In nuclear safety analysis, it is very important to be able to simulate the different transients that can occur in a nuclear power plant with a very high accuracy. Although the best estimate codes can simulate the transients and provide realistic system responses, the use of nonexact models, together with assumptions and estimations, is a source of uncertainties which must be properly evaluated. This paper describes a Rod Ejection Accident (REA) simulated using the coupled code RELAP5/PARCSv2.7 with a perturbation on the crosssectional sets in order to determine the uncertainties in the macroscopic neutronic information. The procedure to perform the uncertainty and sensitivity (U&S) analysis is a samplingbased method which is easy to implement and allows different procedures for the sensitivity analyses despite its high computational time. DAKOTAJaguar software package is the selected toolkit for the U&S analysis presented in this paper. The size of the sampling is determined by applying the Wilks’ formula for double tolerance limits with a 95% of uncertainty and with 95% of statistical confidence for the output variables. Each sample has a corresponding set of perturbations that will modify the crosssectional sets used by PARCS. Finally, the intervals of tolerance of the output variables will be obtained by the use of nonparametric statistical methods.
Being able to simulate accurately the different transients that can occur in a nuclear power plant is one of the main aims in nuclear safety analysis. Transient simulations involve both neutronic and thermalhydraulic calculations which are solved by different computer codes using best estimate models. Even though this approach provides more realistic responses than conservative codes, because of the reduction of the conservatism, the uncertainty in the code predictions of relevant safety system variable resulting from assumptions and code model estimates must be carefully evaluated. Uncertainties may stem from various sources, for example, lack of knowledge, approximate character of physical models, uncertainty in model data, and so forth, and will be statistically propagated to code output parameters which have a probability density function and a range limit.
In order to obtain the uncertainties in the macroscopic neutronic information using a best estimate neutronicsthermalhydraulic coupled code, a Rod Ejection Accident (REA) has been simulatedm and the results are presented in this paper. The REA accident belongs to the reactivityinduced accidents (RIA) category and is part of the licensing basis accident analyses required for pressurized water reactors (PWR). The REA consists of a rod ejection due to the failure of its operating mechanism with the power evolution driven by a continuous reactivity insertion. The main factor limiting the consequences of the accident is the Doppler reactivity effect. The physical description of the reactor response is based on the coupled neutronicthermalhydraulic code RELAP5/PARCSv2.7. Since a coupled code is used for the best estimate analysis, uncertainties from both aspects should be included and jointly propagated. CASMO4SIMULATE3 provides the crosssection sets which are processed using the SIMTAB methodology developed at the Polytechnic University of Valencia (UPV) together with Iberdrola [
With the objective of providing a realistic environment for the development of the dynamic behaviour, monitoring and adjustment procedures, dynamic models for monitoring and control the output system have been developed.
It is possible to quantify the uncertainty by means of statistical measures. For instance, even when the exact values for a given code input are unknown, a range of possible values that it can take may be available. It is then possible to quantify uncertainty by considering the range, and a probability density function (PDF) that assigns a probability to the values inside the range. Sometimes, however, a third factor has to be considered [
In this work, both the tolerance limits and the PDF distributions were an assumption. The real values of the tolerance limits for each kinetic parameter and its PDF distribution type will be known only after a validated uncertainty propagation methodology is applied to the whole process. This process, consisting in neutron kinetic parameters generation for coupled 3D neutronicthermalhydraulic, could be divided into three steps: (1) cell physics (derivation of the multigroup microscopic crosssection libraries and associated uncertainties), (2) lattice physics (derivation of the fewgroup macroscopic crosssection libraries and associated uncertainties), and (3) core physics (core steadystate and transient behavior with the associated uncertainties). Hence, this work is a general approach to the third step (core physics) on the assumption that the uncertainties corresponding to the first two steps are previously obtained.
