Screening of critical parameters influencing thermal-hydraulic reliability of simple passive system

ABSTRACT

Robust safety nature of passive safety systems (PSSs) accounts for their increasing applications. Critical parameters (CPs) which influence reliability of thermal-hydraulic (t-h) PSSs are considered independent in most cases while considering their effects purposely for simplicity which may not be realistic. Findings affirmed reliability of t-h PSSs to be influenced by CPs that are dependent in most scenarios and thus, effects of CPs dependency which can directly/indirectly influence t-h reliability need to be considered. Reliability assessment methodologies (RAM) can thus be improved upon by considering the dependency of CPs in reliability analysis. In this regard, this paper considers the screening of CPs required to justify their dependency consideration in evaluating t-h reliability. The Pearson’s product moment correlation coefficient and covariance method were applied as illustration for the screening of the possible realistic CPs, which affect natural circulation of a passively water-cooled steam generator. The approach was used to determine the combinations of the CPs that are dependent and screens out those adjudged independent. Based on the results obtained, appropriate considerations (dependency/independency) can be made and further analysis of interest (failure/reliability) can be conducted for the system. Incorporation of this screening approach into the existing t-h RAMs will improve their efficiency.

KEYWORDS:

• Reliability
• critical parameters
• thermal-hydraulic passive systems
• dependency screening
• correlation
• reliability assessment methodologies (RAM)

1. Introduction

Recent developments and advances in thermal-hydraulics (t-h) and nuclear reactor safety are directed towards ensuring safe, reliable, operational stability, and economically competitive operations of nuclear power plants (NPPs). In this regard, passive safety systems (PSSs), characterized by inherent safety features, have helped in building public confidence and acceptance of nuclear power generation to some extent. These systems are now being adopted in advanced and innovative reactor designs to ensure safety and boost performance requirements [1].

One of the major issues that must be tackled to achieve the target of safe, stable, dependable, and effective operation of reactors adopting PSSs is the quantification of reliability of the increasingly adopted passive systems [1].

As the knowledge of the operations and principles of PSSs are increasing, discoveries of failure mechanisms (FMs) associated directly or indirectly with operations of passive safety components and phenomena are made. These FMs interfere with the fulfilment of the mission(s) of such systems and thus brought up challenges related to reliability [2]. Special attention is now given to relevant issues of PSSs which are basically epistemic uncertainties (insufficient knowledge) on some t-h phenomena and lower driving forces compared to those of the active safety systems [3].

The t-h passive systems operations depend highly on the natural circulation (NC) phenomena which drives most t-h passive systems. The driving force and resistance are influenced by many uncertain factors/phenomena, which are thus responsible for most physical process failure of PSSs [4,5], which can eventually lead to the failure of overall plant or facility. Generally, PSSs fail when any of their operating parameters or phenomena deviate slightly from their expected values and initial/boundary conditions [6]. The reliability of PSSs, which is the capability of the systems driven by physical phenomena (forces or principles) to satisfactorily perform their safety mission(s) under prevailing conditions of operation for a given period, must therefore be properly investigated.

The reliability of the NC-based passive systems is influenced (either positively or negatively) by some critical parameters (CPs) which are dependent in behaviour [7]. Some of the CPs are non-condensable gas fraction, heat exchanger plugging fraction, undetected leakage area, and valve closure coefficient among others. To effectively evaluate the reliability of the t-h passive systems, the dependency nature of the CPs must therefore be taken into consideration to guarantee the safety and reliability of the systems. Due to the complicated analysis associated with dependency consideration of the CPs, independency approach of handling their effects is considered most times and has been established to be associated with some uncertainties. Though consideration of many associated CPs are essential for the sake of accuracy in dependency analysis of CPs [8], it is also important to minimize as much as possible the number of the identified CPs by screening out those having insignificant influence through dependency consideration in order to reduce the computational burden of the analysis.

This paper is therefore organized in a way to address testing for dependency of CPs using suitable statistical approach (backed up with relevant t-h basics) to ascertain whether or not dependency exists among the possible CPs combinations which is an essential step in the procedures for evaluating the reliability of t-h PSSs based on dependency consideration of the CPs [8]. This dependency, which also can be interpreted as synergistic effect of the CPs, was considered from both mathematical and practical (t-h) perspectives. The issue of ascertaining dependency (screening for dependency) of the CPs influencing t-h reliability and consequently overall reliability results is thus tackled by the analysis presented. This is necessary as adoption or assumption of the independency approach (by virtue of simplification) especially for t-h phenomena may give rise to unrealistic and grossly inaccurate assessments of reliability as discussed above. The existing t-h reliability methods will be improved when they are modified by incorporation of this screening approach especially in terms computation efficiency and accuracy.

2. CPs and FMs of thermal-hydraulic passive systems

2.1 CPs in evaluating the reliability of thermal-hydraulic passive systems

Though a passive system is theoretically expected to be more reliable than an active one as the former does not depend on any external force or energy which may be liable to failure; in reality, several factors (uncertainties) which are depicted by CPs are known to influence the performance and consequently reliability of NC-based passive systems [9].

