# Machine Learning of the Reactor Core Loading  Pattern Critical Parameters

Authors: Krešimir Trontl; Dubravko Pevec; Tomislav Šmuc
Journal: Science and Technology of Nuclear Installations
 (2008)

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## Abstract

The usual approach to loading pattern optimization involves high degree of engineering judgment, a set of heuristic rules, an optimization algorithm, and a computer code used for evaluating proposed loading patterns. The speed of the optimization process is highly dependent on the computer code used for the evaluation. In this paper, we investigate the applicability of a machine learning model which could be used for fast loading pattern evaluation. We employ a recently introduced machine learning technique, support vector regression (SVR), which is a data driven, kernel based, nonlinear modeling paradigm, in which model parameters are automatically determined by solving a quadratic optimization problem. The main objective of the work reported in this paper was to evaluate the possibility of applying SVR method for reactor core loading pattern modeling. We illustrate the performance of the solution and discuss its applicability, that is, complexity, speed, and accuracy.


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## Body

## 1. Introduction

Decrease of the fuel cycle costs is an important factor in nuclear power plant management. The
economics of the fuel cycle can strongly benefit from the optimization of the reactor
core loading pattern, that is, minimization of the amount of enriched uranium
and burnable absorbers placed in the core, while maintaining nuclear power
plant operational and safety characteristics.The usual approach to loading pattern optimization involves high degree of
engineering judgment, a set of heuristic rules, an optimization algorithm, and a reactor physics computer code used for
evaluating proposed loading patterns. Since the loading pattern optimization
problem is of combinatorial nature and involves heuristics requiring large
numbers of core modeling calculations (e.g., genetic algorithms or simulated
annealing algorithms), the time needed for one full optimization run is essentially
determined by the complexity of the code that evaluates the core loading
pattern.The aim of the work reported in this paper was to investigate the applicability of
a machine learning modeling for fast loading pattern evaluation. We employed a
recently introduced machine learning technique, support vector regression (SVR),
which has a strong theoretical background in statistical learning theory. SVR is
a supervised learning method in which model parameters are automatically
determined by solving a quadratic optimization problem.This paper reports on the possibility of applying SVR method for reactor core loading pattern modeling.
Required size of the learning data set, as a function of targeted accuracy,
influence of SVR free parameters, as well as input vector definition were studied.In Section2, the support vector regression method is discussed. Basics of fuel loading pattern
development and optimization as well as the methodology applied for the
investigation of applicability of the SVR method for fuel loading pattern
evaluation are presented in Section 3. Results and discussion are given in
Section 4, while in Section 5 the conclusions based on this work are drawn.


