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9MATHEMATICAL MODEL OF THE RESTORED TECHNICAL SYSTEM RELIABILITY
Khaydarov Sh.A.1, Alikulov T.A.2
1Khaydarov Shamsiddin Abdizhalimovich - Candidate of Technical Sciences, Associate Professor;
2Alikulov Tuigun Avlakulovich - Candidate of Physics-Mathematical Sciences, Associate Professor, DEPARTMENT OF HIGHER MATHEMATICS, KARSHI ENGINEERING-ECONOMICS INSTITUTE, KARSHI, REPUBLIC OF UZBEKISTAN
Abstract: a discrete mathematical model of the reliability of an aging technical system proposed, which studies the wear of working bodies (blades), rotating mechanisms, including hydraulic pumps, the state of which deteriorates over time, which leads to an increase in the probability of its failure. In order to ensure the necessary operability of the system, preventive repairs of various depths carried out at random times. Therefore, it is important to determine such a rule for choosing the periods for such repairs that provide the optimal level of system performance. The article describes a mathematical model that describes the relationship between the reliability parameters of a technical system and random times ofpreventive maintenance, which determine the finite number of system states. Based on the analysis of this model, a rule is determined for choosing the time periods for such repairs that ensure the optimal level of system performance.
Keywords: rotating mechanisms, hydraulic pumps, system failure probability, failure rate, recovery rate, mathematical model of reliability, order Erlang distribution, Kolmogorov equation.
UDC 621.391.175
Introduction. General statement of the problem, relevance and aim of the study.
It is known that all technical systems in the process of operation deteriorate their characteristics, which increases the likelihood of their failure. It can be reduced and brought to an optimal level through preventive maintenance. Typically, such repairs determined by regulations at predetermined points in time. However, for various reasons, their implementation occurs at random times. This determines the relevance of constructing a mathematical model describing the relationship between the parameters of the reliability of the functioning of the technical system under consideration, random times of preventive maintenance and the probabilities of failure in states during repair. The probability of failure is the most important parameter of such a model, since it leads to long system downtime and significant costs for its restoration.
The sequence of moments of carrying out repairs of system transitions from the next state to the next in this setting considered as a process of managing this system.
The aim of the research is to build a mathematical model of the reliability and performance of a controlled technical system with a finite number of states in which the system will be during the next preventive maintenance. The most promising approach to modeling the reliability of technical systems considered, which implies the introduction and definition of intermediate states that determine the level of performance and the probability of failure. This approach makes it possible to take into account the individual characteristics of the system, for example, the rate of wear, which in turn makes it possible to more adequately make decisions on determining the schedule for its preventive maintenance. Existing methods of modeling technical systems [1] to select optimal solutions for determining the schedule of preventive maintenance use statistical data obtained from a large group of systems of the same type. The main quantity in this case is the MTBF. However, the individual characteristics of a particular system are not taken into account in this case. The purpose of the simulation is to maintain the optimal performance of a separate system, taking into account its state for an unlimited time interval. The modeling is based on a Markov random process.
The proposed work develops an approach to modeling the reliability of technical systems, considered in [2].
1. Method. Mathematical model of reliability
Let the stochastic system S be controlled after a random time period 4 = min (4, t), where 4 is the random time to failure, depending on the observed state at the last moment of control;
t is a planned random control period with a order Erlang distribution. At each contro l moment, the system can be in one of the states of a finite set E = {x1, ..., xN,}
We will assume that the best state in which the probability of failure is minimal is x1, and the worst state is xN. The state x1 corresponds to a new system, and the state xn corresponds to the worn-out system. All other states are intermediate, in which the probability of failure is ordered in order of its ascending order from minimum to maximum. It will be convenient for us to expand the set of states by equipping each element x1 G E with a second index s, s = 1, ..., k + 1. Moreover, s = 1, ..., k indicates the phase of the Erlang distribution [2] of period m, and s = k + 1 indicates the state of scheduled control. We denote E" = {xis}, i = 1, ..., N, s = 1, ..., k + 1. Thus, the
E = E"^\xJ [ complete set v J'.
