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������4�4�48�4��6���R�<7<7<7<7<78.E8Y8cReReReReReReR$�Th4Wh�R-��<88�<�<�R��<7<7��Rc@c@c@�<��<7�<7cRc@�<cRc@c@r�KT��oL<707������4>�K�Pt�R0�R�Kz�W#?��WoL�W�oL�e8�Y9�c@:��:�e8e8e8�R�R�?de8e8e8�R�<�<�<�<��D8�%���8%�������������Modelling Uncertainties in Natural Circulation Passive System Stability Analysis

Luciano Burgazzi
ENEA, Italian National Agency for New Technologies, Energy and Sustainable Economic Development
Centro Ricerche �E. Clementel�, Via Martiri di Monte Sole, 4, 40129 Bologna, Italy
Tel. +39 0516098556; fax: +39 0516098279
E-mail address: luciano.burgazzi@enea.it


Abstract

Many research efforts in the field of nuclear safety are now facing problems related to the adoption of passive safety features included in evolutionary and advanced reactors designs. One of the greatest problems concerns the potential passive system performance impairment and consequent failure to accomplish the safety function. In response to the need to evaluate the performance of thermal hydraulic passive systems (i.e. relying on natural circulation) for decay heat removal, probabilistic methodologies are developed to assess the hazard of system stoppage. These methodologies should be able to take into account all the uncertainties inherent in any hazard analysis. Through the practical example of the assessment of a t-h passive system, this article introduces a logical framework that can be used to incorporate the different kinds of uncertainties related to data and models, as well as to specific expert�s choices in the risk analysis process.

Keywords: passive system, uncertainty, reliability, safety factor

Introduction

Many endeavours in the area of nuclear safety research are currently addressing problems related to the implementation of passive safety features included in evolutionary (e.g., ESBWR, Economic Simplified Boiling Water Reactors) and advanced reactors (e.g., Generation IV) designs. Passive systems are broadly treated in several reports as [1] and [2], with reference to natural circulation systems for decay heat removal in water cooled nuclear power plants, and termed thermal-hydraulic (t-h) passive systems. Furthermore Generation IV reactor designs stress the role of passive engineered features to cope with abnormal conditions: this applies, in particular, to the LFR (Lead Fast Reactor) and VHTR (Very High Temperature Reactor) to accomplish decay heat removal safety function, in accident circumstances. While for such systems it is claimed excellence in safety, efficiency and reliability, their implementation, nevertheless, has to face the lack of reliability data for the consequent evaluation of the correspondent reliability, [3]. 
It is stated that, due to the onset of phenomenological factors affecting the relative performance, safety and reliability assessment of these systems is required, since it is recognized that there is a non-negligible likelihood for system to perform below the expectation, and ultimately to fail to achieve the required safety function (i.e., reactivity control, decay heat removal and confinement of radioactive products), [4].
In this area, even though several studies have already been carried out, researchers still go on developing and improving their works in order to evaluate the underlying modes of failure of such systems, and the hazard and then the risk of system instability and stoppage. 
As a result, up to now, a variety of appropriate methodologies for the evaluation of this reliability has been put forward  [5], which can be found in open literature [6]. Today, researchers strive to add credit to the probabilistic models and thus make the approaches proposed so far credible: the main reason of this �Achilles' heel� lies in the paucity of reliability data, both experimental and operational, that forces the analyst to resort to engineering/subjective assessment or expert judgement to a large extent and to found the analysis upon plausible assumptions and realistic simplifying approximations.
This  makes the results strongly conditional upon the expert judgment elicitation process and supporting hypothesis, to be completely credited.
In fact, even though the validity of such methods with the theoretical models is accepted by the scientific community, some open questions make the topic still an open issue [7]. The study in [7] points out the amount of uncertainties associated with passive system performance as one of the most relevant issues, which becomes particularly relevant when innovative or untested technologies are applied, eventually contributing significantly to the overall uncertainty related to the reliability assessment.
The purpose of this article is the analyses of the uncertainties affecting the modelling underlying the passive system reliability assessment approaches. According to [8], uncertainty �refers to a situation in which no reasonable probabilities can be assigned to the potential outcomes�, as compared to the risk, which, conversely, �refers to a situation in which the potential outcomes can be described in objectively known probabilities�. In last years literature about the use of probabilistic methods as a way to to take into consideration uncertainties in t-h passive system analysis, became more and more comprehensive, in an attempt of offering an efficient framework for the incorporation of uncertainty into passive system design, see [9], [10], [11]. Notwithstanding this, it�s worth noticing that if the physical phenomena are relatively well known as described in [1] and [5], their representation within probabilistic studies is relatively rare. 
The study presented herein focuses on the practical assessment of a natural circulation loop in a very simple layout, characterized by a thermo-siphon loop with one heat exchanger dipped in a tank dissipating the heat to atmosphere [9]. The aim is to provide a methodology to incorporate different kinds of uncertainties related to data, models, or specific expert�s choices that may help defining the vulnerability over the system performance and consequently the risk that affects the plant safety.
In the following, at first, the main aspects of uncertainty related to t-h passive systems performance are recalled together with the present state of the art; after that a classical deterministic approach devised for system safety assessment is provided, to be analysed in terms of the related uncertainties  to create the probabilistic model.
Then the implementation of the modeling of the uncertainties is presented together with an application to an illustrative example, and finally some conclusions are drawn.