## 2. Support Vector Regression

Machine learning is, by its definition, a study of computer algorithms that improve
automatically through experience. One of machine learning techniques is the support vector machines (SVMs)
method, which has a strong theoretical background in statistical learning
theory [1].
The method proved to be a very robust technique for complex classification and
regression problems. Although, historically speaking, the first implementation
of SVM was for classification problems [2, 3],
in the last decade, the application of SVM for nonlinear regression modeling is noticeable in different fields of science and technology [4–10], the main reason being robustness and good generalization properties of the method.In the upcoming
paragraphs, we will give a short introduction into the support vector
regression method, stressing only the most important theoretical and practical
aspects of the technique. Additional information can be found in referenced
literature.In general, the starting
point of the machine learning problem is a collection of samples, that is, points,
to learn the model (training set) and a separate set to test the learned model
(test set). Since we are interested in developing a regression model, we will consider
a training data set, as well as testing data set, comprised of a number of
input/output pairs, representing the experimental relationship between input
variables (x→i) and corresponding scalar output value (yi):(1){(x→1,y1),(x→2,y2),…,(x→n,yn)}⊂ℜd×ℜ.In our case, the input vector defines the characteristics of the loading pattern, while the
output value, also referred to as a target value, denotes the parameter of
interest.The modeling
objective is to find a functiony=f(x→) such that it accurately predicts (with ε tolerance) the output value (y) corresponding to a new input vector (x→), yet unseen by the model (the model has not
been trained on that particular input vector) [11].Due to the high
complexity of underlying physical process that we are modeling, the required function
can be expected to have high nonlinear properties. In the support vector
regression approach, the input data vectorx→ is mapped into a higher dimensional feature
space F using a nonlinear mapping function
Φ, and a linear regression is performed in that
space. Therefore, a problem of nonlinear regression in low-dimensional input
space is solved by linear regression in high-dimensional feature space.The SVR technique considers the following linear estimation function:(2)f(x→)=〈w→,Φ(x→)〉+b, where w→ denotes the weight vector, b is a constant known as bias, Φ(x→) is the mapping function, and 〈w→,Φ(x→)〉 is the dot product in feature space, such that Φ:x→→F,w→∈F [12]. The unknown parameters w and b are estimated
using the data points in the training set. To avoid overfitting and maximize
generalization capability of the model, a regularized form of the functional,
following principles of structural risk minimization (SRM), is minimized:(3)Rreg[f]=∑i=1MC(f(x→i)−yi)+λ∥w→∥2, where Rreg[f] denotes regression risk (possible test set
error), based on empirical risk which is expressed through the cost function C
determined on the points of the training set, and a term reflecting the
complexity of the regression model. Minimization task thus involves
simultaneous minimization of the empirical risk and minimization of structural
complexity of the model. Most commonly used cost function (loss functional)
related to empirical risk is the so called “ε insensitive loss function”:(4)C(f(x→i)−yi)={∥f(x→)−y∥−ε,for∥f(x→)−y∥≥ε,0,otherwise, where ε is a
parameter representing radius of the tube around regression function. The SVR
algorithm attempts to position the tube around the data, as depicted in Figure 1 [7], and according to (4) does not penalize data points for
which calculated values (y) lie
inside this tube. The deviations of points that lie more than ε away
from the regression function are penalized in the optimization through their
positive and negative deviations ξ and ξ*, called “slack” variables.Figure 1
The schematic illustration of the SVR usingε-insensitive cost function (tube).It was shown that
the following function minimizes the regularized functional given by (3) [1]:(5)f(x→,w→)=f(x→,α→,α→*)=∑i=1n(αi*−αi)K(x→i,x→)+b, where
αi*αi are Lagrange multipliers
describing w→,
and are estimated, as well as parameter b,
using an appropriate quadratic programming algorithm, and K(x→i,x→) is a so called kernel function describing the dot product 〈w→,Φ(x→)〉 in the feature space. A number of kernel
functions exist [13]. Kernel functions used in this work are described in
more details in the following section.Due to the character
of the quadratic optimization, only some of the coefficientsαi*−αi are nonzero, and the corresponding
input vectors x→ are called support  vectors (SVs).
Input vectors matching zero αi*−αi coefficients are positioned inside
the ε tolerance tube and are therefore, not
interesting for the process of model generation. Support vectors that are
determined in the training (optimization) phase are the “most
informative” points, that compress the information content of the training
set. In most of the SVR formulations, there are two free parameters to be set
by the user: C-cost of the penalty
for data-model deviation, and ε-insensitive zone. These two free parameters and the chosen form of the kernel
function and its corresponding parameters control the accuracy and
generalization performance of the regression model.


## 3. Methodology

One of the key processes of both, safe and economical operations
of nuclear reactor, is in-core fuel management, or to be more precise, fuel
loading pattern determination and optimization. Every method and technique used
for fuel loading pattern determination and optimization tasks, whether based on
engineering judgement, heuristic rules, genetic algorithms, or a combination of
stated approaches, requires a large number of potential fuel loading patterns
evaluation. The evaluation is normally performed using a more or less
sophisticated reactor physics code. Usage of such codes is time consuming.
Therefore, in this work, we are investigating the possibility of SVR method
being used as a fast tool for loading pattern evaluation.However, taking
into account that the SVR method is to be used, a number of factors have to be
addressed prior to creating a model. The first is the setting of the loading
pattern that is to be investigated, including the method by which the
experimental data points are to be generated, the definition of the input space
and parameters used as target values. The second is the choice of the kernel
function and appropriate free parameters used in the SVR model. Finally, SVR modeling
tools have to be addressed.