Let the system be in the state xjk+1, j = 1, ..., N at the scheduled time of control, in which one of the possible preventive repairs is applied. Suppose that the set of admissible preventive repairs, which we call the set of
controls and denote Y = {y1, ...ym}, is finite. The element of this set, control yi, determines the depth of the system update. It determines the intensities j of the system transition from the state xjk+1 to the state xs1, s = 1,.
......, j We assume that the deeper the control provides for the system update, the greater the intensity of the
transition to the state with a lower number s. However, we will take into account that the deeper the system update, the more this management costs.
It is natural to assume that a system failure is possible in any state xjs, j = 1, ..., N, s = 1, ..., k, and the failure rate vj does not depend on the phase s. We assume formally that the failure of the system leads to its transition to the state x0. In state x0, the system is restored to one of the states xj1, with intensity ®j, j = 1, ..., N. Let us denote the intensity of the transition between phases as y.
The main task of modeling will be to select the optimal control strategy, i.e., to select the type of preventive maintenance for each condition at each moment of control.
The system of Kolmogorov equations for the state probabilities of the proposed model has the following form
[4]:
dPn (t) = -(u! + 4 + y)Pn (t )+OP0 (t )+£ Mn Pik (t) dt t!
d n
-PA(t ) = -(ps + 4 + y)Psl(t )+0 P (t +1 (t )+4_n(t), s = 2,..., N -1,
dpm(t) = -(Pn + /)PN1(t)+ ^ Np0 (t) + Mnnpnk+1 (t)+ 4N+1PN-11 (t) dt ,
dPXs (t) = -(ü1 + 4 +r)Pls(t) + r^-t, s = 2,...,k, dt
dPs (t )=-(»q + 4 + r)Pqs (t )+r$qs-1 +vq _fq (t), q = 2,..., N -1, s = 2,..., k,
^Ns {t) = -{Vq +y)PNs (t)+r® Ns-1 +ÜN-1 Pn-1s (t), s = 2,..., k, d N
-Pqk+1 (t )=^MglPqk+1 (t )+rPqk (t) q = 1,..., N,
dt i=1
j N N k
dP (t )=s ® Po (t )+S T vJp j (t),
dt 1=1 1=1 j=1
d N k+1 dt
^J N k+1
dPo (t)+S S Pj (t )=o
1=1 j=i
Note that one of the equations of the system, in addition to the normalization condition, can be omitted when solving it.
Let the system be in state x11 at the initial moment of time. Then the initial probability distribution is as follows:
P11(0) = 1, P0(0) = 0, Pj(0) = 0 for all i, j, besides i = j = 1.
Consider the case when there is a stationary mode of operation of the system [4]. In this case, there are limits of
probabilities of states P1j(t), P0(t) at ^ ^ w and, therefore, in this mode, all derivatives of these probabilities are equal to 0. Then the reduced system of differential equations will transform into an inhomogeneous system of linear algebraic equations. Such a system can be solved, for example, using the MAPLE computer program.
3. Defining system parameters
The state of the system is determined by a set of monitored parameters containing information about the reliability of the system, and in practice it can often be reduced to a scalar value using a certain weighting function. In this case, the dimensionality of the model is significantly reduced. Note that the choice of an informative set of parameters can be carried out, for example, using the methods of the theory of pattern recognition [4].
The process of wear and tear of a technical system is adequately described using the intensity of the
X, i = 1,..., N -1 xu ^ x+1., i = 1, ..., N -1, J = 1, ..., k.
i between neighboring states J J
X j
Indeed, the system J is in the state of random time, distributed according to the exponential law, which corresponds to the absence of wear, and the transition to the state of failure due to "random" factors. Example, with constant conditional probability. An increase in the degree of aftereffect due to transitions to new states reflects an increase in the degree of wear and, hence, an increase in the conditional probability of failure over time [1]. Evaluation of the
X, i = 1, ..., N -1
intensity i , transition between states can be easily obtained by methods of mathematical
statistics based on the available information [6].
v, i = 1,...N,
The intensities can be obtained by methods of mathematical statistics based on the available
information.