For the purpose of simplifying the system’s analysis, it is conventional to consider the reliability of the essential individual components which are parts of the main system while considering the overall reliability of t-h PSS. The failure or performance assessment of the constituting components of PSSs in most studies does not factor-in environmental influence (interactions) and thus could not account for systems or components dependency effects on reliability [6].

Though reliability is intrinsic to a component, sub-system, system or environment; it is usually affected and depends on a number of other factors. The factors referred to the prevailing conditions mentioned in the reliability definition [3]. The influencing factors of reliability, mainly the internal and external parameters of the component or system concerned, affect the reliability in various forms (positively or negatively) which can be directly or indirectly. To quantify the impacts of the influencing factors, the CPs, which can be otherwise referred to as the indicating factors must be used. Five different phenomena which are build-up of non-condensable gases, undetected leakage, heat loss, partially opened valve (POV) in the discharge line, and heat exchanger plugged pipes have been studied with their corresponding CPs [7]. Reliability analysis based on the influencing factors therefore involves defining the failure rate which is a function of the adopted indicators (CPs) values.

Burgazzi [10] attributed the physical parameters directly or indirectly causing the failure modes of passive systems to be CPs while the t-h phenomena on the other hand, are the physical processes which determine the behaviour of the t-h PSSs. Both CPs and t-h phenomena therefore influence the reliability of PSSs.

In this work, a screening approach for the possible CPs in a simple PSS is presented. The approach is based on both qualitative and quantitative points of views of dependency existing in all possible bivariate combinations of the CPs using a simple generic NC-based system. The screening approach makes the dependency analysis of t-h reliability of passive system effective and efficient.

2.2 FMs associated with the thermal-hydraulic passive systems

The FMs, CPs, and the corresponding consequences of failures associated with the natural circulation systems (NCS) which are the commonly adopted t-h nuclear passive systems are presented (Table 1). The CPs are adopted as indicators for the causes or joint causes of the system failure or FM being considered.

Table 1. Failure modes, their effects and the associated critical parameters in common t-h PSSs [11]

S/N	Failure mode	Effects/consequences	PSS	Mission	Critical parameters
1	Fouling (due to surface oxidation)	Alteration of surface characteristics and reduction in heat transfer rate by conduction and convection.	NCS	RHR	Concentration of impurities/contaminants
2	Thermal stratification	Reduction in convective heat transfer.	NCS	RHR	Density, temperature
3	Degradation of piping insulation	Reduction in heat exchange process.	NCS	RHR	Heat loss
4	Stiffness/reduction in motion of moving parts.	Flow restriction (flow rate reduction) and increased friction losses (associated with check valve, and partially opened valve in the discharge line).	PXS^a	RHR, PCCS^b	Valve closure area or coefficient or even ageing effects (fatigue, creep and corrosion)
5	Cracking (due to material defects, localized stresses, and corrosion)	Reduction in heat conduction or even leakage.	NCS Containment building/pressure vessel	RHR FPC^c	Irradiation effects and ageing effects
6	Break/rupture/undetected leakage	Reduction in heat transfer rate and loss of coolant.	NCS	RHR	Leakage area
7	Blockage (heat exchanger (HX) plugged pipes)	Reduction in the heat transfer area of the HX and finally reduction in heat exchange process.	NCS	RHR	HX plugging fraction
8	Boundary/envelope failure (due to thermal loads, material defects)	Leakage (reduction in flow rate/loss of coolants).	NCS	RHR	Undetected leakage and valves locked-open
9	Build up of non-condensable gases	Reduction in heat transfer capability.	NCS	RHR	Non-condensable fraction

^aPXS: passive core cooling system; ^bPCCS: passive containment cooling system;

^cFPC: fission products containment.

Their identification requires detailed investigation of the mission(s) of the PSSs they are associated with [12].

Because some of the CPs are difficult to quantitatively gauge, statistical approaches are adopted to characterize the relationships between failure rates and the CPs as their precise physical relationships are characterized by several epistemic uncertainties [13].

Influencing coefficients are assigned to quantitatively represent the effects of selected relevant CPs which can be positively/negatively discrete or continuous.

Taking time into consideration, the reliability, R(t) for n dependent CPs [14] can thus be generalized as

(1) $\begin{array}{rl}& R=Pr\{X_1(t)\le L_1,X_2(t)\le L_2,...,X_n(t)\le L_n\}\\ & =\underset{L_{0,1}}{\overset{L_1}{\int }}\underset{L_{0,2}}{\overset{L_2}{\int }}...\underset{L_{0,n}}{\overset{L_n}{\int }}f\{x_1(t),x_2(t),...,x_n(t)\}d{x}_{1t}d{x}_{2t}...d{x}_{nt}\end{array}$(1)

where f{x₁(t), x₂(t),…,x_n(t)} = the joint distribution function of X₁(t), X₂(t),…, X_n(t) at time t, with X₁, X₂,…, X_n, being the associated CPs.