### 3.1. Computational Experiment Setup

Taking into account the preliminary and inquiring characteristics of the study, we decided
to use limited fuel assembly inventory for a single loading pattern optimization
as a basis for the development of our regression models. NPP Krško Cycle 22
loading pattern has been used as a reference one. 121 fuel assemblies, grouped
in 7 batches that were used for core loading in Cycle 22 have been used for
generating a large number of randomly generated fuel loading patterns, which
were then divided into training and testing data sets and employed in SVR model
development process. The global core calculations of each of the trial loading
patterns have been conducted using MCRAC code of the FUMACS code package, which
also includes the LEOPARD code for two-group cross-section preparation [14]. The calculation is based on quarter core symmetry,
fixed cycle length, and fixed soluble boron concentration curve.The generation
phase, that is, the definition of the loading patterns, has been based on a semirandom
algorithm. In order to narrow the investigated input space as much as possible,
as well as to stay within the limits of the numbers of available fuel
assemblies per batch, we introduced a limitation for every fuel assembly
regarding the position where it can be placed: fuel assemblies originally placed
on axes positions could be randomly placed only on axes positions, and vice
versa. The central location fuel assembly was fixed for every loading pattern.The most important
issue in the regression model development is the definition of the input space
to be used for SVR model development. Since in a quarter core symmetry setup,
the NPP Krško core is
defined by 37 fuel assemblies, and having in mind the inquiring nature of the
work, we decided to simplify the problem by the assumption of the 1/8 core
symmetry, resulting in 21 fuel assemblies defining the core. Fuel assembly
(position) is defined by initial enrichment, number of IFBAs, and reactor
history, or at least burnup accumulated in previous cycles. Therefore, the
number of potential parameters defining the input space is 63. The high
dimensionality of the input space generally increases the number of training
points and time required for the development of the SVR of certain
generalization properties. Therefore, we decided to reduce the number of
parameters by introducingk-inf at the beginning of the cycle as a new
parameter and representing fuel assembly only by k-inf and number of IFBAs (0
for old fuel, and 32, 64, 92, and 116 for fresh fuel). Thus, the final number
of parameters defining the input space was 42.The SVR model would eventually be used in an optimization algorithm as a fast tool for
loading pattern evaluation. Therefore, the target parameters which we want to
model should be the most important parameters on which such an evaluation is
based. In this work, we used the global core effective multiplication factors
at the beginning and at the end of the cycle (keffBOC and keffEOC),
as well as power peaking factor (FΔHN) as target parameters for which separate SVR models
were built.


### 3.2. Kernel Functions

The idea of the kernel function is to enable mathematical operations to be taken in the input
space, rather than in the high-dimensional feature space [15]. The
theory is based upon reproducing kernel Hilbert spaces
(RKHSs) [16].A number of kernel
functions have been proposed in the literature. The particular choice of the
kernel that is going to be used for mapping nonlinear input data into a linear
feature space is highly dependent on the nature of the data representing the
problem. It is up to the modeller to select the appropriate kernel function. In
this paper, the focus is placed on two widely used kernel functions, namely, radial basis function (RBF),
also called Gaussian and the polynomial function (PF), which are defined by
(6)(6)KRBF(x→i,x→j)=exp(−∥x→i−x→j∥22σ2),KPF(x→i,x→j)=(x→iTx→j+1)d.In the case of RBF
kernel, parameterσ represents the radius of the Gaussian kernel,
while d in the case of PF kernel
represents the degree of the polynomial kernel.As already
mentioned, the behaviour of the SVR technique strongly depends on the selection
of the kernel function, its corresponding parameters, and general SVR “free”
parameters (C and ε). All the parameters used in this study were determined
by a combination of engineering judgement and optimization procedure based on the
application of genetic algorithms [17].


