OPTIMIZATION OF SYSTEMS REPAIR PLANS AND ASSESSMENT OF THE USEFUL LIFE OF NUCLEAR POWER PLANT EQUIPMENT

A. Pidhornyi Institute of Mechanical Engineering Problems of NASU, 2/10, Pozharskyi St., Kharkiv, 61046, Ukraine This paper presents a computer-aided method of planning the volumes of repairs of systems of nuclear power units and a method for calculating their gamma-percentile life. This planning is carried out on the basis of predicting the reliability indicator, the probability of no-failure operation for a certain time period, with the gamma-percentile life of the equipment being determined by solving the corresponding equations. The tasks considered are related to an important energy problem of extending the operation of nuclear power units. Its importance is determined mainly by economic feasibility: it is cheaper to assess the useful life of a nuclear power unit and, on this research basis, extend its operation, than create a new unit. It is also shown that the calculation of the probability of a radiation accident at a nuclear power unit is associated with the results of planning the repairs of its systems, with assessment of its useful life. An optimization problem is formulated: it is required to find such a plan for the volumes of repair of a system that, with limited repair costs, its reliability indicator for a given duration deviates least from the maximum permissible value. The solution to the problem is based on calculating the structural reliability of the system. A graphological image of the system is built in the form of a composition of graphological images of typical structures. After the reliability indicator of typical structures has been calculated, the structures are replaced with individual structural elements, which makes it possible to simplify the initial graphological image of the system in a computational scenario and calculate its reliability indicator. The determination of the repair volume is carried out by applying a version of the coordinate-wise optimization method. To assess the gamma-percentile life, a model is adopted, in which the recoverable equipment components have an unlimited life, although, of course, they "age", and the non-recoverable components spend their life up to the level when their replacement becomes conditioned by the violation of the requirement for the maximum permissible value of the system reliability indicator. Estimates of the gamma-percentile life of the equipment are calculated by planning system repairs on a sequence of intervals of annual energy production by a nuclear power unit.


Introduction
The tasks of planning repairs and assessing the useful life of the equipment of nuclear power units (NPU) are mainly solved on the basis of regulatory instructions of equipment manufacturers. This article proposes an approach based on computing the reliability indicators of NPU systems, with account taken of their structural organization. As the reliability indicator, the reliability function (RF) over a particular time period is used. As the useful life indicator, the gamma-percentile life is used, which is the duration of the working state of a unit with the probability of gamma.
The main NPP quality indicator, that allows it to be operated, is safety. Safety assessment is a multifaceted problem. From the standpoint of systems analysis, one of the ways to obtain quantitative safety assessments is to mathematically model the risk from the operation of NPUs [1]. The magnitude of the risk is determined by the product of the probability of a radiation accident and the magnitude of the consequences of the accident: life loss, environmental pollution, material losses. Knowing the risk allows one to manage it, that is, to provide for compensatory measures.
One of the ways to reduce the risk is to provide high-quality maintenance and repair of NPUs. The paper considers part of this task, namely, the planning of rational volumes of system repairs, carried out during the periods of annual scheduled preventive maintenance overhauls ensuring that NPUs produce costeffective energy and are safe. The solution to the repair planning problem is used to determine the gammapercentile life of equipment.
The plans for repair and useful life assessment are related to the problem of extending the gammapercentile life of NPUs [2]. The importance of the problem is mainly explained by economic feasibility: it is cheaper to assess the useful life of a NPU, and on this research basis, extend its operation, than create a new unit.
The useful life indicators depend not only on the physico-chemical, thermodynamic, vibrational, and other impacts on the equipment, but also on the requirements for the maximum permissible value of the probability of a radiation accident, on the quality of maintenance and repair.
Today, there is increasingly the problem of assessing the individual useful life of NPUs by accumulating reliable information, using diagnostics, witness samples, and accelerated tests as well as modelling the strength and useful life indicators of equipment. The need to carry out special useful life studies of an individual NPU is due to its individual characteristics and operating features. In NPP conditions, a complete study of the individual useful life of an NPU, its numerous equipment, measured in thousands of units, is practically impossible because there will be uncertainty in some of the results due to the reduced volume of measurements and engineering interpretations of incomplete information.
In this article, the determination of the equipment repair plan is based on the analysis of the structural reliability of NPU systems. It is assumed that the values of equipment reliability indicators can be obtained from the registration data of their failures, from expert assessments, or taken from the data of its manufacturers. Repair planning is transferred to the mathematical plane, that is, the problem of optimizing system repairs is formulated based on the calculation of its structural reliability. In this case, a functional relationship is used between the cost of repairing a piece of equipment and the increase in the value of its reliability indicator.
Due to the variety and range of technical systems, there is no and, apparently, will be no universal method for calculating their structural reliability [2]. To date, the main methods for assessing the structural reliability of systems include the logical-probabilistic method, the discrete-state Markov processes method, and the simulation modelling method. They are difficult to apply to systems with a large number of elements [3]. In this regard, other methods that facilitate the solutions of computational problems are of interest too.
One of the approaches used in engineering hand calculations is shown in [4]. It consists in compressing a structural diagram, that is, a graphological image of a system. When there are many elements in the system, the calculations become time-consuming. This process becomes even more labor-intensive if the calculations have to be performed repeatedly with changing data, for example, when optimization problems are being solved.
In order to automate the calculations of the structural reliability of complex systems with a large number of elements, an approach has been developed [5] based on the recognition and compression of structural schemes that change during the calculation on a computer.
The search for a rational plan is carried out step by step by assigning repairs to the equipment, the increase in the reliability of which ensures the greatest increase in the system reliability indicator at each step. The calculated repair plan determines the change in the system reliability indicator over time, which allows one, by solving the corresponding equations, to find the duration of the operative condition of a piece of equipment with the probability of gamma, thereby determining the estimate of its useful life.

