Corresponding author: Arkady I. Pereguda ( pereguda@iate.obninsk.ru ) Academic editor: Yury Kazansky
© 2019 Arkady I. Pereguda.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Pereguda AI (2019) Reliability of the RBMK1000 coolant flow measurement system. Nuclear Energy and Technology 5(1): 8187. https://doi.org/10.3897/nucet.5.34296

An analysis of statistical data of diagnostic measurements of two parameters determining the performance of the RBMK1000 SHADR8A flowmeters – the minimum value of the negative amplitude halfwave at the transistor flow measuring unit (TIBR) input and the meansquare deviation over the flowmeter ball rotation period – made it possible to develop a mathematical model of the flowmeter parametric reliability. This mathematical model is a random process, which is a superposition of two delayed renewal processes. Studying the flowmeter operational reliability model provides an exponential estimate of the probability that the parameters determining the flowmeter performance will not exceed the specified levels. Using the Bernoulli scheme and the probabilityestimating relationship for the flowmeter performance parameters, it is possible to calculate the probability of failurefree operation of both a single reactor quadrant and the coolant flow measurement system. In addition, it becomes possible to estimate the quadrant failure rate. Important for practice is the possibility of predicting the number of failed flowmeters depending on the system operation time. An indicator of the system reliability can be the average number of failed flowmeters, the relation for which is given in the paper. All the research results were obtained without any additional assumptions about the random values distribution laws.
The obtained results can be easily generalized for the cases when the vector dimension of the determining parameters is greater than two. The use of the results of this study is illustrated by calculated quantitative values of the flowmeter parametric reliability indicators and the coolant flow measurement system.
Parametric reliability, coolant flow measurement system, random variables, time between failures, random process, mathematical time expectation, distribution function, exponential estimate
Modern technical systems consist of a large number of elements and are largely automated. The increased complexity of systems has led to increased requirements for their quality and, as a result, to a sharply increasing interest in solving theoretical reliability problems that can provide a quantitative measurement of reliability indicators. Various influences accumulated by a system leads to the evolution of its indicators (changes in parameters), as a result of which a system can pass from normal operation to other qualitative states. Measures to ensure system reliability include: (1) detecting all types of possible transitions from one state to another; (2) determining their causes and consequences; and (3) planning activities to limit the number of failures of technical systems to an acceptable level. Of course, estimating quantitative system reliability indicators is only a small part of the entire complex of practical activities on ensuring the required reliability level, but without a thorough probabilistic analysis of the system operation process it is impossible to elaborate any reasonable decisions.
The reliability indicators of any product can be obtained by studying the behavior of one or several its parameters, which will fully reflect this product quality. If the processes of parameter changes are observable, predictable and manageable, it becomes possible to plan measures to prevent product failures. Failures occur as a result of deviations of the determining parameters from their initial (nominal, calculated) values. Failures are manifested as parameters’ overrunning the acceptable region (i.e., the area of normal operation).
The SHADR8A flowmeters are designed to measure the volumetric water flow in process pipelines and pipelines of the control and safety system channels of the RBMK1000 and RBMK1500 reactors at nuclear power plants. Due to a failed flowmeter, this channel operation is stopped until its operability is restored during routine maintenance. A reactor emergency system contains 240 of more than 1600 flowmeters (
The condition of coolant flowmeters is assessed by the results of measurements of parameters determining their performance, immediately prior to routine maintenance. If the diagnostic parameters deviate from the acceptable values, the corresponding flowmeter is replaced. The criterion for a gradual failure of a system (product) is a deviation of parameters determining its performance from a specified range of values.
As is shown in (
A mathematical reliability model in many cases is the mathematical theory of continuous Markov random processes or the theory of Wiener processes. The determining parameter change is considered as a particle walk along the lattice with a time step Δt and a spatial coordinate step δ = (Δt)^{1/2}. During the time t = nΔt, a particle receives the displacement x (t) equal to the sum of n steps δ = Δx_{k} along the spatial coordinate. The probability of a particle receiving the displacement δ in one step is ½. When Δt → 0, the transition probability of the considered random walk process tends to the Gaussian transition probability density. The Brownian process trajectory is quite cut (Δx ~ (Δt)^{1/2}, Δx/Δt → ±∞), but it is continuous and with probability one is not differentiable at any point (
Let the random process ξ(t) take on a value of x at the initial time t_{0} = 0. The question is: what is the probability that the random process ξ(t) will for the first time reach the specified lower bound a < x, or the upper bound b > x, or just a bound if only one of them is specified?