### 3.3. SVR Modeling Tools

Excellent results
in SVR application to a wide range of classification and regression problems in
different fields of science and technology, initiated creation of a number of
implementations of the support vector machines algorithm, some of which are
freely available software packages. In this work, we decided to test three
often used packages: SVMTorch [18],
LIBSVM [19], and WEKA [20].As stated in the previous subsection, RBF and PF kernel functions have been used. The general
form of the kernels is given in (6). However, practical parameterisation
of the functions, that is, their representation, is somewhat different from
code to code. For example, parameter g in LIBSVM notation for RBF represents 1/(2σ)2. Whenever, a direct comparison of
codes has been performed, general kernel parameters have been set (see (6)), and code specific parameters were modified to reflect on these values.


## 3.1. Computational Experiment Setup

Taking into account the preliminary and inquiring characteristics of the study, we decided
to use limited fuel assembly inventory for a single loading pattern optimization
as a basis for the development of our regression models. NPP Krško Cycle 22
loading pattern has been used as a reference one. 121 fuel assemblies, grouped
in 7 batches that were used for core loading in Cycle 22 have been used for
generating a large number of randomly generated fuel loading patterns, which
were then divided into training and testing data sets and employed in SVR model
development process. The global core calculations of each of the trial loading
patterns have been conducted using MCRAC code of the FUMACS code package, which
also includes the LEOPARD code for two-group cross-section preparation [14]. The calculation is based on quarter core symmetry,
fixed cycle length, and fixed soluble boron concentration curve.The generation
phase, that is, the definition of the loading patterns, has been based on a semirandom
algorithm. In order to narrow the investigated input space as much as possible,
as well as to stay within the limits of the numbers of available fuel
assemblies per batch, we introduced a limitation for every fuel assembly
regarding the position where it can be placed: fuel assemblies originally placed
on axes positions could be randomly placed only on axes positions, and vice
versa. The central location fuel assembly was fixed for every loading pattern.The most important
issue in the regression model development is the definition of the input space
to be used for SVR model development. Since in a quarter core symmetry setup,
the NPP Krško core is
defined by 37 fuel assemblies, and having in mind the inquiring nature of the
work, we decided to simplify the problem by the assumption of the 1/8 core
symmetry, resulting in 21 fuel assemblies defining the core. Fuel assembly
(position) is defined by initial enrichment, number of IFBAs, and reactor
history, or at least burnup accumulated in previous cycles. Therefore, the
number of potential parameters defining the input space is 63. The high
dimensionality of the input space generally increases the number of training
points and time required for the development of the SVR of certain
generalization properties. Therefore, we decided to reduce the number of
parameters by introducingk-inf at the beginning of the cycle as a new
parameter and representing fuel assembly only by k-inf and number of IFBAs (0
for old fuel, and 32, 64, 92, and 116 for fresh fuel). Thus, the final number
of parameters defining the input space was 42.The SVR model would eventually be used in an optimization algorithm as a fast tool for
loading pattern evaluation. Therefore, the target parameters which we want to
model should be the most important parameters on which such an evaluation is
based. In this work, we used the global core effective multiplication factors
at the beginning and at the end of the cycle (keffBOC and keffEOC),
as well as power peaking factor (FΔHN) as target parameters for which separate SVR models
were built.


## 3.2. Kernel Functions

The idea of the kernel function is to enable mathematical operations to be taken in the input
space, rather than in the high-dimensional feature space [15]. The
theory is based upon reproducing kernel Hilbert spaces
(RKHSs) [16].A number of kernel
functions have been proposed in the literature. The particular choice of the
kernel that is going to be used for mapping nonlinear input data into a linear
feature space is highly dependent on the nature of the data representing the
problem. It is up to the modeller to select the appropriate kernel function. In
this paper, the focus is placed on two widely used kernel functions, namely, radial basis function (RBF),
also called Gaussian and the polynomial function (PF), which are defined by
(6)(6)KRBF(x→i,x→j)=exp(−∥x→i−x→j∥22σ2),KPF(x→i,x→j)=(x→iTx→j+1)d.In the case of RBF
kernel, parameterσ represents the radius of the Gaussian kernel,
while d in the case of PF kernel
represents the degree of the polynomial kernel.As already
mentioned, the behaviour of the SVR technique strongly depends on the selection
of the kernel function, its corresponding parameters, and general SVR “free”
parameters (C and ε). All the parameters used in this study were determined
by a combination of engineering judgement and optimization procedure based on the
application of genetic algorithms [17].


