetics with reactors of all types and all generations (from the first to the fourth) prior to the beginning of the twenty-first century, at failure analysis, the basic attention was given to parameter *P*(*τ*) that defined reliability of safety operation of the NPP. Special meaning was added thus to the forestalling and prevention of the heaviest on the aftereffects of catastrophic situations with the peak damages—melting of the core and a radioactivity runout for breaking points of all guard barriers—casings of the fuel element, cartridge, reactor vessel, reactor hall, and containment. In this case, reactor vessel fracture is extremely dangerous. This event concerns the seventh group of limiting states.

Significant aftereffects arise also at fracture of the basic elements of the first circuit of a reactor vessel and collecting channels of steam generators, pumps, volume compensators, bubbler tanks, and also housings and runners of turbines in the second circuit. These fractures amount the sixth group of the limiting states creating threats to the population, the NPP, and the environment.

If while in service of the NPP because of occurrence of damages of parts of the first circuit has arisen a radioactivity outside breaking points of the NPP and there were thus threats of bombarding radiation for the population, then it is necessary to attribute these events to the fifth group of dangerous limiting states.

The leakages caused by partial damages (faults of crack type or depressurizations of connectors) and creating threats for human controllers and the personnel in the NPP concern the fourth group of limiting states.

The third group of limiting states should be bundled to the considerable damages of the above-termed parts of the first and the second circuit without a radioactivity runout for breaking points of an NPP, which are not demanding their mandatory substitution.

The second group of limiting states concern occurrence in bearing structures of the NPP of partial damages without a radioactivity runout for breaking points of the first circuit, not demanding their substitution, but demanding carrying out of repair-and-renewal operations.

The first group of limiting states is amounted by those of them which are bundled to damages and the faults that have fallen outside the limits admissible under inspection norms and calculation, but not demanding mandatory carrying


For a tentative estimation of loss *U*(*τ*), it is possible to use the simplified statistical and expert information on such losses. Generally, values of losses are defined

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture…*

• Losses of human lives or health at occurrence and progressing of unfavorable

• Economical losses (for example, in Rubles or USD) from a loss of life, from maiming to people, and from fractures and damages of technosphere

Direct loss *U*(*τ*) for the LS-7 limiting state interlinked immediately to fracture of

the NPP or full termination of its service. Then, the datum of loss *U*(*τ*) can be accepted to the equal cost of the NPP. In this, the loss can and should include charges *U*(*τ*1) within 1–2 years on a primary elimination of the consequences of disaster or accident (realization of protective measures, evacuation of the population, and termination of infrastructure installation operation). These charges at (*τ*<sup>1</sup> ≥ *τ*) several times (2–4) can exceed the initial loss *U*(*τc*). Decrease of secondary consequences of heavy disasters on an NPP (making of shelters, recultivation, medical examination and the help, and compensating payments) demands comple-

mentary essential annual expenditures *U*(*τ*2) for a long time *τ*<sup>1</sup> < < *τ<sup>c</sup>* ≤*τ*2. In **Figure 20** is displayed schematization of the relative losses *U*ð Þ¼ *τ U*ð Þ *τ<sup>i</sup> =U*ð Þ *τ<sup>c</sup>* depending on time Δ*τ* after the occurrence of heavy disaster (Δ*τ* = *τ<sup>s</sup> = τ*) at reaching

the most dangerous limiting state of the LS-7 type, summarized in **Table 3**.

With the reduction of the hazard level of accidents and disasters (at transition of limiting states from the LS-7 to the LS-1), value *U* (*τc*) and *U*ð Þ*τ* decrease because of

From assemblage of tens methods for definition of risks parameters as the most simple is the statistical or determined-statistical method according to which it is

where *τ<sup>i</sup>* is the time for which one the risk assessment is conducted and *P*(*τi*) and

If under *τ<sup>i</sup>* is fathomed the time of unfavorable event occurrence of *τc*, then

*R*ð Þ¼ *τ<sup>i</sup> P*ð Þ� *τ<sup>i</sup> U*ð Þ *τ<sup>i</sup> ,* (19)

by two basic parameters:

installations and the environment

*DOI: http://dx.doi.org/10.5772/intechopen.88233*

decrease of losses *U*ð Þ *τ*<sup>1</sup> and *U*ð Þ *τ*<sup>2</sup> .