Experimental studies confirm that Markovian models describe well the changes in parameters caused by the degradation processes of aging.
However, in order to apply these mathematical models, it is necessary to make sure that the actual system operation process is Markovian, and then, using the system operation process trajectories, evaluate the coefficients of diffusion and the Kolmogorov equation drift (
For example, in (
The solution to the problem of maintaining uniform reliability and condition levels of the entire coolant flow measurement system is described in (
Since the purpose of the work is to estimate the coolant flow measurement system reliability in RBMK type reactors taking into account the system structure and failure criterion, it can be stated that the Bernoulli scheme can be used for a mathematical model of the system quadrant reliability. According to this scheme, the probability of occurrence of k events with n independent tests is defined as P_{k},_{n} (t) = C_{n}^{k} P^{k} (t)(1 – P (t))^{n}–^{k}, where P (t) is the probability of the failurefree flowmeter operation, which is to be estimated according to the results of diagnostic measurements. To estimate the probability of the failurefree flowmeter operation P (t), we shall use the cumulative reliability model discussed in (
For further presentation, we shall introduce the necessary notations and assumptions presented in more detail in (
Note that if the random variable is equal to τ_{0} = t_{1} – t_{0}, then F (t) ≠ F_{1}(t) = P (τ_{1} ≤ t), i.e., the τ_{0} value is distributed differently than all other random variables τ_{i}, i = 1, 2, ... . Next, we shall assume that the functions F (t) and F_{1}(t) are not arithmetic and each of these random variables has finite first two mathematical moments, i.e., Mτ < ∞ and Dτ < ∞. The sequence {τ_{i}, i ≥ 1} is usually called the delay renewal process, which we shall further denote as {T_{x}}_{x} _{> 0} (
The T_{x} value is the random time between failures of a product at a given determining parameter value of x, which is defined as
where N_{2}(x) = N_{x} is the random number of the determining parameter measurements made for the time before its crossing a specified level x.
Let us consider the second random process corresponding to the determining parameter change. Let γ_{0} denote a random initial value of the parameter determining a product’s performance, which is assumed to be independent of the sequence {τ_{i}, i ≥ 1} and have the arbitrary distribution function G_{0}(y) = P (γ_{0} ≤ y). Let us introduce random variables γ_{I}, i.e., the determining parameter values measured at the timepoints t_{i}, i = 1, 2, ... . Thus, the random process {γ(t)}_{t} _{> 0} at the set T of the real straight line is a process with independent increments, since for any values of t_{0} ≤ t_{1} ≤ t_{2} ≤ ... at the set T, the increments Θ_{k} = γ(t_{k}_{+1}) – γ(t_{k}), k = 0, 1, 2, ... are independent random variables. It is natural to assume that the random increments Θ_{i}, i = 1, 2, ... are distributed with the same function G (x).
The sequence {Θ_{i}, i = 1, 2, ...} generated by the functions G_{0}(x) = P (Θ_{0} ≤ x) = P (γ_{0} ≤ x) and G_{i} (x) = P (Θ_{i} ≤ x) will also be a delayed renewa l process, which will be further denoted as {Θ_{t}}_{t} _{> 0}. Suppose that the mathematical expectations and variances of the random variables Θ_{0} and Θ must satisfy the conditions MΘ_{0} < ∞, MΘ < ∞, DΘ_{0} < ∞, DΘ < ∞.
It is obvious that the total value of the parameter that determines a product’s performance (accumulated load) at the time it crosses the specified bound can be determined by the equality
where N_{1}(t) = N_{t} is the random number of determining parameter measurements that occurred during the time [0, t] or the number of process renewal cycles {τ_{i}, i ≥ 1}.
The purpose of the work is to construct a mathematical model of a product’s parametric reliability and, based on the model analysis, to obtain exponential estimates of the probability that the determining parameters of SHADR8A coolant flowmeters do not go beyond the specified performance bounds. It is also required to estimate the probability of failurefree operation of the flowmeters, their failure rate, the average number of failed RU quadrant flowmeters, and also, using the Bernoulli scheme, to estimate the probability of failurefree operation of the RBMK1000 coolant flow measurement system.
When solving the problem, it is necessary first of all to estimate the probability that the product determining parameters are not beyond the specified performance bounds. For this purpose, we shall use Relations (1) and (2), which are sums of independent random variables, while the number of terms of these sums is random. In these relations, Θ_{i} and τ_{i} are sequences of identically distributed independent random variables with mathematical expectations MΘ_{0}, MΘ, Mτ and variances DΘ_{0}, DΘ, Dτ. It is assumed that the random variable N_{1}(t) = N_{t} is independent of Θ_{i}, and N_{2}(x) = N_{x} is independent of τ_{i}. Let us calculate the first two moments, one of which is the initial moment of the first order (mathematical expectation), and the other one is the central moment of the second order (variance) of the processes {Θ_{t}, t ≥ 0}.