## 3.3. SVR Modeling Tools

Excellent results
in SVR application to a wide range of classification and regression problems in
different fields of science and technology, initiated creation of a number of
implementations of the support vector machines algorithm, some of which are
freely available software packages. In this work, we decided to test three
often used packages: SVMTorch [18],
LIBSVM [19], and WEKA [20].As stated in the previous subsection, RBF and PF kernel functions have been used. The general
form of the kernels is given in (6). However, practical parameterisation
of the functions, that is, their representation, is somewhat different from
code to code. For example, parameter g in LIBSVM notation for RBF represents 1/(2σ)2. Whenever, a direct comparison of
codes has been performed, general kernel parameters have been set (see (6)), and code specific parameters were modified to reflect on these values.


## 4. Results and Discussion

### 4.1. Comparison of Code Packages

The comparison of
three code packages for SVR modeling, namely, SVMTorch, LIBSVM, and WEKA, has
been conducted using a maximum training set size of 15 000 data points while the
test set consisted of 5000 data points. The number of data points for learning
models is typically enlarged until satisfactory results regarding the accuracy
are achieved. In this subsection, only the results of final models comparison
are presented.Preliminary
analyses revealed that preprocessing of the input data is required in order to
allow normal and reasonably fast operation of all SVR code packages. Mainly,
due to the fact that input variables span extremely different ranges, scaling
of the input data has been performed, including the scaling of the target
values (all in the range 0 to 1), using one of LIBSVM codes: SVMSCALE.Models for three
target values (keffBOC, keffEOC and FΔHN) were compared for the model accuracy,
learning and implementation times
(Pentium 4 Mobile CPU 1.7 GHz, 256 MB RAM, Windows XP SP2), and the relative number
of support vectors as the measure of model generalization characteristics. The
implementation time has been measured on 5000 data points. The accuracy of the
model was determined using root mean square error (RMSE) and relative average
deviation (RAD) defined as(7)RMSE=∑i=1n(yi−fi)2n,RAD=∑i=1n(|yi−fi|/yi)×100%n, where fi stands
for predicted value corresponding to the target value yi. The metric of interest was also the percentage of
tested data points which had the predicted value deviate from the target value
by more than 20%:(8)|yi−fi|yi×100%>20%.In the case of RBF
kernel function, the initial values of free parameters were estimated using a
genetic algorithm (GA) on the LIBSVM code. The ranges for every parameter (C, ε, and σ) were set, based on engineering judgement, from
1 to 1000 for C and 0.001 to 2.0, and
1 to 7.07 (50) for ε and σ, respectively. The GA was characterized by 20
populations each consisting of 100 members. The training set consisted of 4500
data points, while the test set had 500 data points. The best result was
obtained for C=371.725, ε=0.05154,
and σ=6.4697.In the case of the
PF kernel function, we decided to set thed parameter to the commonly used value of 3, while for simplicity reasons C and ε were set
to 371.725 and 0.05154, respectively. Comparison results for RBF kernel
function are given in Table 1 while in Table 2 comparison results for PF kernel
function are presented.Table 1
Comparison of results for RBF kernel function.
Target valueCode packageAccuracyLearning/Implem.SV [%]RMSERAD [%]>20%[%]time [s]keffBOCSVMTorch0.0296.7933.44120/145.27LIBSVM0.0297.1793.9618/33.59WEKA0.0286.6213.202250/63.77keffEOCSVMTorch0.0505.0481.9610800/3016.69LIBSVM0.0454.5501.761260/1518.37WEKA0.0454.5701.9828160/3018.22FΔHNSVMTorch0.04015.06020.4213080/1316.91LIBSVM0.03914.81019.641080/1417.97WEKA0.03914.80119.5833362/1417.86Table 2
Comparison results for PF kernel function.
Target valueCode packageAccuracyLearning/Implem.SV [%]RMSERAD [%]>20% [%]time [s]keffBOCSVMTorch0.0306.4183.7650/116.88LIBSVM0.0307.6105.629/34.83WEKA*0.0306.2593.464027/102.43keffEOCSVMTorch0.0727.1474.92840/2019.33LIBSVM0.0585.8563.122113/1130.21WEKA*0.0566.0953.3431120/4530.02FΔHNSVMTorch0.04416.05722.92420/1818.50LIBSVM0.03914.99220.50325/818.43WEKA*0.04215.70122.407000/3017.17*PF kernel in the form KPF(x→i,x→j)=(x→iTx→j)d.The results of preliminary
tests suggest that appropriate regression models using SVM method can be
developed for all target values regardless of the applied code package. The
only difference is the learning time required for the model to be developed.
The implementation or deployment time for the execution of the model (maximum
of 30 seconds for 5000 calculations) is not the issue. The accuracy for thekeffBOC and keffEOC target values is
satisfactory, while additional effort has to be placed on developing the FΔHN model by adjusting SVR parameters and
increasing the training set size.