Moreover, it is known that the fractions of delayed neutrons precursors (and their decay constants) play an important role in output variable variance. This influence depends on how far from prompt critical the insertion of reactivity is. Thus for small insertions of reactivity, delayed neutrons play an important role in the time evolution of the neutron flux, and so the uncertainty in
In any case, the reason why the uncertainties in
The computer code plays the role as the used model and its associated uncertainty is calculated using the presented methodology. Figure
Model with input and output variables, extracted from [
The computer code is a deterministic function that transforms stochastic uncertain in its inputs and models,
The coupled code is represented by the function
To get a proper understanding on what is actually happening in the model, it is also important to perform a sensitivity analysis. This type of analysis reflects the variance of one response random variable when an input related random variable is perturbed. In this work, the sensitivity analysis performed is based on statistical measures of correlation between the input variables selected as sources of uncertainty and those output variables of interest. Two kinds of correlations can be used, those based on regression analysis, for example, Pearson Product Moment, and those based on nonparametric Rank correlation, for example, Spearman regression coefficients. Pearson correlation is most suitable for linear dependencies, whereas Spearman correlation can better quantify nonlinear and linear dependencies. For this work, Spearman correlation has been selected to carry out the sensitivity analysis. Additionally, full and partial correlations have been computed and analysed. Full correlations include the effects of all the input variables simultaneously, and partial correlations can eliminate the effects of the other variables for a given one. A threshold value of
It is important to point out that statistical sensitivity measures only quantify statistical relationships and do not offer quantitative values about the magnitude of the relationship which could be used to further compute linear uncertainties in the output variables of interest. For this, analytical or numerically obtained firstorder derivatives must be computed,
This paper discusses a work based on previous studies [
The first step in the methodology is the characterization of the input variables uncertainty. As a starting point, the user must decide which input variables could be more relevant or sensitive for the output variables. As stated earlier, two factors are used in order to define the uncertainties related to the input variables: the intervals of possible values and the PDF associated. The uncertainty analysis with nonparametric methods can lead to range limits for
Regarding the selection of the distribution, if there is no information or knowledge in order to provide a PDF for a certain
Computer codes, in general, do not accept intervals as
The second step involves performing the simulations a certain
The Wilks’ approach, also known as GRS’s (Gesellschaft für Anlagen und Reaktorsicherheit) method in nuclear safety field, is based on a pure statistical method [
Each of the
The uncertainty of
In order to deal with models which are not clearly linear, simple (SRCC) or partial rank correlation (PRC) coefficients can be used. To calculate these two correlations, the sample values of
If the two “unranked” original series of values are related monotonously, then the ordered series are linearly related. This is true even if the relationship between the unordered series is not linear. Thus the absolute values of SRCC and PRCC will quantify the degree of relationship between the given
For more than twenty years, the Latin Hypercube Lattice Sampling program has been successfully used to generate multivariate samples of statistical distributions. Its ability to use either Latin hypercube sampling or pure Monte Carlo sampling with both random and restricted pairing methods has made it an important part of uncertainty analyses in areas ranging from Probabilistic Risk Assessment (PRA) to complex simulation modelling.
Latin Hypercube Lattice Sampling (LHS) is a stratified random procedure that provides an efficient way of sampling variables from their distributions [
The LHS process consists of three steps: (1) the range of each input variable,
It is well known that some design parameters have a high influence on the response whereas the influence of other parameters can be neglected. In order to optimize the design process, sensitivity analysis techniques and parameter study methods are performed. Both techniques and methods are used for identifying which parameters could be needed and which could be taken as constant parameters. In addition, in a postoptimization role, sensitivity information is useful in determining whether or not the response functions are robust with respect to small changes in the optimum design point [
DAKOTA software, which stands for Design Analysis Kit for Optimization and Terascale Applications, is the selected toolkit for the present sensitivity analysis study. DAKOTA provides a flexible interface between analysis codes and iterative systems analysis methods [
JAGUAR, which stands for JAva GUi for Applied Research, is a graphical wizard program which parses a DAKOTA input specification and serves as a graphical user interface (GUI) providing both graphical and text representations of the problem setup for DAKOTA studies afterwards [
For this specific uncertainty and sensitivity analysis, the studied input variables are the following seven neutronic parameters: diffusion coefficient which determines the leakage for thermal and fast group (
Input file information summary.