Main Part
In this article, the method for planning the repair of a system and the assessment of the equipment useful life are presented according to the following scheme: a brief description of the method for calculating the structural reliability of the system is given; the task of optimizing the system repair plan is formulated; a method for solving the optimization problem is described; the connection of repair plans with the assessment of NPU safety and with the estimates of the equipment gamma-percentile life is shown.

Calculation of Structural System Reliability
The structural diagram of a system is composed of the images of its elements (hereinafter referred to as elements) with the known laws of their reliability indicator. To calculate the system reliability indicator on a computer, simple elements; generalized elements; typical basic graphological images, for example, redundant systems; virtual connectors are introduced into the information image of the system. The following theorem is used: if the graphological image of the system is composed either of typical basic graphological images with simple elements or from generalized graphological images with computable indicators of reliability, then the system reliability indicator can be calculated [5]. Based on this theorem, the process of calculating the structural system reliability is monitored on a computer.
The calculation method developed in [5] is based on the representation of the graphological image of a system in the form of a composition of typical graphological images. Automatic recognition of their images is carried out and the values of their reliability indicators are calculated. The calculation results are assigned to typical graphological images, turning them into elements of a structural scheme, which creates a new structural diagram of the system with new typical structures. Such replacements sequentially compress the initial graphological image of the system to one element with the calculated value of the reliability indicator, which determines the value of the reliability indicator of the initial graphological image of the system.