2.3 Identification and quantification of the associated CPs

In order to quantify (assign numerical values to) the effects of the CPs for a given t-h passive system, they must first be identified in form of precursors (influencing coefficients) such as the non-condensable fraction, valve closure coefficient, etc. Previous works considered similar CPs for systems like isolation condenser [4,7].

2.4 Approaches for considering the CPs

The approaches for treating the CPs influencing reliability of the PSSs are in two forms according to Burgazzi [7], i.e. independency and dependency considerations. The computation of the overall failure probability (from which reliability can be determined) by assuming independency of CPs in most cases has been discovered not in complete agreement with what is obtainable in practice. Attempts to resolve the underlining issues and obtain more realistic results led to the dependency consideration of the CPs. Dependency approach argues the issues associated with the independency consideration to be due to assumption of no interaction among the CPs or associated phenomena and also the way the defined failure criteria are affected by the interactions which are not captured by the independency approach [7].

The findings of Burgazzi [7] on the issues associated with independency consideration of the CPs adopted by Zio and Pedroni [4] led to justification of the need and model development for CPs dependency for t-h passive safety phenomena like NC.

The CP dependency approach considers the interactive nature (influence) of the CPs. The adoption of dependency approach is justified as the influence of interactive behaviours of the phenomena associated with the CPs and consideration of possible changes in the defined failure criteria due to the interactive CPs amongst others can be taken care of by the approach. Though the approach is characterized with uncertainties (inadequacy of knowledge of data, models, etc., characterizing the behaviours) associated with the t-h passive systems, it produces results that are realistic to a greater extent. In dependency consideration, the analysis is complicated due to interactions of the CPs being considered which influence failure or performance and thus reliability. The interactions among the phenomena or parameters are in such manners that two or more of them can influence the overall system performance in terms of the likelihood of occurrence or severity of the resulting consequence(s) [3].

With the nature of dependencies existing among the relevant CPs being known to some extent, a suitable conventional dependency method can be applied to account for dependency influence.

3. Method

3.1 Theory of dependency analysis of CPs (with two key indicators)

Due to the dependency effects of CPs which can trigger the occurrence or non-occurrence of FMs, the dependent parameters can thus possibly have similar contributions to the FMs. Hence, the parameters cannot be combined directly using the independent approach. As aforementioned, the level or degree of the dependency needs to be estimated quantitatively (which translates to the screening procedure) using appropriate approaches as part of the dependency analysis procedures in dependency consideration of t-h reliability. Some of such dependency analysis/multivariate methods are the covariance matrix [14], functional relations, and conditional subjective probability distribution methods [8], all of which effectively put dependency effects into consideration [15].

For ease of computation, the quantitative measures of the time-variant CPs such as X₁(t) and X₂(t) at time, t are usually taken to be time-invariant. Based on that assumption, various relationships [16] can be derived by applying the s-dependent theorem on the combinations of the CPs, with two CPs (bivariate analysis) being the simplest form. One of such relationships is the generalized reliability, R(t) expression for dependent CPs.

Testing for dependency of a set of two different CPs, X₁ and X₂ for instance must satisfy the relation,

(2) $S_{x_i{x}_j}=Cov(X_1(t),X_2(t))=Cov(x_i,x_j)\ne 0,$(2)

and the generalized variance-covariance matrix [17] is of the form,

(3) $\left(\begin{array}{cccc}Var(X_1(t))& Cov(X_1(t),X_2(t))& \cdots & Cov(X_1(t),X_n(t))\\ Cov(X_2(t),X_1(t))& Var(X_2(t))& \cdots & Cov(X_2(t),X_n(t))\\ ⋮& ⋮& ⋮& ⋮\\ Cov(X_n(t),X_1(t))& Cov(X_n(t),X_2(t))& \cdots & Var(X_n(t))\end{array}\right)$(3)

The entries of the above matrix are explained in Section 4.3 where it is applied. It is important to note that for independent parameters, the covariance is zero which implies Equation (2)(2) $S_{x_i{x}_j}=Cov(X_1(t),X_2(t))=Cov(x_i,x_j)\ne 0,$(2) not satisfied.

3.2 Simplified procedure for evaluation of t-h reliability based on dependency consideration of the CPs

A simple procedure that can be followed in the dependency analysis of CPs influencing t-h reliability was presented using a concise flowchart by Olatubosun and Zhang [8] and includes: selection of the desired t-h PSSs; definition and identification of the possible FMs with the selected t-h PSSs; adoption of the key CPs associated with the FMs; quantification of the CPs using the influencing coefficients; definition of the CPs nominal range and failure criteria (FC); testing for dependency of CPs by a suitable approach; development of the joint probability density functions (pdfs) associated with the CPs; and finally, evaluation of reliability using the joint pdfs developed by observing the dependencies.

This paper is organized in a way to address the testing for dependency of CPs, which determines whether or not dependency exists among identified CPs using Pearson’s product moment correlation coefficients and covariance matrix alongside qualitative measure of the influence of CPs in their combinations (practical consideration) and consequently screens out a subset of the CPs that are adjudged independent.