### 4.2. Training Set Size Influence on SVR Model Quality

SVR model quality can be interpreted as the time required for the model learning, accuracy of the
model, and generalization characteristics of the model. As shown in the previous
subsection, model implementation/deployment time is not the key issue.As discussed
previously, the size of the training set influences all factors of the model
quality, and generally thorough analysis of that influence is necessary. Here,
we present the results of preliminary tests conducted forkeffBOC model development using LIBSVM code package (see
Figure 2). The characteristics of applying other code packages on all target
values are qualitatively very similar.Figure 2
Training set size influence on model quality forkeffBOC-preliminary tests.Apart from the
anomaly observed for the RMSE curve at the training set size of 5000 data
points originating in statistical and random characteristic of the training and
testing data sets, the accuracy (RMSE) and the generalization properties (low
SV percentage) of the models increase with the increase of the training set
size. The learning time is also increased exhibiting a nearly linear trend.


## 4.1. Comparison of Code Packages

The comparison of
three code packages for SVR modeling, namely, SVMTorch, LIBSVM, and WEKA, has
been conducted using a maximum training set size of 15 000 data points while the
test set consisted of 5000 data points. The number of data points for learning
models is typically enlarged until satisfactory results regarding the accuracy
are achieved. In this subsection, only the results of final models comparison
are presented.Preliminary
analyses revealed that preprocessing of the input data is required in order to
allow normal and reasonably fast operation of all SVR code packages. Mainly,
due to the fact that input variables span extremely different ranges, scaling
of the input data has been performed, including the scaling of the target
values (all in the range 0 to 1), using one of LIBSVM codes: SVMSCALE.Models for three
target values (keffBOC, keffEOC and FΔHN) were compared for the model accuracy,
learning and implementation times
(Pentium 4 Mobile CPU 1.7 GHz, 256 MB RAM, Windows XP SP2), and the relative number
of support vectors as the measure of model generalization characteristics. The
implementation time has been measured on 5000 data points. The accuracy of the
model was determined using root mean square error (RMSE) and relative average
deviation (RAD) defined as(7)RMSE=∑i=1n(yi−fi)2n,RAD=∑i=1n(|yi−fi|/yi)×100%n, where fi stands
for predicted value corresponding to the target value yi. The metric of interest was also the percentage of
tested data points which had the predicted value deviate from the target value
by more than 20%:(8)|yi−fi|yi×100%>20%.In the case of RBF
kernel function, the initial values of free parameters were estimated using a
genetic algorithm (GA) on the LIBSVM code. The ranges for every parameter (C, ε, and σ) were set, based on engineering judgement, from
1 to 1000 for C and 0.001 to 2.0, and
1 to 7.07 (50) for ε and σ, respectively. The GA was characterized by 20
populations each consisting of 100 members. The training set consisted of 4500
data points, while the test set had 500 data points. The best result was
obtained for C=371.725, ε=0.05154,
and σ=6.4697.In the case of the
PF kernel function, we decided to set thed parameter to the commonly used value of 3, while for simplicity reasons C and ε were set
to 371.725 and 0.05154, respectively. Comparison results for RBF kernel
function are given in Table 1 while in Table 2 comparison results for PF kernel
function are presented.Table 1
Comparison of results for RBF kernel function.
Target valueCode packageAccuracyLearning/Implem.SV [%]RMSERAD [%]>20%[%]time [s]keffBOCSVMTorch0.0296.7933.44120/145.27LIBSVM0.0297.1793.9618/33.59WEKA0.0286.6213.202250/63.77keffEOCSVMTorch0.0505.0481.9610800/3016.69LIBSVM0.0454.5501.761260/1518.37WEKA0.0454.5701.9828160/3018.22FΔHNSVMTorch0.04015.06020.4213080/1316.91LIBSVM0.03914.81019.641080/1417.97WEKA0.03914.80119.5833362/1417.86Table 2
Comparison results for PF kernel function.
Target valueCode packageAccuracyLearning/Implem.SV [%]RMSERAD [%]>20% [%]time [s]keffBOCSVMTorch0.0306.4183.7650/116.88LIBSVM0.0307.6105.629/34.83WEKA*0.0306.2593.464027/102.43keffEOCSVMTorch0.0727.1474.92840/2019.33LIBSVM0.0585.8563.122113/1130.21WEKA*0.0566.0953.3431120/4530.02FΔHNSVMTorch0.04416.05722.92420/1818.50LIBSVM0.03914.99220.50325/818.43WEKA*0.04215.70122.407000/3017.17*PF kernel in the form KPF(x→i,x→j)=(x→iTx→j)d.The results of preliminary
tests suggest that appropriate regression models using SVM method can be
developed for all target values regardless of the applied code package. The
only difference is the learning time required for the model to be developed.
The implementation or deployment time for the execution of the model (maximum
of 30 seconds for 5000 calculations) is not the issue. The accuracy for thekeffBOC and keffEOC target values is
satisfactory, while additional effort has to be placed on developing the FΔHN model by adjusting SVR parameters and
increasing the training set size.


