The input for the integrated assessment of the reactor equipment integrity is information received from

• the ALMS and the HLMS (leak and probable leak detectors, leak rate values and coordinates, as well as values of the acoustic and humidity measuring channel (MC) signals for all piping segments monitored by the system);
• I&C process and radiation monitoring systems (inert gas reactivity sensor signals).

It is expected in the development of the algorithm under consideration that the reactor facility global noise is recorded identically by all of the acoustic system’s MCs. Since the cause for this noise is common for the MCs in each loop (which is confirmed by the signal cross correlation coefficients the average value of which amounts to 0.65), the signals in the presence of noise can be presented in a vector form:

s (t) = q(t) + d(t), (1)

where s(t) is the system MC readings; q(t) is the global noise; and d(t) is the local signal.

Therefore, the global noise filtering problem is reduced to finding the unknown function q(t) and deducing same from the known signal s(t). The values of any MC readings s_i (t) for the i-th segment can be expressed through the MC readings for the adjacent segments using a regression model. Such approach was proposed in (Skomorokhov et al. 2010) where a group data handling method (GDHM) was used as the regression model. Two types of regression models were explored in the algorithm proposed below: a two-layer neural network based on which the nonlinear regression was built, and a linear Bayesian robust regression model (both are described hereinafter). As compared with (Skomorokhov et al. 2010), good results have been achieved in predicting the global acoustic noise thanks to using exactly the Bayesian robust regression model and Kalman residual signal filtering.

Further, we shall consider the principles of building regression models separately for the primary circuit and the secondary circuit.

A very high correlation is observed in both circuits for nearly all MC signals, and the correlation coefficient represents a significant value. A regression was so built in the study for each MC signal in the particular reactor facility portion depending on the signals from all MCs in other loops (three loops), as well as on the signals from MCs in other segments in the same loop. As a result, there were four regressions obtained for each MC. This is quite enough to ensure the stability of the algorithm to a change in the state or to the failure of other MCs (predictors) since too many MCs may not fail during one life cycle.

Using the proposed grouping, one can describe the common part of the signal from each MC, q(t), as a function of the signals from other MCs in four different ways. Therefore, it is possible to predict the signal from any MC based on signals from the MCs in other loops (belonging to one pipeline type), as well as based on signals from the MCs in the same loop but in the segment in another room.

Therefore, there were four dependences obtained for each signal. A neural network (Goodfellow et al. 2016; Nikolayenko et al. 2018; Chollet 2017) with one inlet layer, one hidden layer with a dimension of 30, and one outlet layer was used initially to determine these dependences. Here

q _i (t) = net ((s_i (t), ∑_j s_j (t)) , (2)

where s_i (t) is the signal in the i-th MC for which the regressions were built; s_j (t) is the signals from MCs in another loop or another sections based on which the regression was counted; q_i (t) is the global noise vector for the i-th MC calculated using different regressions; and net is the trained neural network consisting of two fully connected layers with 30 neurons each, into which the values of the MC signals s_j (t) are plugged.

This regression model is nonlinear and, therefore, even minor changes in the operation of the measuring channels may lead to major deviations in its application results. Fig. 1 presents the results of applying two different regression models to the signal from the first acoustic MC for the life cycle that started after the preventive maintenance as from 20.04.2018.

Shown in dark-grey is the regression obtained using formula (2) based on data for the period of 20 May through 20 June 2018 and propagated to all of the currently available data. It can be seen that it describes the original signal not in the best way possible. At the same time, the light-grey curve obtained for the period of 3 May through 3 June 2018 describes pretty well the entire set of data. This curve was obtained as the result of using the linear robust Bayesian regression model (Kruschke 2013; Cameron 2016; Barber 2017)

q _i (t) = ∑_j β_j s_j (t) + ξ(t), (3)

where ξ(t) is the regression error having a Student distribution.

Student distribution has thicker “tail areas” than normal distribution and it is therefore more stable to different bursts in data (Kruschke 2013). Unlike Gaussian distribution when the problem solution is reduced to finding the pseudoinverse matrix in the least square method, the problem of minimizing the negative log-likelihood from the Student distribution needs to be addressed directly here. The result however justifies such computational efforts.

All of the further analysis was based on regression model (3). Calculations have shown that it is enough to build the regression using model (3) for one day to reproduce in an acceptable manner the lifetime data.