4. Dependency screening of a passively water-cooled steam generator

For the purpose of illustrating the dependency screening procedure, a simple NC loop being adopted in most t-h PSSs was used. The simple system, passively water-cooled steam generator, is a generic form of the NCS adopted in most nuclear reactors with the safety mission of removing residual heat from the reactor core after a reactor emergency shutdown. Other NC-based passive systems in this group of moving working liquid PSSs [18] beside the passively water-cooled steam generator are passive residual heat removal heat exchangers (PRHR-HX) and passively cooled core isolation condensers.

4.1 Description of the passively water-cooled steam generator

The case study – the passively water-cooled steam generator – was characterized to obtain the CPs responsible for almost all possible FMs in this screening analysis. The system is commonly adopted in advanced pressurized water reactor (PWR) designs with its mission being the removal of the reactor core decay heat through the steam generator (SG). The system has the simple NC operating principle. The pressurized water getting to the steam generator rises through the hot leg (inlet of the PRHR) and is condensed in a heat exchanger (HX) immersed in a pool of water-condenser (the heat sink) at an elevated height (Figure 1). The cold coolant returns to the SG through the cold leg (outlet of PRHR) connected to the SG at a lower elevation relative to the hot leg [18].

Figure 1. Schematics of core decay heat removal by a passively water-cooled steam generator [18].

System-integrated Modular Advanced ReacTor (SMART) and Hualong-1 reactor (HPR1000) are examples of the nuclear reactors that operate based on the principle of passively water-cooled steam generator as described above [18].

4.2 Characterization of the FMs and CPs of the illustrative system

The possible CPs influencing the failure/performance behaviour of the simple NC loop together with the FMs have been extracted (Table 2) from Table 1 that shows most of the common CPs with passive systems. The CPs to a greater extent are responsible for FMs associated with the illustrative system such as pipe break, blockage of HX pipes, closure of valves, etc. Their identification involves thorough study of the safety mission(s) the system is designed to fulfil [12] and thus the selected CPs are the ones that may likely have influence on performance from all possible points of views. Impurity was not considered in Table 2, though can have negative impact on the mission of the system as the system itself can be affected when its concentration is high and also depending on its chemical composition. But in the case study, practically, impurities are rare and their effect when present may not be significant qualitatively. In addition, pipings inclination which appears in Table 2 is not a FM and thus not listed in Table 1 but capable of causing form losses (especially in NC-based systems with low driving forces), which affect the flow/the system’s mission and indirectly t-h reliability.

Table 2. Range of critical parameter and failure threshold associated with the system [7,12,19]

S/N	Critical parameter	Range of indicator (discrete value)	Nominal value	Failure threshold	PDF of failure (P(x))
1	Non-condensable fraction	0.01–0.8	0	0.50	D-TND
2	Heat exchanger plugging fraction	0–0.15	0	0.10	D-TND
3	Valve closure coefficient	0–0.5	0	0.30	D-TND
4 5 6	Undetected leakage area (cm^2) Heat loss due to degradation of thermal insulation of pipes (kW) Piping layout/inclinations (rad)^a	0−10 0−100 0–0.175	0 5 0	1.00 60 -	D-TND LND Exponential

Non-condensable fraction: 0 – absence of non-condensables, 0.8 – presence of about 80% non-condensables … as the presence of 100% non-condensables is not realistic.

HX plugged pipes: 0 – total absence of plugged pipes, 0.15 – unacceptable condition requiring replacement/fixing

Valve closure coefficient: 0 – valve completely open, 1 – valve completely closed

Undetected leakage: 0 cm² – no leakage, 10 cm² – maximum area of undetected leakage, failure threshold of 1.0 cm² (about 3/8 in) is adopted as it is approximately the lower limit for Small Break Loss of Coolant Accident (LT-SBLOCA) in nuclear reactors

D-TND: Doubly Truncated Normal Distribution

LND: Log-normal distribution

^aPiping layout: range of 0–10° (0–0.175 rad) considered, 0° being the design layout and 10° considered not easily apparent elevation responsible for head losses

Due to inadequacy of experimental and real operational data associated with the operations of the t-h PSSs, expert judgment and engineering assessment are commonly adopted to characterize these systems (as in Table 2) [19]. Expert judgment involves the application of approaches/models such as analytic hierarchy process (AHP) to scenarios like NC. AHP is a structured technique for organizing and analysing complex decisions based on existing knowledge (such as mathematics) and experience. With AHP, different properties of the system or the scenarios concerned are characterized and ranked on qualitative basis of high, low, medium, etc. On the other hand, engineering assessment involves application of basic engineering principles such as basic physics and t-h concepts to systems or phenomena concerned. Other necessary steps in dependency analysis are usually followed like the formulation of suitable pdfs. The status of the system under normal operating condition is depicted by the nominal parameters (Table 2).

