=Paper=
{{Paper
|id=Vol-1995/paper-07-965
|storemode=property
|title=
Simulation of Xenon Transition Processes Based on Data of Reactor
Experiments and Metric Analysis
|pdfUrl=https://ceur-ws.org/Vol-1995/paper-07-965.pdf
|volume=Vol-1995
|authors=Alexander V. Kryanev,Alexander A. Orekhov,Anatoly A. Pineguin,Sergey V. Semenov,David K. Udumyan
}}
==
Simulation of Xenon Transition Processes Based on Data of Reactor
Experiments and Metric Analysis
==
https://ceur-ws.org/Vol-1995/paper-07-965.pdf
53
Simulation of Xenon Transition Processes Based on Data
of Reactor Experiments and Metric Analysis
Alexander V. Kryanev∗ , Alexander A. Orekhov∗ , Anatoly A. Pineguin† ,
Sergey V. Semenov† , David K. Udumyan∗‡§
∗
National Research Nuclear University “MEPhI”
†
National Research Centre “Kurchatovsky Institute”
‡
People’ Friendship University of Russia (RUDN University)
§
University of Miam
Email: a_v_kryanev@mtu-net.ru, orehovsasha@mail.ru, Pinegin_AA@nrcki.ru, Semenov_SV@nrcki.ru,
mathudum@gmail.com
The report presents a new scheme for modeling xenon transient processes, based on the use
of accumulated information on the parameters of the core state in the reactor operation. The
proposed scheme uses an interpolation algorithm for functions of many variables, constructed
on the basis of metric analysis. Approbation of the developed algorithm for searching for
the most probable set of model parameters was carried out using the example of solving a
series of model problems. The search for a posteriori probability density was carried out by
the Monte Carlo method according to the Markov chain scheme. To construct a posteriori
probability density, the construction of a Markov chain involving several hundred thousand
links is required. For each link of the Markov chain, it is necessary to calculate the values of
the functionals. When constructing the Markov chain, the values of the sets of functionals
corresponding to different sets of parameters were determined by using the interpolation
algorithm between the reference points, based on the metric analysis scheme. To improve
the accuracy of the interpolation procedure, on one hand, and to maintain an acceptable
volume of calculations, on the other hand, a procedure was developed to supplement the set
of reference points with new calculation points. The main idea of such a procedure is that the
asymptotic density of points in the Markov chain corresponds to the density of the unknown a
priory probability density. The scheme allows us to refine the values of the initial parameters
of the calculation model on the basis of the information accumulated during the operation
of the reactor and, thereby, to calculate the transient processes with greater accuracy.
The publication was financially supported by the Ministry of Education and Science of the
Russian Federation (the Agreement number 02.A03.21.0008)
Key words and phrases: xenon transients, mathematical modeling, accumulated in-
formation, interpolation of functions of many variables, metric analysis, Bayesian method.
1. Introduction
Many researchers developed procedures for using experimental data on xenon tran-
sients to refine the values of the physical parameters of the calculation model. Such
procedures relate to the solution of inverse problems of mathematical physics, which
have been intensively developed in recent decades [1].
The simplest options for restoring the values of the parameters of the core model are
usually based on the method of least squares. However, their practical use for solving
the problems of restoring the parameters of the core model on the basis of experimental
data on xenon transient processes faces a number of problems. This is due, in particular,
to the nonlinearity of xenon processes. In addition, using the method of least squares
does not take into account the available information on the accuracy of the calculation
of individual parameters.
Copyright © 2017 for the individual papers by the papers’ authors. Copying permitted for private and
academic purposes. This volume is published and copyrighted by its editors.
In: K. E. Samouilov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VII Conference
“Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Moscow, Russia, 24-Apr-2017, published at http://ceur-ws.org
54 ITTMM—2017
2. Simulation scheme
To solve the problem of restoring the parameters of the model on the basis of exper-
imental data on xenon transient processes, we propose to use the Bayes method [2, 3]
in this paper, which makes it possible to construct a posteriori (taking into account
the available experimental information) the probability density of the values of the re-
quired parameters, taking into account the nonlinearity and possible degeneracy of the
tasks. For a nonsingular problem, we can assume that the combination of the unknown
parameters, which corresponds to the maximum probability density, is the desired set
of model parameters. For a singular problem there is a whole set of possible sets of
model parameters that are realized for equal or close probability densities. Using the
Bayes method, the available (a priori) information on the accuracy of calculation of
individual parameters is taken into account. Since the purpose of the work is to restore
the integral parameters of the active one, for practical implementation of the method
it is necessary to distinguish several functionals from the distribution of energy release,
the values of which essentially depend on the desired parameters.
To date, effective procedures for constructing the posterior probability density have
been developed. One of the promising is the DRAM algorithm [4, 5], which was used
in the developed procedure. However, practical implementation of the Bayes method
requires carrying out multiple (several hundred thousand) calculations of xenon tran-
sients. Therefore, a direct solution to the problem under consideration without prelim-
inary modifications is impossible because of the limited computing resources.
To reduce the number of direct calculations of transient processes, an interpolation
procedure based on metric analysis schemes was used [3], which allows to restore cal-
culated values of the monitored functionals from the energy release field at a small
computational cost, taking into account the nonlinearity of the process under consider-
ation with a limited amount of preliminary calculations.
In this paper, we present the results of solving test problems that demonstrate the
operability of the developed procedures.
To simulate the xenon transients, the NOSTRA software was used [6].
