APPLICATION OF NEURAL NETWORKS FOR THE ANALYSIS OF VVER-1000 REACTOR PRESSURE VESSEL HYDRODYNAMICS

Katkovsky Е.А.1, Katkovsky S.E.1, S. Nikonov2, I. Pasichnyk3, K. Velkov3

1“Energoautomatika” Ltd, Moscow, Russia katkovsky@mail.ru

2NRC „Kurchatov Institute“, Moscow, Russia niks@vver.kiae.ru

3Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) mbH, Garching, Germany {Ihor.Pasichnyk, Kiril.Velkov}@grs.de

ABSTRACT

The paper presents a methodology which will enable in the future performing of fast transient analysis of NPP with VVER-1000 reactors. It is based on artificial neural networks (ANN) method. For the training of the network the best-estimate system code ATHLET (GRS) is used with detailed nodalization (multi-channel model) of the Reactor Pressure Vessel (RPV). The present work is dedicated to the training procedure which is connected at this stage of development only with the setting in the system the RPV hydraulic resistances. Training samples are created by random variation from a double sided 95% confidence interval of 50 different hydraulic resistance coefficients in the RPV. For each set of hydraulic resistance coefficients a test calculation is performed with ATHLET to obtain the corresponding thermo-hydraulic distributions (mass flow, pressure, temperature) within the RPV. For this training procedure the total and the assembly power distributions used in all calculations remain unvaried and equal the nominal values, e.g. at this stage a dynamic 3D neutron-physics model is not being taken into account. As a result of the neural network training an application program is created. This program is capable to give almost simultaneous answer about all thermo-hydraulic fields at each point of the RPV. A comparison of the system results with results obtained from the AHTLET simulation gives a good correspondence for such nodes (fluid objects) of the active core, which have not been taken into account in the training sequence of the artificial neural network. The paper contains also some ideas how to apply the artificial network modeling of the RPV in the current practice. This work is a step forward to create an artificial neural network system for performing of fast NPP analysis. Further training of the system will be done with the help of coupled neutron-physics/thermal-hydraulics codes based on ATHLET.

1. INTRODUCTION To model the thermo-hydraulic processes in the nuclear power plant (NPP) using system codes, regardless of the coolant type and the type of simulation method (one-, two-, pseudo three- or three-dimensional), it is required an input data in the form of hydraulic resistance coefficients, thermal coefficients, closing relations, etc. And it does not matter whether they

are obtained experimentally or analytically, for different ambient conditions or different parameters of a facility, they always contain uncertainties. In order to verify and validate any system code one has to investigate the response of the simulated systems on the variation of the above mentioned coefficients and relations. This requires performing of a set of calculations with varied tested parameter, sometimes in a rather wide range. This approach is time consuming and leads to a considerable number of performed calculations. Taking into account the fact that the wide spread system codes such as ATHLET, RELAP, TRACE or CATHARE have a high CPU simulation time of a single run. The assessment process for all values of the tested parameter may require enormous CPU costs. In the case of testing simultaneously several parameters or relations the calculation time grows exponentially. Moreover the probability of introducing failures and human errors increases and that may distort the results of the analysis. Such a multiple varied analysis in this case may not lead to the desired result. As an alternative one can apply the method of factor analysis. Factor analysis is a multidimensional method and can be applied to study possible correlations between variables. Under the assumption that the variability of the system can be described by a lower number of unobserved variables, the result of the factor analysis could be such unobserved variables, which are named factors. Factor analysis helps to solve two important problems of the researcher: to describe the system: a) fully and b) compact. It allows identifying of unobserved latent factors, which are responsible for the linear statistical correlations between the observed variables. Thus it is possible to highlight two objectives of factor analysis:

1) Identification of interdependencies between variables, or variables classification [1-2]; 2) Reduction of the number of variables needed to describe the system.

Practical application of the factor analysis begins from testing the correctness of the input data. To the obligatory conditions of factor analysis belong the following:

1) The system description must be given numerically; 2) The number of observations must be at least twice larger than the number of variables; 3) A sample must be homogeneous, i.e. a sample which describes the system must

contain only data related to only one property of the system; 4) The input data must be distributed symmetrically, i.e. to each input corresponds one

output; 5) Factor analysis is applied to correlated variables [3]. If there is no correlation between

inputs and outputs, the result of the factor analysis is random value. The present paper is devoted to the development of a new technique for performing factor analysis of NPP hydraulic variables, which is based on ANN-method. ANN possesses a number of unique properties to identify the latent interdependencies in the investigated systems and to highly compress the numerical representation of these interdependencies. ANN automatically satisfies all requirements needed to perform factor analysis. After discussing the method and the system used for testing of the ANN results are presented which confirm the applicability of the method to the studied system.