Uncertainties in Passive System Performance

For performing their accident prevention and/or mitigation functions, t-h passive safety systems rely exclusively on natural forces, that is gravity and density difference, not generated by external power sources. Since the magnitude of the natural forces, which drive the operation of passive systems, is relatively small, counter-forces (e.g. friction) can be of comparable magnitude and cannot be ignored as is generally the case with pumped systems. 
The uncertainties in t-h passive systems address the deviations of the system performance from the expectation, mainly because of the onset of thermal-hydraulic phenomena that may challenge the underlying physical principle and impair the performance of the system itself, so that the passive system may fail to meet the required function. 
Actually this point is well acknowledged and addressed over the studies conducted up to now, and will be briefly summarized here below.
For instance, the approach described in [9] allows identifying the uncertainties pertaining to passive system operation in terms of critical parameters driving the modes of failure, as, for instance, the presence of non-condensable gas, thermal stratification and so on. 
Actually there are two facets to uncertainty that, because of their natures, must be treated differently when creating models of complex systems. They have recently been termed aleatory and epistemic uncertainty. The aleatory uncertainty is that addressed when the events or phenomena being modelled are characterized as occurring in a "random" or "stochastic" manner, and probabilistic models are adopted to describe their occurrences. It is this aspect of uncertainty that gives PRA the probabilistic part of its name. The epistemic uncertainty is that associated with the analyst's confidence in the predictions of the PRA model itself, and it reflects the analyst's assessment of how well the PRA model represents the actual system being modelled. This is referred to as state-of-knowledge uncertainty. 
Ref. [9] points out, as well, the difference between the uncertainties related to passive system reliability and the uncertainties related to the t-h codes (e.g. RELAP), utilized to evaluate the performance itself, as the ones related to the coefficients, correlations, nodalization, etc.: these specific uncertainties, of epistemic nature, in turn affect the overall uncertainty in t-h passive system performance. 
A further step of the matter can be found in [10], which attempts to assign sound distributions to the critical parameters, to further develop a probabilistic model and a failure distribution function. References [6] and [10] provide a clear prospect of the uncertainties, taking into account that the overall uncertainty are both of aleatory and epistemic character, as shown in Table 1. 
As highlighted above, clearly the epistemic uncertainties address mostly the phenomena underlying the passive operation and the parameters and models used in the thermal-hydraulic analysis of the system (including the ones related to the best estimate code) and the system failure analysis itself. Some of the sources of uncertainties include but are not limited to the definition of failure of the system used in the analysis, the simplified model used in the analysis, the analysis method and the analysis focus of failure locations and modes and finally the selection of the parameters affecting the system performance.