Three different suitable distributions (truncated normal distribution (TND), lognormal normal distribution (LND), and exponential distributions) were adopted based on expert judgment/engineering experience as the best fit for the associated FMs as used for similar systems in literature. For instance, the selection of TND is on the premise that the standard deviation is very small when compared to the mean value and in addition, it is a simple and commonly applied practical engineering tool. The associated Equations for these distributions are shown in the Appendix.

4.3 Dependency screening of the CPs by correlation and covariance approaches (mathematical perspective)

The six identified CPs are denoted as x₁: NCF (non-condensable fraction), x₂: HXP (heat exchanger plugging), x₃: VCC (valve closure coefficient), x₄: UL (undetected leakage area), x₅: HL (minor heat loss due to piping insulation degradation), and x₆: PL (Piping layout/inclinations). A set of five observations each for the CPs was considered. The selection of the observed values, though randomly selected, is justified as they all fall within the range of the permissible values (below failure threshold) of the CPs selected (as in Table 2); they are likely expected discrete nominal parameter values under normal operation (though they are time-variant variables in practice), i.e. the observations are success cases as x_i < x_max, with x_max being the maximum permissible thresholds.

Similar values to the chosen ones (randomly selected possible CP values within the permissible indicator range) have been used in literatures for similar systems to this generic passively cooled steam generator. The interesting fact about this case study is that the identified CPs and their fractional indicator values (as in Table 2) can be adopted for other systems with similar configurations (NC system loops). The failure thresholds are defined on the basis of experiments, simulation and engineering/expert judgment [4,12,19] with the allowable range of the indicators for the systems very realistic and practically applicable though may still be associated with some epistemic uncertainties. A very similar instance is that of an isolation condenser [12], which is a two-phase flow NC system loop with the main safety mission of rejecting core decay heat to the heat sink through condensation of primary fluid into the tubes of its heat exchanger. The heat exchanger is submerged in a pool at an elevated position and connected by pipes (riser and downcomer) to the pressure vessel (location of the heat source). The system has a gate valve that switches the system into action just like case study adopted and other passively cooled steam generators as in SMART.

For the identified CPs (x_i), a set of five observations each for the CPs on the basis of randomly taking permissible values for normal operating conditions which are all below failure thresholds (Table 2) for the system can be considered. With the set of observations x_i for each of the CPs, a matrix can be obtained for the system which is denoted as:

$A=\left[\begin{array}{cccccc}{x}_1& x_2& x_3& x_4& x_5& x_6\\ 0.30& 0.035& 0.25& 0.60& 3.0& 0.005\\ 0.40& 0.050& 0.20& 0.80& 5.0& 0.100\\ 0.20& 0.025& 0.18& 0.50& 80.0& 0.070\\ 0.10& 0.040& 0.08& 0.40& 20.0& 0.120\\ 0.08& 0.015& 0.04& 0.20& 10.0& 0.040\end{array}\right].$

Similar values have been adopted in the literature for related passive systems among which are isolation condensers and PRHR systems which are validated using experiments, simulations, and engineering experience as aforesaid [7,10,12].

Using the covariance dependency approach, the mean vector and the variance-covariance matrix (needed to test for dependency) for the selected CPs are determined. For the observations,

the mean vector is, $\bar{\mu }=\left[\begin{array}{cccc}0.216& 0.033& 0.150\text{break}& 0.500\begin{array}{cc}23.600& 0.067\end{array}\end{array}\right]$,

and the variance,

${\sigma }^2=\left[\begin{array}{cccc}0.01828& 0.00018& 0.00760& 0.05000\text{break}\begin{array}{cc}1037.30000& 0.00212\end{array}\end{array}\right].$

With the Ms Excel statistical function tools (PEARSON and COVAR), the Pearson’s correlation coefficients, r and the covariances were obtained for all the 15 possible combinations (i.e. ⁶C₂ = 6×5/2) of the key CPs (Figure 2 and Table 3, respectively).

Table 3. The COVAR values and product of standard deviations for the combinations of the critical parameters

Index	Combination of critical parameters	Cov (xi, xj)	Product of standard deviations (σiσj)
1	x1x2	0.001012	0.001826
2	x1x3	0.008040	0.011787
3	x1x4	0.023200	0.030232
4 5 6 7 8 9 10 11 12 13 14 15	x1x5 x1x6 x2x3 x2x4 x2x5 x2x6 x3x4 x3x5 x3x6 x4x5 x4x6 x5x6	−0.757600 –0.000332 0.000460 0.002000 –0.117800 0.000254 0.013000 0.090000 –0.001040 –0.640000 0.001300 0.201800	4.354520 0.006225 0.001178 0.003021 0.435095 0.000622 0.019494 2.807754 0.004014 7.201736 0.010296 1.482928

Figure 2. The Pearson’s correlation coefficient, r for the possible combinations of the CPs.