Sharp peaks occur in acoustic MC signals during the reactor warm-up and cooldown, this being immediately connected with the RCP operation in the reactor primary circuit. These acoustic bursts caused by the RCP startup at a low pressure in the circuit are highly significant (they have a value of about 5000 μV). With such signals, the acoustic sensor response turns heavily nonlinear, and it becomes impossible to find out if there is a leak. The thing is that the integral signal that comes from acoustic MCs is, in fact, a dispersion of the actual signal and, therefore, the signal from the leak is quadratically added to the background signal, that is

U _tot ² = U_back² + U_leak², (4)

where U_back is the background signal; U_leak is the leak signal; and U_tot is the total signal.

The obtained regression dependences for each MC were used to calculate the values of acoustic signals with the filtered global noise d_i^j (t) = s_i (t) – q_i^j (t), where i is the MC number; and j is the regression dependence number. Therefore, if there is a leak in any piping segment, two different situations are possible.

• The leak occurred in the segment with the MC under consideration. Then all values d_i ^j (t) will vary according to the noise caused by the leakage, and d_i (t) = | min _(i)d_i ^j (t) | is the sought-after leak signal with the filtered global noise which is compared with the given setpoints.
• The leak occurred in another segment, j. Then the regression dependence q_i ^j (t) for given MC i will be wrong, but the others, q_i ^k (t), where k ≠ j, will give correct values, and the expression d_i (t) = | min _(k)d_i ^k (t) | will remain valid, and the value d_i (t) for the i-th MC will not contain the leak signal. It should be noted here that the more regressions are considered, the more reliable will be the result. Therefore, if regressions are considered not cumulatively from all MCs in the adjoining loops but from all MCs in each segment in all loops, we shall then have 11 regressions for each MC in the loop.

The resultant quantity d_i (t) is an observed random signal which contains a measurement error, as well as a global noise filtering (regression dependence) error. In order to obtain the optimum signal from this, we shall consider the problem at hand in a state space model where the optimum solution is achieved thanks to the use of a Kalman filter (Najim 2008; Durbin and Koopman 2012; Haykin 2014; Grewal and Andrews 2015).

It is assumed in the system model that the actual hidden state at the time t+1 results from the state at the time t according to the state equation

x (t+1) = A·x(t) + ε(t), (5)

where A is the state space matrix; and ε is the state space error vector, and the measurement vector d_i is connected with the state vector of the equation system

d _i (t) = C·x(t) + ω(t), (6)

where C is the measurement matrix; and ω is the measurement error vector.

The Kalman filter iteration is divided into two phases: prediction and registration of observations. The prediction phase uses the state calculated at the previous step for obtaining the evaluated state at the current step through the system model. At the observation registration phase, information on the measurements performed at the current step is used to update information on the system state, which, as a result, makes this information more accurate. Kalman filter is the best way to make the signal obtained by measurements in a linear system with Gaussian noise as close to the actual value as possible.

Fig. 2 presents diagrams of the initial signals s_i (t) and the signals d_i (t) filtered of global noise using regressions and optimized by the Kalman filter. Evidently, the frequency of false responses with signals cleared of global noise decreases considerably both for the acoustic system and the humidity system.

Let us consider how the quantity d_i (t) will change in the j-th piping segment if the acoustic MC fails elsewhere in another segment. In this case, the regression dependence of global noise for acoustic MCs in the segment j on the segment with the failed MCs will be wrong. However, as a result of the operation of the algorithm proposed above, the resultant quantity d_i (t) will not differ greatly from the case when all acoustic MCs in all segments operated normally. The result of the considered situation is presented in Fig. 3.

Fig. 3a shows a standard situation, and the readings in Fig. 3b for acoustic MC 32 in the U-shaped bend in loop 4 have been artificially changed by 500 μV. It can be seen that the quantities d_i (t) obtained for this MC based on four and three regressions do not practically differ. Further, it is planned to consider the dependence of the signal from each MC not simply on the signals from MCs in other loops but from MCs in other segments. We shall have then 11 regressions instead of four (since there are three segments in each loop) which will make it possible for the algorithm to achieve a much greater reliability and stability of results in the event of the MC failure in other segments.

We shall note that it is possible to filter global noise exactly in the same way for the HLMS as well. Absolute humidity is compared here with the threshold value (the threshold value is 0.375 kg/m³). It is even easier to filter global noise in the event of a humidity system than for an acoustic system since the values of the signals from the HLMS MCs are rather smooth and do not have bursts throughout the life cycle.