2. DESCRIPTION OF THE MODEL AND OF THE INPUT PARAMETER SPACE

To test the ANN methodology the ATHLET model of VVER-1000 RPV is considered with a detailed nodalization scheme. This model is used in the analysis of the OECD/NEA coolant transient Benchmark (K-3). The studied transient in the Benchmark is a switch off of one main coolant pump (MCP) in the Kalinin NPP Block 3 at nominal reactor power (98.6% of nominal power, fuel assemblies (FA) TVSA, first load, 130.6 effective days) [4]. In the present work is considered the analysis only at steady state conditions. The detailed description of the nodalization scheme is given in [5]. Using the real geometric information from in-vessel objects, it resolves the hydraulic properties on the level of hydraulic resistance coefficients (HRC). Such a model allows simulating the response of the system with respect to variation of all these HRC. The complete list of all varied HRC with their statistical properties (art of probability distributions and their parameters) is given in [6]. In total there are 50 varied HRC in the RPV generated with the help of SUSA package. An input data for the ANN training consists of all varied HRC, three dimensional coordinates of the control volume centre point or its section centre point for each fuel assembly (FA), and the results of ATHLET calculation (e. g. pressure, coolant temperature and mass flow at these coordinates). Hence the training is performed using 53 training parameters (inputs). The size of the training sample is taken as 𝑁 = 100 ATHLET runs. In total the training sample contains approximately 5 ⋅ 105 data rows. Due to the nature of the simulation method of ATHLET code, which is essentially a Finite Volume method, the coordinates of the mass flow values are shifted with respect to the given coordinates of the temperature and pressure (staggered grid). ANN is capable to convert all values to a single mesh (sections of FAs). That is possible since after the training phase ANN is capable to “understand” and continuously reproduce all hydraulic fields at any coordinate (so-called ANN response). The problem to be solved is very complex. It is enough to mention that the ANN construction is carried out by a search for a global error minimum of the neuronal network approximation function in 53-dimensional space. It is done without prior information about the location of the minimum, using only the training samples of the ATHLET simulation runs with 50 varied HRCs. To tackle this problem it is required to select correctly the architecture of ANN. After defining the training set as a set of pairs {𝑥𝑛,𝑑𝑛} with {𝑥𝑛} as input and {𝑑𝑛} as output, (𝑛 = 1, … ,𝑁),𝑁 – sample size used for the training, the training procedure is controlled by a number of several error estimators. The normalized mean squared error is a mean squared error estimator normalized to the deviations in the desired output vectors. It is defined as:

𝑁𝑀𝑆𝐸 = ����𝑦𝑖𝑗 − 𝑑𝑖𝑗�2

𝑁

𝑖=1

𝑃

𝑗=1

� ⋅ ����𝑑𝑖𝑗2𝑁

𝑖=1

−1𝑁��𝑑𝑖𝑗

𝑁

𝑖=1

�

2

�𝑃

𝑗=1

�

−1

(1)

where 𝑦𝑖𝑗 is the network output, 𝑑𝑖𝑗 is the desired output for sample 𝑖 at processing element 𝑗 and 𝑃 is the number of output processing elements. Most simple estimator for the error 𝐸 is the percent error and is determined as

𝐸% =1𝑁��

�𝑑𝑦𝑖𝑗 − 𝑑𝑑𝑖𝑗�𝑑𝑑𝑖𝑗

⋅ 100𝑁

𝑖=1

𝑃

𝑗=1

(2)

where 𝑑𝑦𝑖𝑗 is the denormalized network output and 𝑑𝑑𝑖𝑗 is the denormalized desired output for sample 𝑖 at processing element 𝑗. The network performance for the training data is estimated using Akaike information criterion (𝐴𝐼𝐶) as a measure of the relative goodness of fit of a network:

𝐴𝐼𝐶(𝑛𝑐) = 𝑁 ⋅ ln(𝑀𝑆𝐸) + 2𝑛𝑐 (3) where 𝑛𝑐is the number of connections in the network. 𝐴𝐼𝐶 is used to measure the tradeoff between the complexity and the performance of the network. The goal is to minimize 𝐴𝐼𝐶 value. The criterion which is similar to 𝐴𝐼𝐶 has the name of Rissanen's minimum description length 𝑀𝐷𝐿 which combines the model error and the number of degrees of freedom to determine the level of generalization. The 𝑀𝐷𝐿 is calculated by this formula

𝑀𝐷𝐿(𝑛𝑐) = 𝑁 ⋅ ln(𝑀𝑆𝐸) +12𝑛𝑐ln (𝑁) (4)

The training is considered successful if 𝑁𝑀𝑆𝐸 ≤ 5 ⋅ 10−3, 𝐸 ≤ 0.1%, 𝑟 ≥ 0.99. For ANN construction the package Neurosolution 6.30 is used [10].

3. CHOICE OF ANN ARCHITECTURE AND ANN TRAINING The simplest choice of the ANN architecture is given by Multi-Layer Perceptron (MLP). It is the most common supervised neural network. It consists of multiple layers of processing elements (PE) connected in a feedforward fashion. Such ANN was applied to study of Balakovo NPP parameters in [9]. Unfortunately for the current study this network is not suitable. It gives a large error and its correlation coefficient 𝑟 does not exceed 50%. Hence it is not possible to use this network even for rough approximations. Jordan and Elman networks which extend the multilayer perceptron with context units [11], as well as Radial basis function (RBF) networks [11] do not even reach the convergence during the ANN training. Support Vector Machines (SVM) which are implemented using the kernel Adatron algorithm [12] has a slow convergence but delivers better results compared to previously mentioned networks. Nevertheless even SVM architecture does not reach the required accuracy.

A qualitative leap in the accuracy and correlation of the results is achieved by using one of the types of Modular Feedforward Network (MFN) [11]. It is a special case of MLPs, such that layers are segmented into modules. In contrast to the MLP, MFN networks do not have full interconnectivity between the layers and process their input using several parallel MLPs recombining the results. A modular network will generally train faster than a MLP, due to the fact that it has “short-cut” connections to the output, aiding in the weight adaptation for the hidden and input layers. There are many ways to segment a MLP into modules. In the present work MFN is represented by two branches (or modules) of two hidden layers. One module is used to train the network from HRC variations and the second module is trained from coordinates (nodes) in RPV. An optimization of the number of neurons and synapses in the hidden layers is carried out by the Genetic Algorithm (GA) approach. GA determines the best network parameters by successively trying and testing different combinations of parameters. Like an evolution, the good parameter sets are more likely to survive from one population to the next [11]. In the Neurosolution 6.30 the genetic optimization takes place by repeatedly training the network over and over with various parameters and calculating the best 𝑁𝑀𝑆𝐸 for each network. GA provides two categories of optimization algorithms: solution-based optimization and population-based optimization. Solution based algorithms include Attribute Selection and Simulated Annealing. The population-based optimization currently includes Genetic Algorithm, which is further classified in to Single Objective Genetic Algorithm and Multi-Objective Genetic Algorithms. Single Objective Genetic Algorithms include Generational and Steady State. Multi-Objective Genetic Algorithms currently include Non-dominated Sorting with multi-constraints. Furthermore, it is possible to select between several types of selection, crossover, and mutation [9]. In the training process to go out from local minima the switch from the batch mode to the online mode is applied. This is triggered by the absolute values of 𝐴𝐼𝐶 and 𝑀𝐷𝐿 (see Eq. (3) and Eq. (4)). As a result Figure 1 presents the topology of the MFN chosen for the K-3 transient analysis. The description of all MFN components is given in Table 1. The calculation of one state in K-3 transients takes on one core of CPU i7-3930 (OS Windows 8) approximately 6 ⋅ 10−4𝑠𝑒𝑐 by the trained ANN. Comparing it with the typical calculation of one state by the ATHLET code, which is approximately 7 sec, the speed up is 105 times.

Figure 1. MFN architecture of the trained network. Explanation of symbols is given in Table 1

Table 1 Description of all modules of MFN (see Figure 1)

Icon Name Description Primary Usage

Axon Layer of PE's (processing elements) with identity transfer function.

Can act as a placeholder for the File component at the input layer, or as a linear output layer.