Table 1	Categories of uncertainties associated with T-H passive systems reliability assessment [6]
				       
 

R-S Approach 

With reference to the two-phase natural circulation system relevant to the current technology as the isolation condenser [9], let�s consider the failure criteria in terms of a safety variable (for instance a physical parameter as the flow rate or the exchanged heat across the heat exchanger) falling below a failure threshold value as 80% of its nominal value , coming out from a deterministic evaluation by best estimate code, [12]. Let�s denote S as the achieved value of the parameter during system operation and R its required value, i.e. 0.8 of its nominal value, so that safety condition is given by S/R>1, while system fails if S <R. It�s worth noticing that all these considerations are conditional upon the successful inception of the natural circulation, so that we disregard one of the factors that could challenge the system operation start-up, as for instance the opening of the valve placed on the condensate line. We have also to take into account that these evaluations are valid for each instant t of operation, since the system operation is a time dependent process.
Let�s consider the notion of Safety Factor (SF) borrowed from the fracture mechanics as an indicator of the stability of the system, already adopted in [13]. This safety factor is defined as the ratio of the system state S over the system requirement R. Theoretically an SF value greater than 1 means that the system performs satisfactorily, while a SF lower than 1 means it is unstable, so that the previous expression for the safety condition becomes S=SF*R. In practice a threshold value greater than the unity is used at the design level to incorporate uncertainties. 
As previously mentioned if R is known as a fraction of the nominal value, S is evaluated from the analysis of the output of a t-h analysis conducted on a set of different cases by deterministic code simulation. In particular the set of cases defined by relevant system parameters - as the design parameters defining the initial conditions and the critical parameters as failure drivers, [14] -, randomly selected from the probability distributions (Monte Carlo simulation), brings to the distribution of the safety variable. 
By performing Monte Carlo simulations, it is possible to propagate model and parameter uncertainties and calculate the distribution of e.g. the clad temperatures and thus the probability of observing a temperature value above the defined limit.
This procedure is practically followed within REPAS (Reliability Evaluation of Passive Systems) and RMPS (Reliability Methods for Passive Safety) approaches. 
In particular, the methodology known as REPAS [15], developed in late 1990s, cooperatively by ENEA, the University of Pisa, the Polytechnic of Milan and the University of Rome, represents the earliest significant effort to quantify the reliability of such systems. This methodology is based on the evaluation of a failure probability of a system to carry out the desired function from the uncertainties of those physical and geometric parameters which can cause a failure of the system.
REPAS paves the way for the multinational EU (European Union) funded RMPS project, started in 2001 till 2004, [16]. The RMPS methodology, is developed to address the following problems: 1) Identification and quantification of the sources of uncertainties and determination of the important variables, 2) Propagation of the uncertainties through thermal-hydraulic (t-h) models and assessment of passive system unreliability and 3) Introduction of passive system unreliability in accident sequence analyses. In this approach, the passive system is modelled by a qualified t-h code (e.g. CATHARE) and the reliability evaluation is based on results of code runs, whose inputs are sampled by Monte Carlo (MC) simulation. This approach provides realistic assessment of the passive system reliability, thanks to the flexibility of the MC simulation, which adapts to t-h model complexity without resort to simplifying approximation. In order to limit the number of t-h code runs required by MC simulation, alternative methods are proposed such as variance reduction techniques, first and second order reliability methods and response surface methods. Within the RMPS project the methodology is successfully applied to passive systems utilizing natural circulation in different types of reactors (BWR, PWR, VVER). The RMPS methodology tackles also an important problem, which is the integration of passive system reliability in a Probabilistic Safety Assessments (PSA).
On the other hand, as compared to the previous approaches, in [13] and [17] a more simplified and direct method is adopted, based on the concept of functional failure, defined as the probability of the passive system to fail to achieve the specified safety function in terms of a given safety variable crossing a failure threshold.
This makes it possible the uncertainty pertaining to the selected safety variable, as the water mass flow rate, being evaluated in terms of suitable probabilistic distributions (as normal distributions), chosen upon subjective/engineering judgement. Thus the uncertainty related to this variable would represent the whole set of uncertainties about the input parameters. 
However a commonality between the two alternative approaches can be found in the fact that in both cases one has to resort to subjective assessment or specialist elicitation with reference to the deterministic code simulation as regards parameter distribution and range selection and performance parameter probabilistic characterization in the second case.
According to the previous section, the overall uncertainty includes mainly two classes of uncertainties:

The uncertainty linked both to the variability of geometrical and mechanical data 
The uncertainty related to a lack of knowledge, that makes it difficult to assess exactly the parameter values used in models

While the first category refers to temporal or spatial variability and to individual heterogeneities, the second category comprises model uncertainty, parameter uncertainty in used models and uncertainty in decision making from modelling.
This article deals principally with the second type of uncertainty, that, according to the distinction put forward  in the previous section, is usually named epistemic uncertainty.
A very detailed treatment of all these uncertainties that impact on the performance of  passive systems is presented in [11], with reference to a passive decay heat removal system for a gas- cooled fast reactor. In particular the uncertainties related to the thermal hydraulic model used to find the steady-state condition and evaluate the system performance concern both the models, i.e. the physical correlations, and the physical variables inputted in the code, as pressure and power. In this study the epistemic uncertainties are propagated to calculate the probabilities of functional failures, i.e., the possibility that the loads will exceed the capacity in a reliability physics framework. 
Conversely, presently the uncertainty is treated as lumped element, and the implicit assumption here is that it encompasses the whole set of uncertainties related to the value of the safety variable, as in [13].