*U*(*τi*) are the probabilities and losses for time *τi*.

according to Eq. (19), it is possible to obtain

*The time-history and schematization of losses* U*(*τ*).*

possible to write

**Figure 20.**

**213**

situations

**Table 3.** *Groups of limiting states for the analysis of the NPP safety.*

out of repair-and-renewal operations and that can be admitted to prolongation of service before the next examination.

These facts allow to execute summary classification by groups of limiting states for the NPP equipment (**Table 3**) from the most dangerous admissible (the seventh group of limiting states LS-7) to the least dangerous admissible (the first group of limiting states LS-1).

For the groups of limiting states indicated in **Table 3** taking into account summarizing of great volume of normative and technical materials and results of the executed researches, it is possible to describe demanded (admissible) probabilities [*P*(*τ*)] occurrence of unfavorable events. To such probabilities there correspond their actual levels obtained from statistics of their occurrence while in service of NPPs of all generations. Each severe accident or disaster on an NPP, happening at the moment *τc*, was accompanied by comprehensive analysis of their reasons and sources, and also realization of considerable on volumes and expenditures of activities for safety improving. Eventually, at *τ<sup>s</sup>* > *τc*, after such accidents or disasters, decrease of probabilities from *P*(*τc*) to *P*(*τs*) was observed.

For values of probabilities *P*(*τc*) and *P*(*τs*) for all reactors operated in the world at *τ* ≤ *τ<sup>c</sup>* and *τ* ¼ *τs*, it is possible to estimate on ratios

$$P(\mathbf{r}\_c) = \frac{N\_d}{N\_{tc} \cdot \mathbf{r}\_c} ; P(\mathbf{r}\_s) = \frac{N\_d}{N\_{ts} \cdot \mathbf{r}\_s} , \tag{18}$$

where *Nd* is the quantity of the reactors that have obtained damages at the given *i*-th type of limiting state under **Table 3**; *Ntc* is the total of reactors to the time *τ<sup>c</sup>* of occurrence of the given *i*-th type of damage; *Nts* is the total of reactors to the time *τs*; *τ<sup>c</sup>* is the mean time (years) of service of one reactor to the time of reaching of the given *i*-th type of limiting state; and *τ<sup>s</sup>* is the mean time of the service of one reactor.

As it was already mentioned, unfavorable events on an NPP (disasters, accidents, failures, and disruptions) are accompanied by corresponding losses *U*(*τ*) both at the moment of occurrence of these events *τ<sup>c</sup>* and after them (*τ* ≥ *τc*). These losses are caused to the person (to human controllers, the personnel, and the population), to technosphere installations (to an NPP and other installations of its infrastructure), and also to the environment. Now while miss direct legal and normative documents by the quantitative definition of these losses. Some suggestions on this problem are stated below.

### *Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture… DOI: http://dx.doi.org/10.5772/intechopen.88233*

For a tentative estimation of loss *U*(*τ*), it is possible to use the simplified statistical and expert information on such losses. Generally, values of losses are defined by two basic parameters:


Direct loss *U*(*τ*) for the LS-7 limiting state interlinked immediately to fracture of the NPP or full termination of its service. Then, the datum of loss *U*(*τ*) can be accepted to the equal cost of the NPP. In this, the loss can and should include charges *U*(*τ*1) within 1–2 years on a primary elimination of the consequences of disaster or accident (realization of protective measures, evacuation of the population, and termination of infrastructure installation operation). These charges at (*τ*<sup>1</sup> ≥ *τ*) several times (2–4) can exceed the initial loss *U*(*τc*). Decrease of secondary consequences of heavy disasters on an NPP (making of shelters, recultivation, medical examination and the help, and compensating payments) demands complementary essential annual expenditures *U*(*τ*2) for a long time *τ*<sup>1</sup> < < *τ<sup>c</sup>* ≤*τ*2. In **Figure 20** is displayed schematization of the relative losses *U*ð Þ¼ *τ U*ð Þ *τ<sup>i</sup> =U*ð Þ *τ<sup>c</sup>* depending on time Δ*τ* after the occurrence of heavy disaster (Δ*τ* = *τ<sup>s</sup> = τ*) at reaching the most dangerous limiting state of the LS-7 type, summarized in **Table 3**.

With the reduction of the hazard level of accidents and disasters (at transition of limiting states from the LS-7 to the LS-1), value *U* (*τc*) and *U*ð Þ*τ* decrease because of decrease of losses *U*ð Þ *τ*<sup>1</sup> and *U*ð Þ *τ*<sup>2</sup> .