Let us obtain estimates of Y the intensity of the transition between phases and of the order of a k random variable
1 b t,U,...t„
' having an Erlang k -order PAD distribution. Let 12 n - the realization of the period of time between the
^ k
planned shutdowns of the system for preventive maintenance. Since Y is the variance, the method of
Di= k
moments Y gives the following estimates of the parameters Y andk : where:
1 ^ -2 1 "\2 o =-Yli, -1)
r
t
— 2 a
-2 t
— 2 a
- -2 1 -V
t=-zt. a = nn ? -t'
Square scores mean the whole part.
x
The intensity of the repair in the state 0 can be estimated by the methods of mathematical statistics. It should be noted here that it is convenient to estimate separately the intensity of ^ the exit from the state X° and the probability of
n
n Xi,i = 1, N = 1
the transition from this state 1 to the states , '-1 then the required intensities should be
= ® .
set equal 1 .
Let us further define the control parameters. Management, i.e. preventive repairs of various depths of system
xik+1,i = 1, ..., N
renewal, as a rule, cannot worsen its condition. Therefore, we will assume that from a state ,
Xi1, j = 1, ..., i
under the action of control, a transition is only possible to a state i1 . Similar to the previous one,
n
P (y), j = 1, ..., N zPi =1
statistical methods easily make it possible to estimate the probabilities j such that i=1 , for
y ^ Y x u, x
each y , each state of prevention lk+1, as well as the intensity of ^ the exit from the state lk+1. Then the required
intensities for leaving the state Xrk+1 are equalUi] (y) UU>] ^^ j 1 i.
The operation of the system is associated with some costs that ensure its operation, in particular, with the implementation of preventive repairs. In addition, the normal operation of the system usually brings a certain income per
x
unit of time, possibly depending on the state. Let us denote the income of the system per unit of time 11 in the state through ). Let the cost of carrying out routine maintenance work per unit of time in the state x'k+1 is the value,
v eZ
where is the control ^ , which determines the depth of the preventive repair (control number), and the cost of
x r
restoring the system from the state of failure per unit of time in the state 0 is the value 0. System optimization
The mapping E ^ Y will be called a decision function and will be denoted f, and the sequence of decision functions ^ = f 1 f ) will be called a strategy. A view strategy _ f 1 f2 • • •} is called stationary. For a
ro
given stationary strategy ^ , the average income per unit time in a stationary mode for the system under consideration is determined as follows:
W í k \ N
Lwj)=£w(xu-)I£Pv - £r(x<k+i'f(xik+i))p»k+i-roPo i=1 v i=1 J i=1
f (x )e Y f
Where v ik+1' is the management determined by the decisive function f in the strategy. The challenge is to
ro j ro
find a strategy ^ that maximizes the functionL . There is a finite set of stationary strategies ^ , therefore such a strategy exists. In our case, it can be easily found using the MAPLE computing program. Next, consider a model example. All cumbersome calculations are supposed to be carried out in the MAPLE V R4 system [7]. 4. Sample
Let, as a result of processing statistical data, it was obtained that the duration of the period of routine preventive maintenance is a random variable distributed according to the Erlang law of the k order with parameters^ = 1.5 andk =2. This determines the set of states of the systemE = {x0,x^ }i = 1,...3, j = 1, ...3 . In
states xi3, i = 1, ...,3,, preventive repairs are carried out, and in a state, the system is restored after a failure.
The system of equations for the probabilities of states in a stationary mode will be obtained from (1). It will take the following form:
-(Pl + ¿1 + r)Pn + LP23 + f1 P0 + L P13 + LP33 = 0
P13 = 0
-(M21 +M22 )P23 = 0,
- (lL31 + L + M33 )P33 + 7P32 = 0,
- (P2 + ¿2 + r)P21 + LP33 + f2P0 + M22P23 + M31P33 + ¿1P11 = 0,
- (U31 + r)P,1 + M21P23 + f3 P0 + LP33 + ¿2P 21 = 0,
-(f + f2 + f3 )P0 (P11 + P12 )+»2 (P21 + P22 )+^3 (P31 + P32 )= 0,
- (u2 + ¿2 + y)P22 + P + ¿P12 = 0,
+r)P32 +¿2P 22 = 0
P0 + P11 + P12 + P13 + P21 + P22 + P23 + P31 + P32 + P33 = l .