Before calculating the moments of the random variable Θ_{t}, it is necessary to write the Laplace–Stieltjes transformation. Since all the random variables in (2) are independent and equally distributed (perhaps, except for Θ_{0}), the desired transformation can be written as:
where the sum on the righthand side is the generating function of the random variable Θ [8, 16]. Differentiating the function Θ^{*}(s) with respect to the variable s and inserting s = 0 into the resulting derivative, we obtain the mathematical expectation Θ_{t}:
MΘ_{t} = MΘ_{0} + MN_{1}·MΘ, (3)
where MN_{1} = H_{1}(t) is the process renewal function {Θ_{t}, t ≥ 0}.
When calculating the variance of the random variable Θ_{t}, it is necessary to twice differentiate the function Θ^{*}(s) with respect to the variable s and subtract the square of Relation (3), as a result of which we obtain
Using the strengthened elementary renewal theorem (
It is worth reminding that all the results of the renewal theory obtained asymptotically are valid for each initial distribution F_{1}(t). For this reason, in Relation (5) and further, the random variable t_{1} will be absent:
If the limiting value of the determining parameter change is specified,
it is possible to obtain from (6) the average time between failures of a product T_{Θ} before the specified bound is crossed:
Thus, using Relation (2) and the Laplace–Stieltjes transformation, we obtained the expectation and variance of the random variable Θt as well as the average time between failures of a product T before the specified performance bound is crossed.
It is much more difficult to solve the problem of calculating the probability of failurefree operation of a product affected by a periodically varying load. It proved to be impossible to calculate the probability of failurefree operation of a product operating in the above conditions, even under the assumption that all random variables have an exponential distribution. Therefore, there is a need to obtain relations that will make it possible to approximately estimate the probability that the determining parameter will cross the specified bound of a product’s performance. Note that the normalized sum of a large number of independent random variables has a distribution that is close to the Gaussian one (
It is known (
P (L_{t} > x) ≤ Mg (L_{t})/g (x),
where L_{t} is the random accumulated load; g (x) is the nondecreasing nonnegative function defined on the interval [0, ∞), and the function and its derivatives are continuous and differentiable in this neighborhood and g(x) > 0.
Suppose that the function g (x) = exp(λx), where λ is a constant, then for any s ≥ 0 we have
The estimate of the mathematical expectation of a random variable is written as:
where L_{t} – ML_{t} ≤ A (A is a constant). Assuming that λ < 2/A, we shall rewrite the estimate as:
Inserting the obtained estimate of the random variable M exp(λL_{t}) into (8), we obtain
Estimate (9) can be somewhat improved, for which it is necessary to minimize the function
where B = 0.5σ^{2}_{Lt} = 0.5(σ^{2}_{0} + H (t)σ^{2}_{Θ} + (MΘ)^{2}σ^{2}_{Nt}); B_{1} = 0.5Aσ^{2}_{Lt} = 0.5A (σ^{2}_{0} + H (t)σ^{2}_{Θ} + (MΘ)^{2}σ^{2}_{Nt}); x_{a} = x – MΘ_{0} – H (t)MΘ.
Then the value of the λ_{0} parameter, which ensures the minimum of the function F (λ), is optimal and is found as a solution of the algebraic equation –x_{a} + 2λ_{0}B + 3λ_{0}^{2}B_{1} = 0. Therefore,
λ_{0} = –B/3B_{1} + B/3B_{1}(1 + 3Ax_{a}/B).
In this case, the probability of a product’s failurefree operation in the conditions of discrete degradation will be determined by the relation:
where B, B_{1}, λ_{0} and x_{a} are the values entered earlier.
If the value of 3Ax_{a}/B is small, the λ_{0} parameter can be written as:
λ_{0} ≈ – B/3B_{1} + B/3B_{1}(1 + 3B_{1}x_{a}/2B^{2}) = x_{a}/2B_{1}.
In this case, the estimate of the probability of a product’s failurefree operation will take a simpler form:
Estimate (10) is a pessimistic estimate of the probability of a product’s failurefree operation under the influence of a periodically varying load. Note that the estimate is calculated quite simply and provides accuracy that is sufficient for practical use.