Statement of the Repair Planning Problem
For the mathematical formulation of the problem, the following is accepted: S is a system formed from systems S 0 , S 1 , S 2 , …, S M , each of which consists of elements a i m , i m =0, 1, 2, …, I m , m=0, 1, 2, …, M, (further a i , i=0, 1, 2, …, I); t 0 is the repair start time; ∆t=t-t 0 is the duration of the annual operation of the system; Q(S 0 ), Q(S 1 ), Q(S 2 ), …, Q(S M ) are the probabilities of failures of the systems S 0 , S 1 , S 2 , …, S M over time period ∆t; V is the repair volume of the system S; * S Q is the maximum permissible value of the probability of failure of the system S over time period ∆t; * S Z is the maximum allowable value of the cost of repairing the system S. The task of planning is to find such a plan for the repair volume V of the system S that the probability of no-failure operation R S (∆t, V) over time period ∆t deviates least from the maximum permissible value At the same time, the costs Z S of implementing the plan should not exceed the maximum permissible value * S Z . In a symbolic form, the problem can be written as follows: it is required, for given values ∆t, For the mathematical formulation of the problem, it is necessary to determine the maximum permissible value * S R and the volume of repair V.
The value * S R can be obtained by calculating the reliability indicator of the system S, based on its structure and the reliability indicators r(a i ) of the elements a i , i=0, 1, 2, …, I, taken from the nominal data of equipment. Also, the value * S R can be determined by applying expert estimates.
To determine the repair volume V, the specific value of the reliability indicator δr i of the element a i , i=0, 1, 2, …, I is introduced as the value of increase of the reliability indicator of the element a i per one cost unit.
Let r i (t 0 ) be the RF of the element a i over its operating time t * ∆ preceding time t 0 . As a result of repairs in the volume of costs c i , the element can obtain the RF ) ( 0 * t r i over time period t * ∆ . The difference between these RFs at time t 0 , divided by the value of incurred costs c i , determines the value of the specific RF δr i (t 0 ) of the element a i : In this formula, r i (t 0 ) and c i are assumed to be known quantities and ) ( 0 * t r i , i=0, 1, 2, …, I are unknown quantities. The specific value of the RF δr i (t 0 ) can be estimated as follows. Suppose that the element a i is replaced with an identical new element that has not spent its reliability indicator over time period t * ∆ . Therefore, it is assumed that its RF 1 ) ( 0 * = t r i over time period t * ∆ . Let the cost of installing a new element a i equal z i . Then the specific RF of the new element (the value of the increase in RF per unit cost) will be . Therefore, if the costs c i of repairing the element will be z i , it will mean replacing the element a i with the RF r i (t 0 )=0 with an identical element a i with the RF 1 ) ( 0 . Then, according to expression (2), the value of the RF increase as a result of replacing the element a i will be 1 0 If the repair is carried out with costs c i , then the value of the RF increase over time period t * ∆ is determined by the ratio: , that is, it is assumed that the increase in the RF of the element during repair at time t 0 is directly proportional to the costs of repair c i , which determines the repair volume v i of the element a i .
Then the performed repair volume V of the system S will be determined by the sum . It is assumed that the costs of repairing the elements c i , i=0, 1, 2, …, I are known and can be measured in various units, for example, in monetary units or in labor costs.
The RF R S (t 0 ) over time period t * ∆ of the system S depends on the scheme of its structure and the RF r i (t 0 ) over time period t * ∆ depends on its elements a i , i=0, 1, 2, …, I. Since the RF values over time period t * ∆ of the systems S 0 , S 1 , S 2 , …, S M are computable, the FBG R S (t 0 ) over time period t * ∆ of the system S will be computable as well. To simplify the notation and emphasize that the RF of the system S is a function of the reliability indicators r i (t 0 ) of the elements a i , the notation R S (t 0 )=φ(r i (t 0 )), i=0, 1, 2, …, I is used.
As a result of repairs, within the allotted time, the RF increment ∆R S (t 0 ) over time period t * ∆ of the system S with the RF increments of its elements a i , i=0, 1, 2, …, I, ∆r i (t 0 ) over time period t * ∆ , can be written as follows: Due to the fact that the RF R S (t 0 ) over time period t * ∆ of the system S without the application of repair costs will be R S (t 0 )=φ(r 0 (t 0 ), r 1 (t 0 )), …, r I (t 0 )), as a result of the application of the costs for the repair of the elements a i , i=0, 1, 2, …, I, we will receive the RF ) since the RF increment of the elements a i , i=0, 1, 2, …, I, over time period t In formula (3), the values δr i (t 0 ), i=0, 1, 2, …, I determine the specific RF over time period t * ∆ of the elements, which are calculated by the formula , and the unknown repair costs c i , i=0, 1, 2, …, I can take any real non-negative values.
The result of the repair carried out at time t 0 will affect the RF r i (∆t) over time period ∆t=t-t 0 of the elements a i , i=0, 1, 2, …, I, and the RF ) ( t R S ∆ , over time ∆t of the system S. In expression (4), the RF R S (∆t) of the system S over time period ∆t depends on the RF values r i (t 0 ) over time t * ∆ of the elements a i at time t 0 , on the laws of the RF r i (∆t) over time period ∆t of the elements a i ; on the costs c i incurred during the repair of the elements a i ; on the specific RF , i=0, 1, 2, …, I. Using (4), problem (1) can be formulated as follows: it is required to find such a distribution of costs (5) Thus, in problem (5), the uncertainty of problem (1) is eliminated: the values * S R and V are determined. The desired repair plan, as follows from (5), will consist of the values c i =C i , i=0, 1, 2, …, I.
If one supposes that the events of failures of the systems S 0 , S 1 , S 2 , …, S M are independent, problem (5) can be solved for each of the systems S 0 , S 1 , S 2 , …, S M separately, and then the results can be combined, or the problem can be solved immediately for the entire system S. Such solutions may be different [4].

Repair Planning Problem Solution
The RF function )) ( , , ( is continuous and differentiable in time t in the domain of its definition as it is continuous and differentiable by the parameters of c i , i=0, 1, 2, …, I. It is assumed that the maximum permissible values * S R and * S Z are such that a solution to problem (5) exists. However, as is easy to see, considering the serial connection of two elements, it is not unique. Problem (5) is solved using the coordinate descent method. The efficiency of descent depends on the ratio between the costs of repairing the elements and the magnitude of the increase in the system reliability indicator for these costs. The descent is carried out along that coordinate c j , j=0, 1, 2, …, I, for which at the end of the interval [t 0 , t] the partial derivative of the RF )) ( , , ( with respect to the parameters of c i gets the highest value at the end of the interval [t 0 , t]: Since for a complex system it is practically impossible to compose the RF )) ( , , ( in analytical form, the values of its partial coordinate derivatives c i , i=0, 1, 2, …, I are calculated by the approximate formula , where ∆c i is the value of the applied costs for the repair of the element a i , i=0, 1, 2, …, I.
The value of the partial derivative can be interpreted as the partial specific RF of the system S along the coordinate c j . It is the value of the change in the RF over time period ∆t of the system S per unit of applied costs. Naturally, the descent along the coordinate c j having the greatest value of the partial specific RF of the system S will be most effective. With the same costs for the repair of the elements, the costs (volume of repair) c j of the element are selected, the repair of which increases most the RF over time period ∆t of the system S. The coordinate descent is carried out until a solution to problem (5) is obtained.
Since it was assumed that the repair costs c j can be any real non-negative numbers, the solutions to problem (5) may require interpretation in terms of the traditional discipline of repairs -capital, average, current, maintenance. If there is information on the costs of these types of repairs, the solution to problem (5) in these terms can be interpreted automatically.