The values of Pearson product moment correlation coefficient, r range between −0.3384 and 0.9592 for all the combinations of the CPs (Figure 2). Seven of the possible combinations have correlation coefficients above +0.4, which implies seven of them have a strong linear dependence/synergistic relationships. In addition, the remaining eight combinations with correlation coefficients between +0.4 and −0.4 depict either a low linear dependence (low correlation) or possibly strong non-linear dependence relationships which call for further investigation. Hence, it is reasonable to conclude that the eight combinations of the CPs have a relationship which is a mixture of linear and other components (probably non-linear or low correlation). The correlation coefficient of +0.4 which delineates the strong linear and low linear dependence in this case is based on statistical principles associated with correlation coefficients in general, though some school of thought based it on a correlation coefficient of +0.5.

In summary, except for the outlier of index 9 that has a r-value of 0.5104, all the small magnitude r-values from Figure 2 (and also coloured cells in the legend to Figure 2 above) involve either CPs x₅ or x₆, and the r-value of index 15 [r(x₅ x₆)] is also very small, 0.1701. An observation of the trend of values of the Pearson’s correlation coefficient, r (Figure 2) for CPs 5 and 6 possible combinations thus suggests that they may likely be unimportant parameters in this analysis even though they might have been selected based on expert judgment and engineering experience. This trend of results is in line with the t-h knowledge that minor heat loss through degradation of piping insulation (x₅) and minor inclination change of piping layout (x₆) have a second – or third-order significance, relative to the other CPs, in this screening analysis of natural circulation phenomena (flow rate and heat rejection rate) of the study system. In addition, in practical sense, considering index 2 [r(x₁ x₃)] in which non-condensable gas degrades heat transfer efficiency of the system and valve closure fraction being a type of flow restriction (degradation of mass flow rate) impact different aspects of NC (heat transfer and friction losses limit flow rate); this is depicted by a relatively high r-value of 0.8527; though statistical coincidence can also not be explicitly ruled out due to available small sample size.

Furthermore, covariances for the combinations are between −0.7576 and 0.2018 (Table 3), which indicates that there exist some kinds of dependency and the few values close to zero does not necessarily indicate independence as in the case of r in Figure 2. Applying the Cauchy-Schwarz inequality rule, since none of the covariance values attain the highest obtainable value of the products of standard deviations (σ₄σ₅ = 7.201,736) as in Table 3, then the variables can be said to have purely linear relationship. However, with some of the covariances very close to zero, the relationships between the parameters can be assumed to either be of some non-linear type, or else the parameters are independent.

Applying the rules governing the generalized variance-covariance matrix [16] as earlier stated (Equation (3)(3) $\left(\begin{array}{cccc}Var(X_1(t))& Cov(X_1(t),X_2(t))& \cdots & Cov(X_1(t),X_n(t))\\ Cov(X_2(t),X_1(t))& Var(X_2(t))& \cdots & Cov(X_2(t),X_n(t))\\ ⋮& ⋮& ⋮& ⋮\\ Cov(X_n(t),X_1(t))& Cov(X_n(t),X_2(t))& \cdots & Var(X_n(t))\end{array}\right)$(3) ), the variance-covariance matrix, V obtained for the combination is,

$V=\left[\begin{array}{cccc}0.018280& 0.001012& 0.008040& 0.023200\\ 0.001012& 0.000183& 0.000460& 0.002000\\ 0.008040& 0.000460& 0.007600& 0.013000\\ 0.023200& 0.002000& 0.013000& 0.050000\\ -0.757600& -0.117800& 0.090000& -0.640000\\ -0.000332& 0.000254& -0.001040& 0.001300\end{array}\begin{array}{cc}-0.757600& -0.000332\\ -0.117800& 0.000254\\ 0.090000& -0.001040\\ -0.640000& 0.001300\\ 1037.300& 0.201800\\ 0.201800& 0.00212\end{array}\right].$

The matrix V above has its entries obtained as follows; all the major diagonal entries are the variances of the key CPs 1, 2, 3,…, 6, and the other entries V(i,j), for instance V(1,2) is the covariance of the CPs 1 and 2 which is also same as V(2,1) and so on.

Through the results of the dependency screening, further investigations and analysis such as the failure probability or phenomenological (t-h) reliability of the system (as desired) can be carried out with the shortlisted combinations of CPs (x₁x₂, x₁x_3, x₁x_4, x₂x_3, x₂x₄ and x₃x₄, i.e. non-coloured outliers in the legend to Figure 2) using desired suitable multivariate analysis (dependency consideration) methods.

4.4 Explanation of the synergistic influence of the identified CPs (physical perspective)

A comprehensive explanation of the practical ways the CPs influence one another on a qualitative basis is given as follows (Table 4), with the six CPs as earlier denoted.