Statistical material for measuring the control parameters of the flowmeters makes it possible to determine the predicted value of the average time between failures before any of the determining parameters crosses the specified level and calculate the quantitative values of the reliability indicators of both the flowmeters and coolant flow measurement system. Thus, the coolant flow measurement system of RBMK type reactors is a rather cumbersome, consisting of 240 homogeneous elements, each of which can be in one of three possible conditions, when
– both determining parameters has not reached the specified levels;
– the first determining parameter has crossed the specified level;
– the second determining parameter has crossed the specified level.
Since random processes corresponding to changes in the determining parameters are independent, the task of calculating the reliability indicators of the coolant flowmeters is somewhat simplified, but it also becomes necessary to consider two problems of estimating the flowmeter reliability indicators for each of the determining parameters separately.
For the analysis, we took the data obtained as a result of annual measurements (from 1999 to 2013) for 50 SHADR8A flowmeters. The analysis of statistical data obtained from diagnostic measurements made it possible to estimate the mathematical moments of the determining parameters, i.e., the minimum value of the negative amplitude halfwave at the input of the transistor flow measuring unit (TIBR) and the meansquare deviation over the flowmeter ball rotation period. The results of estimations of the mathematical moments necessary for further calculations are shown in Table
Mathematical expectation and variance of the minimum negative amplitude halfwave value at the TIBR input.
MA _{min}  DA _{min}  MA _{min 0}  DA _{min 0} 

– 4.552  131.221  120.728  387.654 
Mathematical expectation and variance of the meansquare deviation over the flowmeter ball rotation period.
M_{T}  D_{T}  M_{T} _{0}  D_{T} _{0} 
1.588·10^{–4}  4.568·10^{–6}  7.067·10^{–3}  1.988·10^{–6} 
Mathematical expectation and variance of the overhaul period.
M  D 
8645  34590 
Using the parameters of the random variables of the minimum negative amplitude halfwave at the TIBR input and those of the overhaul period (see Tab.
The value of the specified level for the first parameter, which determines the performance of the flowmeter, is A_{0} = 10 mV; the specified level for the second parameter is σ^{2}_{0} = 0.02. Since the criterion for a coolant flowmeter’s failure is the crossing of the specified performance level by any of the determining parameters, the probability of the flowmeter failurefree operation is P (t) = P_{1}(t)P_{2}(t). The time dependences P (t), P_{1}(t) and P_{2}(t) are shown in Fig.
Note that the failure criterion for the reactor quadrant is failure of 10 or more flowmeters in one quadrant. Consequently, the quadrant will function properly if less than 10 out of 60 flowmeters are in failure mode. Let F (t) denote the probability of the quadrant failurefree operation. To obtain the formula by which F (t) can be calculated, the Bernoulli scheme can be applied (
Since a system failure occurs when at least one quadrant fails, the probability of failurefree operation Q (t) of the reactor coolant flow measurement system is calculated by the formula Q (t) = (F (t))^{4}. The calculated probability of failurefree operation of the quadrant and the system is shown in Fig.
Along with the probability of failurefree operation of elements and systems, other reliability indicators play an important role in system analysis. For example, using the probability of failurefree operation of the quadrant F (t), one can estimate the quadrant failure rate by the formula (by definition) λ(t) = – dF (t)/dt × 1/F (t). Figure
Of practical interest is the possibility of predicting the number of failed flowmeters depending on the system operation time. Such an indicator of reliability can be the average number of failed flowmeters defined by the formula
where k (t) is the random number of failed flowmeters in the quadrant. A graph of Mk (t) versus operation time is shown in Fig.
A mathematical model of parametric reliability has been developed that takes into account the statistical data of diagnostic measurements of two parameters that determine the efficiency of the SHADR8A flowmeters of the RBMK1000 reactor, i.e., the minimum value of the negative amplitude halfwave of the TIBR input signal and the meansquare deviation over the flowmeter ball rotation period. The mathematical model of the flowmeter reliability is a random process, which is a superposition of two delayed renewal processes. Studying the mathematical model of the coolant flowmeter reliability made it possible to obtain an exponential estimate of the probability that both parameters determining the flowmeter performance did not cross the specified levels. The probability of failurefree operation of one reactor quadrant and the coolant flow measurement system was found. The estimated quadrant failure rate and the relation for calculating the average number of failed flowmeters depending on the system operation time were obtained. In studying the mathematical model of parametric reliability, no assumptions were made about the random values distribution laws.
The author is grateful to V.L. Mironovich for his very useful comments and suggestions in the preparation of this paper.