Sample size  146 
Input model variables  7 
Lower bound  −0.003 
Upper bound  +0.003 
Sampling method  Random/LHS 
Variables distribution  Uniform/normal 
After settingup the input model file, the perturbation matrix file is generated. Both processes are part of the socalled
Once the 146 simulations, with the corresponding perturbations, are performed, nonparametric statistical methods are applied for studying the influence of the uncertainty on the macroscopic neutronic information (
DAKOTAJaguar followed methodology.
The modification of the matrix is one of the main parts of the sensitivity analysis and it can be performed following two different approaches. The first one is based on maximum values so the extra column contains the maximum value for each output variable (power, enthalpy, or reactivity) for a given case. The second approach is based on the time step values and it has been repeated three times, one for each output variable. The extra column contains the desired output variable value for each time step for a given case. Furthermore, for this second approach, some interpolations may be required since not all the simulations have the same time steps.
This paper describes the two approaches and the sensitivity analysis performed based on them. Henceforth, the first approach will be named as scalar sensitivity analysis and the second approach as indexdependent sensitivity analysis.
The reactor core contains 157 fuel elements. Each fuel element has 264 fuel rods, 24 guide tubes, and 1 tube for the instrumentation. Therefore, the core has been modelled with 157 thermalhydraulic flow channels, with a onetoone correspondence with the fuel elements. The initial Hot zero power (HZP) steadystate conditions are: temperature equals to 565.58 K, initial density of 740.74 kg/m^{3}, and a total inlet mass flow rate through the core of 13301 kg/s which is distributed among all the channels depending on the crosssectional area. The transient is started by the ejection of a control rod with the maximum reactivity worth.
RELAP5 is the selected system code for modelling the 157 thermalhydraulic channels, which are connected with branches (
SNAP representation of the RELAP5 model.
Radially, the core is divided into 21.504 cm
The neutronic nodal discretization consists of 157
The control rods are grouped in 6 banks which initially all are fully inserted. In a REA simulation, one of the main parameters that needs to be determined is the control rod with the maximum reactivity worth. Table
Values of
Case 

Control rod reactivity worth (pcm/$)  Coords./bank 

ARI  0.0052775  1057.8/2.0043581  14–10/5 
As seen in Table
Control rod banks configuration.
The transient is started by the ejection of the rod 14–10 which is fully extracted in 0.1 s. The transient is simulated following the sequence of the events showed in Table
Sequence of the events during the simulated transient.
Time  Event 

0 s  Start of simulation 
2.0 s  The control rod with maximum reactivity worth begins to withdraw 
2.1 s  The control rod with maximum reactivity worth is fully extracted 
500 s  End of simulation 
Recalling what was discussed previously, the sample size which guarantees double tolerance limits with a 95% of uncertainty and with 95% of statistical confidence for the output variables is equal to 146. The uncertainty and sensitivity analysis is performed assuming uniform distribution, on one hand, and normal distribution with three different deviations (0.1%, 0.5%, and 1%) on the other hand. Both cases have been simulated using Random and LHS sampling methods.
Figures
Partial rank correlation coefficient for maximum power, maximum enthalpy, and maximum reactivity for LHS normal 0.1% case (1:
Partial rank correlation coefficient for maximum power, maximum enthalpy, and maximum reactivity for random normal 0.1% case (1:
For simplicity, it has been only presented the comparison between the sampling methods for the normal PDF. In case of uniform PDF, the results are similar to normal PDF but with partial rank correlation slightly increased, that is, all output parameters are more sensitive with respect to input variables uncertainty. In next figures, it is seen that the fast absorption crosssection (
Figure
Scalar sensitivity of the scattering crosssection for all simulated cases.