Table 4. The ways the CPs influence one another (their synergistic effects on failure) from t-h point of view

Index	Combination of CPs	Explanation of the CPs influence	Significance/qualitative rated degree of influence
1	x1x2	Non-condensable gas decreases heat transfer coefficient/efficiency of the system. In the same vein, heat exchanger plugging reduces flow of the coolant and consequently heat transfer rate is reduced.	x1x2 has a significant influence^1 on flow rate of the coolant and thus cooling capability is reduced with the two factors.
2	x1x3	Non-condensable gas decreases heat transfer coefficient/efficiency of the system; and valve closure fraction being a type of flow restriction, decreases mass flow rate. As the friction losses limit flow rate, presence of non-condensable gases magnifies reduction of heat transfer in the system.	x1x3 has a significant influence^1 on flow and heat rejection rate and was also displayed by the screening analysis with high r-value.
3	x1x4	Non-condensable gas decreases heat transfer coefficient/efficiency of the system and undetected leakage area reduces the coolant inventory.	x1x4 has a significant influence^1 on cooling capability of the coolant/heat removal system.
4	x1x5	Non-condensable gas decreases heat transfer coefficient/efficiency of the system and piping insulation degradation is somehow complicated on its influence of cooling.	x1x5 has a low significance^2, relative to the other CPs combinations.
5	x1x6	Non-condensable gas decreases heat transfer coefficient/efficiency of the system; and piping layout/inclinations contribute to head losses of likely small reduction in flow rate.	x1x6 has a very low significance^3, relative to the other CPs combinations.
6	x2x3	Heat exchanger plugging reduces flow of the coolant; and valve closure fraction being a type of flow restriction, decreases mass flow rate.	x2x3 has a significant influence^1 on flow rate of the coolant and thus cooling capability is reduced with the two factors.
7	x2x4	Heat exchanger plugging reduces flow of the coolant; and undetected leakage area reduces the coolant inventory.	x2x4 has a significant influence^1 on flow rate of the coolant and heat removal capability of the coolant is reduced with the two factors.
8	x2x5	Heat exchanger plugging reduces flow of the coolant; and piping insulation degradation is somehow complicated on its influence of cooling.	x2x5 though a bit complicated (with insulation degradation), can be considered to have some degree of significance^2 (especially with HX plugging).
9	x2x6	Heat exchanger plugging reduces flow of the coolant; and piping layout/inclinations contribute to head losses of likely small reduction in flow rate.	x2x6 can be considered to have some degree of significance^2 (especially with HX plugging), relative to the other combinations.
10	x3x4	Valve closure fraction being a type of flow restriction (decreases mass flow rate); and undetected leakage area reduces the coolant inventory	x3x4 has a significant influence^1 on flow and coolant inventory; thus heat removal capability of the coolant is reduced with the two factors.
11	x3x5	Valve closure fraction being a type of flow restriction (decreases mass flow rate); and minor heat loss is associated in practice with degradation of piping insulation.	x3x5 has some degree of significance^2 on flow, though a bit complicated (with insulation degradation).
12	x3x6	Valve closure fraction is a type of flow restriction which decreases mass flow rate; and piping configurations contribute to head losses of likely small reduction in flow rate.	x3x6 has some degree of influence^2 on flow (especially with valve closure fraction) with less contribution of piping layout.
13	x4x5	Undetected leakage area reduces the coolant inventory and minor heat loss is associated in practice with degradation of piping insulation.	x4x5 has some degree of influence^2 on heat removal capability especially with undetected leakage.
14	x4x6	Undetected leakage area reduces the coolant inventory while piping layout contributes to head losses of likely small reduction in flow rate.	x4x6 has some degree of influence^2 on heat removal capability especially with undetected leakage.
15	x5x6	Minor heat loss is associated in practice with degradation of piping insulation and minor inclination change of piping layout.	x5x6 can be considered to have a very low significance^3, relative to the other CPs combinations, in this screening analysis of NC phenomena (flow rate and heat rejection rate).

Degree of influence: ¹first-order significance; ²second-order significance; ³third-order significance.

5. Issues in dependency screening of the CPs and outcome of further analysis

As afore-mentioned, scarce real plant and especially inadequate experimental data on t-h phenomena necessitate the use of engineering experience and expert judgment in selecting the key CPs, assigning the CP indicator values, defining the possible (realistic) range for the indicators and setting of the failure thresholds. This practice will have some implications (uncertainties) on the exact nature and value of the quantified results and further possible results like probability of failure or overall reliability [20].

In practice, the dependency of CPs is just one of the several other environmental influence and factors which are most times not considered and this can make the quantified reliability or failure probability inaccurate and less representative. Due to the dependency effects existing among the relevant CPs, a given FM may be initiated even when a CP value is within its permissible range, which often occurs when the FC is set by considering the influence of the CPs independently. Thus, dependency consideration among the different influencing parameters would lead to complications in the selection of the FC [7].

The time-invariant nature (discrete values) of the CPs assumed for simplification will also cause some discrepancies between the estimated and the real value of final results.

Finally, assignment of the suitable pdfs among the selected key CPs cannot be fully certified as true representatives (since they are based on expert judgment and operating experience) as several factors are involved in practice due to the complex nature of the operation and phenomena of the t-h PSSs.

In spite of these issues and challenges, reasonable, and realistic results are expected through dependency consideration of the CPs compared to independency consideration.