From assemblage of tens methods for definition of risks parameters as the most simple is the statistical or determined-statistical method according to which it is possible to write

$$R(\pi\_i) = P(\pi\_i) \cdot U(\pi\_i),\tag{19}$$

where *τ<sup>i</sup>* is the time for which one the risk assessment is conducted and *P*(*τi*) and *U*(*τi*) are the probabilities and losses for time *τi*.

If under *τ<sup>i</sup>* is fathomed the time of unfavorable event occurrence of *τc*, then according to Eq. (19), it is possible to obtain

**Figure 20.** *The time-history and schematization of losses* U*(*τ*).*

out of repair-and-renewal operations and that can be admitted to prolongation of

These facts allow to execute summary classification by groups of limiting states for the NPP equipment (**Table 3**) from the most dangerous admissible (the seventh group of limiting states LS-7) to the least dangerous admissible (the first group of

For the groups of limiting states indicated in **Table 3** taking into account summarizing of great volume of normative and technical materials and results of the executed researches, it is possible to describe demanded (admissible) probabilities [*P*(*τ*)] occurrence of unfavorable events. To such probabilities there correspond their actual levels obtained from statistics of their occurrence while in service of NPPs of all generations. Each severe accident or disaster on an NPP, happening at the moment *τc*, was accompanied by comprehensive analysis of their reasons and sources, and also realization of considerable on volumes and expenditures of activities for safety improving. Eventually, at *τ<sup>s</sup>* > *τc*, after such accidents or disasters,

For values of probabilities *P*(*τc*) and *P*(*τs*) for all reactors operated in the world at

*; P*ð Þ¼ *τ<sup>s</sup>*

where *Nd* is the quantity of the reactors that have obtained damages at the given *i*-th type of limiting state under **Table 3**; *Ntc* is the total of reactors to the time *τ<sup>c</sup>* of occurrence of the given *i*-th type of damage; *Nts* is the total of reactors to the time *τs*; *τ<sup>c</sup>* is the mean time (years) of service of one reactor to the time of reaching of the given *i*-th type of limiting state; and *τ<sup>s</sup>* is the mean time of the service of one reactor. As it was already mentioned, unfavorable events on an NPP (disasters, accidents, failures, and disruptions) are accompanied by corresponding losses *U*(*τ*) both at the moment of occurrence of these events *τ<sup>c</sup>* and after them (*τ* ≥ *τc*). These losses are caused to the person (to human controllers, the personnel, and the population), to technosphere installations (to an NPP and other installations of its infrastructure), and also to the environment. Now while miss direct legal and normative documents by the quantitative definition of these losses. Some

*Nd Nts* � *τ<sup>s</sup>*

*,* (18)

*Nd Ntc* � *τ<sup>c</sup>*

decrease of probabilities from *P*(*τc*) to *P*(*τs*) was observed.

*P*ð Þ¼ *τ<sup>c</sup>*

*τ* ≤ *τ<sup>c</sup>* and *τ* ¼ *τs*, it is possible to estimate on ratios

suggestions on this problem are stated below.

**212**

service before the next examination.

*Probability, Combinatorics and Control*

*Groups of limiting states for the analysis of the NPP safety.*

limiting states LS-1).

**Table 3.**

$$R(\mathfrak{r}\_{\mathfrak{c}}) = P(\mathfrak{r}\_{\mathfrak{c}}) \cdot U(\mathfrak{r}\_{\mathfrak{c}}) = P^\*\left(\mathfrak{r}\_{\mathfrak{c}}\right) \cdot U(\mathfrak{r}\_{\mathfrak{c}}).\tag{20}$$

Risk *R*(*τc*) is possible to consider as risks of the implemented unfavorable events at *τ<sup>i</sup>* = *τ<sup>c</sup>* and to use them for prediction of events for times *τ<sup>i</sup>* ≥ *τc*. One such prospective risk appears as the risk for the current phase of service *τ<sup>i</sup>* = *τs*. In this case, on the basis of Eq. (19), it is possible to write

$$R(\mathfrak{r}\_s) = P^\*\left(\mathfrak{r}\_s\right) \cdot U(\mathfrak{r}\_s),\tag{21}$$

for negative consequences of accidents and disasters, it is possible to build a line