Uncertainty modelling

In reliability engineering, a very efficient way of taking into account uncertainty on input parameters is to define statistical distribution functions for each of these parameters. Unfortunately, as stated before, available information is frequently poor, as in case of passive system reliability [6] and test data are insufficient to characterize the statistical moments required in the distribution functions. When test and operational data are unavailable, those parameters are inferred from the literature or judgement or experience of experts. Once distribution functions are defined, MC simulations may be used to propagate uncertainties through the deterministic model. The principle of such simulations is to take one random value from the distribution of each input parameter and to calculate the output value of the function. Each calculation gives one outcome, and a great number of simulations allow one to obtain a frequency distribution of the output values: mean value, standard deviation, or other statistical moments of the studied parameters may then be estimated. The greater the number of simulations, the more precise are the computed statistical moments.
In the present case one has to evaluate the probability that the safety factor SF is less than the target value of stability, that is the unit, or in other terms:

Pf  = P(SF< 1)

Then one needs to evaluate the distribution of the SF, intended as the ratio of the actual value of the performance parameter (e.g. the flow rate) to the required one, SF=S/R. 
Note that R and S are considered independent of each other. This treatment is similar to the approach proposed in [18], where the probability of failure of the system is assessed in terms of the safety margin or reliability index, computed as the ratio of the difference of the mean values over the square root of the of the sum of the variances, assuming normal distributions for both R and S. In that case the objective is to define the minimal requirement for the passive system, in terms of its performance significant parameter towards a reliability target. In the present analysis the expected outcome is to evaluate the system failure probability through the quantification of the uncertainty of the safety factor. 
We can evaluate the uncertainty in the safety factor by MC simulation. Using the previously defined method, the statistical analysis of a large number of computed simulations makes it possible to estimate the mean value and the standard deviation of the safety factor. 
From a practical viewpoint, uniform and normal distributions are chosen to fit the parameters. Tables 2 and 3 show the parameters of interest of the uniform and normal distributions respectively, with reference to the flow-rate W, across the system where Ws is the actual flow rate value, Wr is the minimal flow rate value required for natural circulation. The numerical values are inferred directly from [17], since no finalized study has been carried out to propose the relevant probability distributions, coming out from the t-h analysis. A uniform probability would be suitable to describe the parameter if one considers the design range of the parameter and additional information is not available. The normal distribution is considered for its relative simplicity and familiarity to engineers. It represents a good approximation in case the standard deviation is small as compared to the mean value. It is worth noticing that the ranges defined by 2 standard deviations roughly cover the 95% confidence interval, considering that the two-sided 95% confidence interval lies at +1.96s from the mean value. 

Table 2 Uniform pdf characteristics
ParameterRange(a-b, kg/sec)Characteristics (kg/sec)Ws12-28� = 20
� = 4.61Wr8-24 � = 16
� = 4.61
Table 3 Normal pdf characteristics
ParameterRange(a-b, kg/sec)Characteristics (kg/sec)Ws12-28� = 20
� = 4Wr8-24 � = 16
� = 4
The results of the simulations (1000) for both cases in terms of mean, median and standard deviation of the resulting distribution of the safety factor SF are shown in table 4.