In states, the systemxi3,i = 1,...,3, is idle, preventive repairs are carried out in them. The total rate of exit from each state xi3, i = 1,..., 3, is equal to JLI = 3 . The probabilities of transitions from these states to operating states depend on the depth of preventive maintenance. Let us assume that management updates the system v1 to the greatest extent, management v to a lesser extent, and management v 3 to the least extent.
The intensity Lij determines the transition from xi3 state to state xv1, j = 1,...,i, . Intensity values for control y1 : L11 = LL21 = 0,9L , L22 = 0.L, L31 = 0.75^, L32 = 0.15l, L33 = 0.L ; For driving y2 : L=L,L= 0,5l , L22 = 0.5l, L31 = 1/3L, L32 = 1/3l, L33 = 1/3l ; For driving y3 : L = L,L21 = 0,15L , l21 = 0.15L, L22 = 0.85^, L31 = 0.L, L32 = 0.15L, L33 = 0.75^ .
The cost of preventive maintenance per unit of time in a state xi 3 is the value r (xi3, y j ) if control is applied. These values are chosen as follows:
rV1 ) = 5, r(x23,V1 ) = 5 5, r(xзз,V1 ) = 6; r(xn,V2) = 4 r(x23,y2) = 4 2, r(xзз,V2)= 4.6;
r{xi3,Уз) = 3, r(x23,Уз, ) = 3.2, r(x33,Уз) = 3.4 y e 7, j = 1,...,3.
Let the unit of measurement of income and costs be selected. The system brings income per unit of time in states xn, x12 equal to 30, in states x21, x22 equal to 22, in states x31, x32 equal to 14 units. Recovery of a failed system costs
r0 = 18 conventional units per unit of time.
For the rest of the parameters, the following values are selected \ = 0.5, ^ = 2 . The intensity of the transition
between states. The failure rate in the states: for i = 1 and j = 1,2 is equal to ц = 0.1 for i = 2 and
j = 1,2 Ц = 0.2 for i = 3 and j = 1,2 ц = 1. The rate of recovery x i of the system from failure to state is equal to
= 0.6 ,state x31 equal to state Ф3 = 0.3 .
The following result is obtained. The best strategy is one that prescribes control in all conditions in which preventive maintenance is envisaged. The use of the MAPLE V R4 program for solving the assigned task gave the following result. Application of management leads to income 15.25437722, management - to income 14.53419191, management - to income 14.00128186 per unit of time with unlimited operating time of the system. 5. Conclusion
A new approach to modeling aging technical systems, the probability of a possible failure of which increases with time, is proposed. It consists in the fact that to describe the evolution of the aging process of a technical system, a sequence of states is introduced, which it passes in turn. At the same time, such an indicator of the system as the failure rate increases monotonically. It is shown that under certain conditions the evolution of the system can be described by a Markov process and, therefore, the model can be based on the system of Kolmogorov equations. The aim of the simulation was to find a strategy for preventive system updates that would optimize its performance and reliability. The practical significance of the results obtained is that the solution of this topical problem can be easily obtained using mathematical computational programs, for example, MAPLE. It should be noted that the model adopted a random control period. The latter expands the range of systems that can be adequately described by the proposed model.
It is advisable to teach the topic "Creating a mathematical model of reliability and stability of controlled technical systems" on the subject "Repair and reliability of technical systems" in higher educati onal institutions in the form of self-study. This requires the creation of electronic resources on this topic [11]. In the classroom, this topic should be carried out in the form of a problematic modern lecture [12]. As a result, future engineers will have theoretical knowledge and practical skills at the level of qualification requirements for creating models of reliability and stability of technical systems.
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