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture…*

From stated above follows that the major problems which have been not decided while to the full for a NPP there are problems of provision of their protectability and safety on the basis of new scientific fundamental and application researches on mechanics, hydrodynamics, economics, mathematical and physical modeling of dangerous processes resulting to heavy disasters, and also development of detailed

Results of the fulfilled scientific researches and developments in this direction, integrated [3–8, 15–17] in the serial of monographic publications on strength, life time, and safety of power nuclear reactors, are initial scientific baseline for the

protectability of the NPP equipment from heavy disasters on the basis of criteria of

The above-mentioned results of analytical and experimental researches can be considered in the capacity of a theoretical basis for the subsequent development of practical models of the computational analysis of risks for strategically relevant installations of a nuclear energetic on the basis of the complex Eqs. (1)–(24). Development of such models, and the most important—their filling up statistically reliable probability distribution of fractures on groups of limiting states (see **Table 3**) on the one hand, and economical computations of losses, with another, it is necessary to consider as the major task for a solution of a problem of safe

At up-to-date and subsequent stages of evolution of power engineering in Russia in the capacity of a basic recommended position, it is necessary to use the position about provision of an acceptable risk level of occurrence of accidents and disasters. In this connection, it is not obviously possible to ensure from social-economic and technological stands the declared principle of absolute safety with null risks (*R* (*τ*) = 0). Owing to it, the solution of the delivered problem is brought together to determination of scientifically well-founded admissibility of occurrence of the emergency situations with possible minimization of loss caused by them, with an estimation of the greatest possible, acceptable, and controlled risk both at probable occurrence of global and national accidents and disasters, and their realization at

applicable normative, designer, technological solutions on provision of

of negligible risk parameters ½ � *P*ð Þ *τ*<sup>Σ</sup> min � ½ � *U*ð Þ *τ*<sup>Σ</sup> min.

*DOI: http://dx.doi.org/10.5772/intechopen.88233*

methods of the analysis of risks for heavy disasters.

development of power supply of human community.

**5. Conclusion**

acceptable risks.

regional and local levels.

**215**

where *τ<sup>s</sup>* is the time after unfavorable event (*τ<sup>s</sup>* ≥ *τc*).

This time can be situated in the interval *τ<sup>c</sup>* ≤ *τ*<sup>1</sup> ≤ *τ*2. Then, for one operated unit of the NPP, the common risk at reaching the given i-group of limiting state from the LS-7 to the LS-1 will constitute

$$\mathcal{R}(\mathfrak{r}\_{\mathfrak{c}})\_{\mathfrak{T}} = \sum\_{i=1}^{7} \mathcal{R}(\mathfrak{r}\_{\mathfrak{c}})\_{i} \tag{22}$$

If at loss estimations to consider not only direct losses at occurrence of unfavorable event *U*(*τ*к) together with complementary losses *U*(*τ*1) and *U*(*τ*2), then it is possible to define common (integral) losses as

$$U(\tau)\_{\Sigma} = U(\tau\_{\varepsilon}) + U(\tau\_1) + U(\tau\_2) \tag{23}$$

These integral losses respond to the appropriate risks

$$R(\boldsymbol{\pi}\_{\Sigma})\_{\Sigma} = \sum\_{i=1}^{7} U(\boldsymbol{\pi}\_{\Sigma})\_{i} P^\* \left(\boldsymbol{\pi}\_{i}\right)\_{i}. \tag{24}$$

On the basis of results of an estimation considered above risk components, it is possible to build dependences between basic parameters of risk for the NPP probabilities *P*(*τ*) occurrence of unfavorable situations and losses *U*(*τ*) from them (**Figure 21**).

The line had above and design points in the **Figure 21** belong to probabilities *P*(*τc*) and to losses *U*(*τc*) for the moment of accident or disaster occurrence on the NPPs. The lower line made like overhead characterizes a negligible zone of risk parameters *P*ð Þ *τ<sup>c</sup>* ½ �min � *U*ð Þ *τ<sup>c</sup>* ½ �min and the midline characterizes a zone of acceptable risks *P*ð Þ *τ<sup>c</sup>* ½ �� *U*ð Þ *τ<sup>c</sup>* ½ �. If to allow common (near-term and long-time)

**Figure 21.** *Parameters of risks for the NPP with reactors of VVER types.*

for negative consequences of accidents and disasters, it is possible to build a line of negligible risk parameters ½ � *P*ð Þ *τ*<sup>Σ</sup> min � ½ � *U*ð Þ *τ*<sup>Σ</sup> min.