Table 4 Simulation results
MeanMedianStandard DeviationUniform1.371,240.55Normal1.351,240.55
The graphics representing the frequency distribution functions of the safety factor, in the form of histogram, and the cumulative distributions are represented in the following figures. The probability of system failure that is SF<1 are respectively 0.3 and 0.25.
Since designers and engineers are required to give answers, as accurate as possible, to problems full of uncertainties, one of their greatest needs is thus to reduce the uncertainty of the results they provide (in this study the standard deviation of the calculated safety factor is about 0.55) in a very effective way. Therefore each source of uncertainty, both parameters and models, should be investigated separately: it is obvious that one can not study all the sources of uncertainties but can only try to focus on some of them and to identify the factors whose uncertainty has the greatest impact on the results. Uncertainty in risk analysis modelling can also be attributed to some of the expert�s choices. Assessing a risk generally requires using specific methodologies and mathematical models: in our case, for instance, the choice has been made to compare the values of the achieved and requisite flow rate. Nevertheless, as the variance of the computed safety factor are the same in both cases of normal and uniform distributions, it appears that, in our study, the choice of a specific distribution function has only a weak influence on the results. Clearly, as previously stated, the uncertainties of epistemic nature, that is parameter and model uncertainties, are deemed to have the greatest impact on the results of the variance: the practical meaning of this statement is that if one wants to reduce the variability of the results and consequently to be more precise in the analysis, the involved analysts should concentrate on providing a more accurate estimation of model parameters.




Figure 1 Frequency distribution of the Safety Factor (Uniform distribution)




Figure 2 Frequency distribution of the Safety Factor (Normal distribution)




Figure 3 Cumulative distribution of the Safety Factor (Uniform distribution)



Figure 4 Cumulative distribution of the Safety Factor (Normal distribution)

Another question that arises from the analysis concerns the interpretation of the results presented as a probabilistic number: what does it mean for a t-h system to have a probability of failure of e.g. 20%? We should consider that for natural circulation a probability of failure (computed as probability of �performance� failure) of x% means that under certain conditions dictated by the transient under consideration, the system could be expected to fail x times out of 100 times its operation is required or alternatively x out of 100 similar systems could be expected to fail at some time during the period of operation. We could interpret the probabilistic figure respectively as a spatial or temporal frequency. However the �real� meaning of the results depends on how the modelling assumptions are made; specifically on how the total uncertainty is divided between aleatory and epistemic (parametric). To the extent that all the uncertainty is assumed aleatory, the probability refers to a spatial or temporal fraction, while to the extent that the uncertainty is assumed epistemic, the probability refers to a chance of complete failure. In our case, since the majority of uncertainties associated to the t-h code to derive the variable distributions are related to the lack of knowledge, we can state that the epistemic uncertainties are prevailing with respect to the aleatory, so that the probabilities in this case are just a tool to quantify the epistemic uncertainties. 
The practical interpretation of the probability of failure is rather delicate. Figures 1 and 2, for example, present the results of two different analysis that have been undertaken considering two different input distributions, characterizing the uncertainty of input data. Although the distributions have the same mean safety factor, the distribution in Figure 1 presents a larger spreading (is much more scattered), so that we are more confident in the results of Figure 2.  

Summary and Conclusions

The incorporation of passive engineered systems to accomplish safety functions raises specific issues for the design and the safety analysis, especially as regards the required improvement for the demonstration of their effectiveness.
This article attempts to provide a logical and straightforward way to take into account several kinds of uncertainties that can be encountered in a risk analysis study, which is very relevant in case of passive systems because of the scarcity of both operational and experimental available data. Defining statistical distribution functions for input parameters may help engineers to deal with the problem of �data uncertainties� inherent to the study of a t-h passive system relying on natural circulation.
It can also be pointed out here that the definition of the characteristics of those distribution functions requires the expert to make some choices (e.g., evaluation of mean values and maximal errors), so that the �expertise uncertainty� is also involved. 
In our study, both normal and uniform distributions are assumed to describe the uncertainties relative to the system performance parameters, taken as a whole without factoring into stochastic and epistemic, for instance; it has been shown that such a choice has only a weak or even negligible influence on the estimation of the probability of failure, compared to large spreading in the input of the analysis. 
The distributions relative to the thermal hydraulic performance parameters describe principally the uncertainties relative to the thermal-hydraulic models, resulting from the numerical simulations by the best estimate thermal-hydraulic code. The hazard relative to the passive system is evaluated in terms of the analysis of the safety factor distribution, coming out from a number of simulations on the probabilistic model. Great attention must be paid to the conditional results and their interpretation, since their dependence upon the assumptions taken in the analysis. Finally the so called epistemic uncertainties due to the lack of knowledge are identified as the main source of uncertainty relative to the assessment process.
































.
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 In the following by ��nominal�� condition, we mean the underlying condition characteristic of the reference system configuration and performance.













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