## 4.2. Training Set Size Influence on SVR Model Quality

SVR model quality can be interpreted as the time required for the model learning, accuracy of the
model, and generalization characteristics of the model. As shown in the previous
subsection, model implementation/deployment time is not the key issue.As discussed
previously, the size of the training set influences all factors of the model
quality, and generally thorough analysis of that influence is necessary. Here,
we present the results of preliminary tests conducted forkeffBOC model development using LIBSVM code package (see
Figure 2). The characteristics of applying other code packages on all target
values are qualitatively very similar.Figure 2
Training set size influence on model quality forkeffBOC-preliminary tests.Apart from the
anomaly observed for the RMSE curve at the training set size of 5000 data
points originating in statistical and random characteristic of the training and
testing data sets, the accuracy (RMSE) and the generalization properties (low
SV percentage) of the models increase with the increase of the training set
size. The learning time is also increased exhibiting a nearly linear trend.


## 5. Conclusions

This work introduces a novel concept for fast evaluation of reactor core loading pattern,
based on general robust regression model relying on the state of the art research
in the field of machine learning.Preliminary tests
were conducted on the NPP Krško reactor core,
using the MCRAC code for the calculation of reference data. Three support
vector regression code packages were employed (SVMTorch, LIBSVM, and WEKA) for
creating regression models of effective multiplication factor at the beginning
of the cycle (keffBOC), effective multiplication factor at the end
of the cycle (keffEOC), and power peaking factor (FΔHN).The preliminary
tests revealed a great potential of the SVR method application for fast and
accurate reactor core loading pattern evaluation. However, prior to the final
conclusion and incorporation of SVR models in optimization codes, additional
tests and analyses are required, mainly focused on the parameters defining
input vector, thus influencing its size, the required size of the training set
and parameters defining kernel functions.In the case of the
scenario involving machine learning from the results of more accurate and time
consuming 3D code, we do not anticipate any major changes in the learning stage
of SVR model development, as well as it its implementation. However, generation
of training and testing data sets would be more demanding (time consuming and
requiring more hardware resources).These are the issues that are within the scope of our future research.



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*Source: 695153-2008-08-27.xml*