To reduce the number of direct calculations of transient processes, an interpolation
procedure based on metric analysis schemes was used [2, 3], which allows one to restore
calculated values of the controllable functionals from the energy release field with small
calculation costs, taking into account the nonlinearity of the process under consideration
with a limited amount of preliminary calculations.
Approbation of the developed algorithm to search for the most probable set of model
parameters was carried out using the example of solving a series of model problems.
We denote the set of recoverable deviations of the model parameters from their nominal
values by the vector θ,~ and the set of functionals by the vector f~.
At the first stage of solving the model problem, a xenon transition process was sim-
ulated, initiated by varying the power and moving the working group of the regulating
bodies. The values of the reactivity coefficient for the fuel temperature, the coefficient
of reactivity for the density of the coolant and the value of the xenon-135 absorption
cross section in this calculation deviated from their nominal values corresponding to
the constant file data and the magnitude of the deviations were specified by the vec-
tor f~exp . The functionals from the energy release field obtained during this calculation
were considered as their “experimental” values f~exp . Calculations at the first stage were
carried out with the help of the NOSTRA software [6].
At the second stage of the solution of the model problem, the corresponding devia-
tions in the parameter values of the model were reconstructed using the Bayes method
Kryanev Alexander V. et al. 55
on the basis of using the specified “experimental” values of the functionals from the en-
ergy release f~exp , the error in determining the “experimental” values of the functionals
and the a priori error of the model parameters.
The calculated values of the functionals from the energy release for an arbitrary
set of deviations of the reactivity coefficient from the fuel temperature, the reactivity
coefficient for the coolant density and the xenon-135 absorption cross section from their
nominal values were determined by direct calculations using the NOSTRA software
(∼ 100 calculations) and by interpolation methods based on schemes of metric analysis.
At the third stage of solving the problem, the restored and actual values of the model
parameters were compared.
3. Numerical Results
In order to evaluate the effectiveness of the developed algorithm, series of test prob-
lems were solved, in each of which, when the model parameters were replaced by θ~exp ,
a xenon transition process was modeled and a set of functionals f~exp was calculated.
Next, a set of deviations of the model parameters θ~res was determined, which corre-
sponded to the maximum probability density for the functionals obtain f~exp d.
The values of the functionals from the energy release were based on the axial offset
of the energy release. The axial offset of the energy release is defined as the difference
in energy release in the upper and lower parts of the core, referred to the energy release
of the entire core.
Four functionals were considered as functionals from the energy release: the damping
decrement of the deviation of the axial offset of the energy release from its stationary
value, the time between the achievements of the axial offset of the energy release of the
maximum values, the difference in the values of the axial offset of the energy release
at neighboring points of the maximum and minimum, and the value of axial offset at
one of the instants. As parameters of the model, deviations from the nominal values
of the reactivity coefficients for the fuel temperature, coolant temperature and relative
displacement of the absorption cross section were considered.
Through direct modeling of the xenon transient in the NOSTRA software and with
the help of metric analysis of the deviations of the model parameters by values θ~ from
their nominal values, the calculated values of the functionals were determined. The
indicated procedure in the operator form can be written in the form f~cal = Ã(θ),~ where
~ is a nonlinear operator. The problem of determining parameters θ~ is a classical
Ã(θ)
inverse problem and Bayes’ method is applicable to its solution.
The change in the axial offset of energy release in the case of xenon oscillations is
shown in Fig. 1 for the initial and restored values of the parameters.
Table 1 shows the results of the calculation of functionals and their comparison with
empirical values.
Fig. 2–4 show the distributions of conditional probability densities for two of the
parameters, provided that the third parameter belongs to the interval indicated in the
figure.
4. Conclusion
The refinement of the parameters of the computational model using simple math-
ematical schemes, for example, the method of least squares, faces problems of non-
linearity and degeneracy. In this paper, the Bayes method was used to solve the prob-
lem, using metric analysis and the DRAM algorithm. The obtained numerical results
56 ITTMM—2017
Figure 1. Dependence of axial offset of energy release on time
Table 1
The values of the energy release functionals for the “experimental” values f exp
and for the restored parameter values f~ res
Parameter f~res f~exp Relative error
Swing range 95.70 97.89 0.022
Decrement damping −22.42 −22.49 0.003
The period of oscillations 28.95 28.88 0.002
Axial offset at t = 5 hours −23.59 −23.84 0.010
show the possibility of restoring the parameters of the computational model ensuring
the matching of the calculated values of the functionals with their empirical values.
References
1. V. G. Romanov, Inverse problems of mathematical physics. Leningrad: Nauka, 1984
[in Russian].
2. A. V. Kryanev, G. V. Lukin, D. K. Udumyan, Metric Analysis and data processing.
Leningrad: Nauka, 2012 [in Russian].
3. A. V. Kryanev, D. K. Udumyan, Metric Analysis, Properties and Applications as
a Tool for Interpolation, International Journal of Mathematical Analysis (2014),
Vol. 8, no. 45, pp. 2221–2228.
4. Heikki Haario, Marko Laine, y Antonietta Mira, Eero Saksman. DRAM: Efficient
adaptive MCMC.
5. Jim Demmel, Communication avoiding algorithms, 2012 SC Companion: High Per-
formance Computing, Networking Storage and Analysis. IEEE, 2012.
Kryanev Alexander V. et al. 57
Figure 2. Conditional probability density of parameters (θ1 , θ2 )
Figure 3. Conditional probability density of parameters (θ1 , θ3 )
6. The NOSTRA software (version 5.0). Attestation passport of the software.
58 ITTMM—2017
Figure 4. Conditional probability density of parameters (θ1 , θ3 )