Connection of Repair Plans with the Safety Assessment of a NPU with the Estimates of the Gammapercentile Equipment Life
The system repair plan is connected with the safety assessments of a NPU. One of the ways to calculate the probability of a radiation accident is to construct a set of chains of events consisting of the occurrences of the initial events and failures of the systems that handle them. Let H 0 , H 1 , H 2 , …, H M be independent events determined by the failures of the systems S 0 , S 1 , S 2 , …, S M , which are a chain of events consisting of the occurrence of the initial event H 0 (failure of the system S 0 ) and the way of development of an accident (failures of the systems S 1 , S 2 , …, S M . The probabilities of system failures over time period ∆t can be computed, for example, using nominal data of system elements. Then the probability of a radiation accident P(A) of this chain of events will be defined as the product of the probabilities of system failures ) ( The repair plan c i =C i , i=0, 1, 2, …, I, obtained as a result of solving problem (5), will change the probabilities of system failures Q(S 0 ), Q(S 1 ), Q(S 2 ), …, Q(S M ) over time period ∆t for new probabilities of system failures Q 1 (S 0 ), Q 1 (S 1 ), …, Q 1 (S M ) over time period ∆t. Then the probability of an accident P(A) associated with a chain of events will acquire a new value ) ( ...
. This determines the connection between the system repair plans and the NPU's safety assessments.
The solutions to problem (5) also determine the assessments the useful life of NPU systems equipment. Indeed, since the constraint on the maximum permissible value of the system reliability indicator * S R is determined, then there is such a plan for the system repair on a sequence of annual operation intervals that will not satisfy the conditions of problem (5) without replacing the non-recoverable equipment or nonrecoverable components of recoverable equipment.
In other words, when planning the repairs i k i k C c = , where the index k=1, 2, 3, … denotes the interval number, and the index i=0, 1, 2, …, I denotes the number of the element a i it is necessary to solve the inequality relative to the variables ∆t k , which represent the duration of one k time interval.
If the value of the interval ∆t=const, then, in this case, ∆t k =k·∆t is the duration k of the time intervals on each of which there is a solution to problem (5) and, therefore, inequality (6). Suppose that on the time interval k+1, Then there is such an unrecoverable element a i in the repair plan for the time interval k+1, the replacement of which provides a solution to problem (5). Then the duration ∆t k =k·∆t of the system's serviceable condition with the probability * S R = γ will be the gamma-percentile life of the element a i to be replaced.

Conclusions
NPP accidents have unacceptable consequences for people and their environment. One of the ways to reduce the likelihood of accidents and the scale of their consequences is to calculate risk assessments in order to manage them on the basis of this knowledge, while improving the measures to ensure the safety of an NPU. Therefore, the problems related to the risk of their operation are studied comprehensively. An important aspect of this problem is reflected in the organizational and technical activities of maintenance and repair, part of which is planning the volume of repairs of a NPU, monitoring the spending of the equipment useful life, and its timely replacement.
For the planning of repairs, a method of calculating the system reliability indicator is used, based on the graphological scheme of the system, consisting of basic structures with information images recognizable on a computer and with computable reliability indicators. A schematic compression process is applied, which makes it possible to evaluate the reliability indicator of a system with a large number of elements.
The solution of the problem of planning repairs, based on the mathematical modelling of the system reliability indicator, allows planning repairs based on taking into account their structure and maximum permissible values of reliability indicators. It is shown that there is a relationship between the system repair plan, the safety assessment and the gamma-percentage life of the NPU equipment. The presented approaches to planning system equipment repairs and assessing their gamma-percentage resources can be used to solve problems of extending the operation of NPUs.
The software, oriented to the application of the presented approaches in practice, can be included, as an independent software complex, in the information and analytical systems that ensure the solution of problems of effective management of the maintenance and repair system of NPUs.