TanhAxon Layer of PE's with hyperbolic transfer function (output range –1 to 1).

Used as hidden or output layer.

FullSynapse Full matrix multiplication.

Connects two axon layers.

L2Criterion Square error criterion.

Computes the error between the output and desired signal, and passes it to the backpropagation network.

BackAxon Layer of PE's with identity transfer function.

Attaches to "dual" forward Axon, for use in backpropagation network.

BackTanhAxon Layer of PE's with transfer function that is the derivative of the TanhAxon.

Attaches to "dual" forward TanhAxon, for use in backpropagation network.

BackCriteriaControl Input to backpropagation network.

Attaches to Criterion, for use in backpropagation network. Receives error from Criterion.

Momentum Gradient search with momentum.

Updates weights. Momentum increases effective learning rate when weight change is consistently in the same direction.

StaticControl Static forward controller

Controls the forward activation phase of network.

BackStaticControl Static backpropagation controller

Controls the backward activation phase of network (backpropagation).

File File input For network input and desired data from a file.

ThresholdTransmitter Thresholded transmitter

For controlling one component based on the values of another.

BarChart Bar chart probe Displays data bar graph style.

DataGraph Graphing probe Displays data versus time.

MatrixViewer Numerical probe Displays numerical values at the current instant in time.

DataWriter Numerical probe Displays numerical values across time. Also allows for the saving of data to a file.

4. RESULTS The following section is devoted to the performed testing for the ANN after its training. Tested is its ability to predict the results on a sample of 400 ATHLET runs which have not been applied in the training procedure. These 400 runs are generated using the same probability distributions as for the 100 training runs. All results are given in terms of relative error of the ANN result with respect to the ATHLET calculation. Figure 2, Figure 3, Figure 5, Figure 6 and Figure 8 show the mean value and the standard deviation of the relative error over a whole sample for mass flow, coolant temperature and pressure. The values are presented for all nodes in RPV and compare all local core parameters. In the case of mass flow the mean value of the relative error does not exceed 1% and the maximal mean value is in the center of the core in a last axial layer. The standard deviation of the mass flow relative error has a maximal value of approximately 1.25%. These maximal values appear at the periphery of the last axial layer (see Figure 2). The value of the standard deviation gives a hint that the relative errors for a total sample set has some outliers, which cannot be represented only by the two first moments (mean and standard deviations) of the distribution. Thereby the maximal value of the relative error reaches 9%. at the periphery of the last axial layer (Figure 4).

The similar picture is observed also for the nodal temperature distribution (Figure 5, Figure 6 and Figure 7). Statistically the ANN gives very god results: the maximal mean value is approximately 1.3% and the maximal standard deviation is approximately 0.5% The largest irregularities are observed at the last axial layer where the maximal relative error reaches approximately 10%. Finally, the pressure prediction of the ANN is perfect: the maximal values of mean and standard deviation do not exceed 0.05%.

Figure 2. Mean relative error of mass flow and standard deviation of the relative error. The facets represent the axial layers (in m) of RPV

Figure 3. Mean value of the mass flow relative error. The facets represent the axial layers (in m) of RPV.

Figure 4. Maximal relative error of mass flow. The facets represent the axial layers (in m) of RPV.

Figure 5. Mean relative error of coolant temperature and standard deviation of the relative error. The facets represent the axial layers (in m) of RPV.

Figure 6. Mean relative error of coolant temperature.

Figure 7. Maximal relative error of coolant temperature. The facets represent the axial levels layers (in m) of RPV.

Figure 8 Mean relative error of pressure and standard deviation of the relative error. The facets represent the axial layers (in m) of RPV.

5. SUMMARY AND CONCLUSIONS The work describes an application of ANN methodology to predict the main hydraulic parameters of a steady state K-3 Benchmark WWER-1000 core. The training procedure is based at this stage only on 100 ATHLET runs sample sets generated by variation of the RPV hydraulic resistances at nominal power. The ANN shows very promising results on a large testing sample set, which contains 400 ATHLET runs. The lower performance at the boundary nodes of the last axial layer can be caused by a lack of interpolation points in this region. The next step will be to improve the performance of the ANN for the last core axial layers boundary nodes and to include in the ANN construction additional training parameters like the total reactor power, the assembly wise powers and their axial distribution. Additional studies are on-going to determine the optimal selection of the available randomized data sets which should be applied by the learning procedure.

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