6. Conclusions

Dependency screening of the identified CPs that can influence the performance of a simple generic thermal-hydraulic PSS (passively water-cooled steam generator) with randomly assigned realistic permissible values was carried out. Pearson product moment correlation and covariance matrix approach applied to the identified CPs for dependency screening revealed that some of the possible combinations of the CPs are truly dependent while some whose relationships cannot be explicitly determined can be screened out prior further desired failure or reliability analysis. For the generic passively water-cooled steam generator considered in this work, the CPs observed to be dependent from mathematical approaches also show dependency/synergistic effects that can be reasonably explained practically using basic thermal-hydraulics concepts. In addition, for the CPs screened out, their independence or non-correlation can be argued to some extent from practical perspectives too. Furthermore, the effect of the CPs on each other was analysed qualitatively in this study. It is essential to note that a CP that is not correlated with the others can still have a significant influence on the physical phenomena being study, the correlated cases in this study just perfectly fit theoretical and practical explanations in unison.

Though the CPs are dependent, they may be capable of independently degrading the NC phenomena and there is a high possibility of joint degradation impacts of same parameters in such a way that they can have amplifying effects on the associated phenomena when they occur simultaneously.

For the combinations of the CPs observed to be dependent (four out of the total six CPs subjected to the screening), the dependency effect will definitely influence subsequent results of further analysis.

This screening approach justified the need to put the dependency behaviour of the CPs into consideration as much as possible in order to reduce uncertainties and make performance analysis (functional reliability assessment) of the t-h PSSs more realistic, dependable and of practical significance/application. The output of the screening analysis will be useful as inputs for various t-h reliability methods and application of artificial neural network (ANN) – involving building and training of the selected network parameters. Furthermore, incorporation of this screening approach into the existing t-h RAM will therefore improve their computational efficiency and accuracy.

It is recommended that other possible screening methods should be considered (as supplements to the correlation and covariance approaches) to systems of this nature as well as investigating their synergistic or dependency nature practically (from t-h perspective) to make the screening procedure much more dependable and realistic. Furthermore, screening of CPs or other influencing factors can also be improved on by quantifying synergistic effects of the associated CPs/influencing factors especially in cases where there are reliable experimental data for the study system.

Acknowledgments

This study was supported in part by Nuclear Power Plant Living-PSA and Online Risk Monitor and Management Technology Research, under National Science and Technology Major Project of China (2014ZX06004-003).

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by the Nuclear Power Plant Living-PSA and Online Risk Monitor and Management Technology Research, under National Science and Technology Major Project of China [2014ZX06004-003].

Appendix

Important statistical variables

1. Pearson’s product moment correlation coefficient, r [16]:

(1`) $r_i=\frac{S_{xy}}{S_x{S}_y}=\frac{\sum \left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)}{\sqrt{\sum {\left(x_i-\bar{x}\right)}^2{\left(y_i-\bar{y}\right)}^2}},$(1`)

where $S_{x_i{x}_j}=Cov(x_i,x_j)=\frac{1}{n}\sum _{k=1}^n(x_{ki}-{\bar{x}}_i)(x_{kj}-{\bar{x}}_j),$

and S_i is the standard deviation, n is the sample size, x_i and x_j are sets of the observed CPs and k is individual terms in vectors i and j.

2. Continuous Distribution Functions

(i) The s-normal pdf, otherwise called the central limit theorem is:

(2a`) $f(x)=\left(\frac{1}{\sigma \sqrt{2\pi }}\right)\ast e^{-\left[\frac{{(x-\mu )}^2}{2{\sigma }^2}\right]},$(2a`)

while the curtailed s-normal or truncated normal distribution (TND), is given as

(2b`) $f_{DTN}(x)=\left\{\begin{array}{cc}0,& -\mathrm{\infty }\le x\le x_L\\ f(x)/[F(x_R)-F(x_L)],& x_L\le x\le x_R\\ 0,& x_R\le x\le +\mathrm{\infty }\end{array}\right.,$(2b`)

Where x_L and x_R represent the lower and the upper thresholds for parameter x.

The associated statistical equations are:

Location parameter = estimator of the mean,

(3`) $\mu =\frac{\sum x_i}{n}.$(3`)

And estimator of the variance,

(4`) ${\sigma }^2=\frac{1}{n-1}\sum (x_i-\mu {)}^2,$(4`)

For parameters observed from a sample x_i with population n.

(ii) The log-normal distribution (LND), having a better fit to reliability data is expressed as

(5`) $f(x)=\left(\frac{1}{\sigma x\sqrt{2\pi }}\right)\ast e^{-\left[\frac{{(lnx-\mu )}^2}{2{\sigma }^2}\right]}forx\ge 0.$(5`)

The associated statistical equations are:

Location parameter (mean),

(6`) $E[X]=exp\left(\mu +\frac{{\sigma }^2}{2}\right);$(6`)

And variance,

(7`) $V[X]=[exp(2\mu +2{\sigma }^2)-exp(2\mu +{\sigma }^2)],$(7`)

for parameters observed from a sample x_i with population n.

(iii) The exponential distribution is of the form,

(8`) $f(x)=\lambda exp(-\lambda x),$(8`)

where $\lambda$ is the failure rate.