The indexdependent sensitivity analysis for normal PDF and the two sampling methods are presented in this subsection. For the particular case in which the number of runs is equal to 146, critical values for Spearman’s coefficient are
First the results of Partial rank correlation coefficient (PRCC) for the output variable Power are presented. As shown in Figures
Partial rank correlation coefficient for LHS normal 0.1%, output variable : Power.
Partial rank correlation coefficient random normal 0.1%, output variable : power.
As seen, there are changes in the sensitivities of these four neutronic parameters once the insertion of positive reactivity has occurred. Also, the sensitivities change slightly to reach stability when the rods have already been extracted leading to their final value, corresponding to the sensitivity of those values for the reached steady state. All these results and conclusions can be extended to the random sampling normal distribution with 0.1% deviation case as seen in Figure
Figure
Partial Rank Correlation Coefficient LHS Normal 0.5%, output variable : Power.
In Figure
Partial rank correlation coefficient LHS normal 0.1%, output variable : enthalpy.
In Figure
Partial rank correlation coefficient LHS normal 0.5%, output variable : reactivity.
A final comment is needed with respect to the PRC coefficient variations. All output variables show an abrupt variation in PRC values at time 2 and around 25 seconds. As expected, the variation at time 2 s is due to the rod ejection, this variation expands its effect until time 6 s, and then a plateau is reached. The variation at time 25 s could be explained as follows: the PRC is a relative magnitude, so when all input variables are almost constant, a small variation in a thermalhydraulic or neutronic variable could produce an important relative change in the PRC.
The effect of the rod ejection and the reactivity variation are explained in Figure
PRC coefficient related to the reactivity and the reactivities.
Figure
When Doppler reactivity becomes important (2.15 s), absorption parameters increase their influences and fission parameters have opposite tendency. This effect could be explained as follows: the power increases as a consequence of the rod ejection, then the fuel and moderator temperature rise. This is followed by a reduction in moderator density, hence, moderation, and the multiplication factor decrease.
From the point of view of the uncertainty analysis, the results demonstrate that deviations about 0.1% have the smallest influence on the output variables of interest as expected. Figure
Power upper tolerance limit (UTL) for LHS sampling method.
Enthalpy upper tolerance limit (UTL) for LHS sampling method.
Reactivity upper tolerance limit (UTL) for LHS sampling method.
Finally, similar conclusions can be extracted for the uncertainty in power, enthalpy, and reactivity for Random sampling method.
This paper has described a Rod Ejection Accident (REA) simulated using the coupled code RELAP5/PARCSv2.7 with a perturbation on the crosssectional sets in order to determine the response of the computational system to uncertainties in the macroscopic neutronic information.
For all cases, the most influential uncertainties obtained by the scalar sensitivity analysis were the fast diffusion coefficient (1) with a positive influence on power, the scattering cross section (3) and both fission crosssections (6 and 7) with mixed positive/negative influence. The absorption crosssections together with the thermal diffusion coefficient could be neglected regarding the influence for the selected output variables. Therefore, the performed sensitivity analyses have shown that the influence of the uncertainties is not dependent on the selected sampling method.
The indexdependent sensitivity analysis showed the same influence for the different neutronic parameters. Moreover, there were sign changes in the most important neutronic parameters which are produced once the insertion of negative reactivity has been occurred. However, in all cases the tendency is to reach a steadystate condition for “long” time simulations. Regarding the sampling methods used, there were no significant differences. Furthermore, deviations greater than 0.1% showed smoother behavior; however, perturbations with standard deviations greater than 1% could lead to a run failure due to the high heat flux generation and consequently properties can be overpassed in steam tables.
The authors of this paper have no conflict of interests to declare.
This work has been partially supported by the Spanish Ministerio de Educación y Ciencia under Project ENE200802669, the Generalitat Valenciana under Project ACOMP/2009/058, and the Universitat Politècnica de València under Project PAID05094285.