Introduction

Digital system gives us enormous benefit, on the other hand, it also brings us challenge. Once accident is taken place by the digital system, tremendous disaster will be occurred, for instance1 the accident at Three Mile Island (TMI) in 1979, the nuclear accident at Chernobyl in1986. Although the reliability of hardware and software systems is gradually improved with social progress, human beings have uncertain features, then an accident arising from human errors is very easily to occur. Just data shows that 60–90% accidents can be attributed to human error2.

Current industry digital systemic design is converting physical and perceptive types into psychological and cognitive types. Digital Human-Computer Interaction is very important and has effects on the occurrences of accidents. Its reliability greatly depends on human inner factors and crew influence. Digital systems have many advantages that are listed as follows3: Firstly, there are some navigation systems that can instruct operators to get into a certain graphic interface; Secondly, it has good query function; thirdly, it has the function of adjusting pictures; Finally, there are shortcut ways that can easily manage interface management tasks. Digital Control System of TianWan nuclear power plant in China has successfully been applied, As well as the extending project of other nuclear power plant will introduce digital control system. On the other hand, computer console brings a new challenge that focuses on the change of cognitive behavior for operators. Thus, previous cognitive models or influencing factors can be not suitable for digital system cognitive reliability evaluation. In traditional reliability research, Fault Tree Analysis4 (FTA) generally considers researched objects as only two states (absolute failure or absolute reliability). It is very simple to divide system states into success and failure. Many scholars have proposed some methods about multi-state’s systematic reliability and dynamics5,6,7,8. Recently, Bayesian theory has been successfully applied to fault diagnosis, data mining, artificial intelligence and reliability research. Bayesian Networks can easily signify randomness, uncertainty and relevance, and reliability evaluation. Bayesian Networks (BN) can better make up for the deficiency of previous evaluating methods9.BN can do well in combining with conditional probability distributive tables, can better express the relation between system and each factor, and can make the expression of many states simpler and clearer.

For the above-mentioned situation, taking nuclear power plants as the research background, this paper presented human cognitive reliability model in digital system based on Bayesian theory. This model can provide methodological guidance for operators’ cognitive reliability analysis of nuclear power plant and provide assistance in analyzing human error probability of cognitive process. This research has some distinct characteristics that are listed as follows: Firstly, the proposed cognitive reliability model considered the crew effects on operators; Secondly, this proposed model is symmetrical in term of failure and success; Finally, the condition probabilities were obtained through Bayesian inference and the contribution rate of influencing factor level corresponding to a cognitive stage of a task.

Related works

Cognitive science is at the leading edge of current scientific research. It primarily researches the human cognitive process of psychological inner mechanism and is involved with philosophy, psychological, information science, neurology, linguistics, computer science, evolutionary biology and animal behavioral biology etc. Baron et al.10 proposed process-oriented crew model that consists of three members by simulating the crew behavior. The model mainly researched crew communication processes and performance influence factors. In 1984, EPRI11 developed the HCR (Human Cognitive Reliability) model that is one of models that can quantitatively analyze human reliability. HCR model divides the behavior of operator into skill, rule and knowledge types and offers a tool of human reliability analysis for man-machine interface in a NPP (Nuclear Power Plant). As perception, analysis and decision making usually are made up of a human various abilities, the model is very difficult to category the skill, rule and knowledge types. On the other hand, the model only ponders permitting time and executing time, neglects specific task features and absolute time effect on human error probability. Philips et al.12 proposed STAHR method. This method has the following characteristics: (1) it relies on subjectivity and psychological conditions; (2) it has strong sensitivity. Straeter13 proposed an accident analysis and cognitive reliability method (Cognition Assessment Human Research, CAHR) in his PhD thesis. The method is based on the incidents occurred in German NPPs, it can quantitatively analyze human errors probability according to a psychological model. Blom et al.14 proposed ATM operational risk assessment based on Scenario and Monte Carlo event hazards. In fact, the security assessment is decided through the authorization of the operator. Later, he defined the hazard of relevant operation and gathered the impact factors. The model is based on human cognitive reliability analysis of cognitive lever in correlation context and is combined with analyzer impact risk model. Goldberg et al.15 suggested a computational model, this model is different from the statistical and mathematical model, because the model considers the influencing factors of human behavior. The model simulates the behavior characteristics by computer program. Shu et al.16 advanced team behavior network model. The model consists of four parts: the task execution model, the initial event model, posteriori event exploitative model and team model of human-machine interface. The model mainly focuses on team cognitive process in learning and recognizing representation. The same year Gazzaniga et al.17 presented a calculation model in cognitive process. In the model the computer must perform internal factors of operators when inputting. Researchers may estimate the match degree of thinking and behavior by observing the change in behavior. Avi Ma’ayan et al.18 proposed Comprehensive Dominant Cognitive Method. The method described various forms of creative cognitive and computer integration and the ways to obtain the signal path. Andrei khrennikov19 raised quantitative cognitive models about decision-making and information process. The model mainly researched digital measurement and decision-making quantitative analysis based on psychological context and psychological exchange. Hyun-ChuL Lee et al.20 advanced the computational model of effective assessment on the attention, memory, and mental status for nuclear power plant operators. Furthermore, the method used the BN in evaluation and posed the important influence factors of operator’s assessment, such as attention, the recession of work memory, mental condition. In 2017, Zwirglmaier et al. illustrated how Bayesian networks (BNs) can be used to qualitative causal paths to provide traceability, the proposed BN structure addressed the need for causal traceability and strong scientific basis in HRA, and illustrated how the model could be quantified with a combination of expert-probabilities and information21.Jiang et al. propose a cognitive reliability model with influencing factors based on Bayesian network, this model provides a simple and feasible approach to analyze cognitive reliability of operating process in digital human–computer interface22.Alvarenga and Melo23 surveyed how HRA models stand in face of the cognitive psychology models and their internal structures in human information processing and proposes the training of parameters of a specific cognitive architecture (ACT-R) with the scenarios of human activities in the external world to provide human error probabilities. Liu et al.2 presented a cognitive reliability model for the DCS + SOP system in the Ling’ao Phase II Nuclear Power Plant’s main control room, the model is based on the cognitive process with respect to considering the coordinator’s accident recovery effect. Liu et al.3 proposes a method for human reliability analysis (HRA) of different cognitive Stages. This method constructed the influencing factors of three cognitive processes and evaluates the weights of the influencing factors through an analytic hierarchy process (AHP). In 2021, Wang et al.24 proposed a method for the reliability assessment of MMPMS (man-machine phased-mission systems) to address the phase dependencies of human cognitive error. A decision tree is used to quantify the dependence level, the Bayesian network is used to construct the system reliability model. Based on the CREAM method and Bayesian network, Chen et al.25 proposed a human reliability evaluating method and risk prediction technology for deep submergence operating system. This method is based on CREAM method, it can well identify and interdict the key risk points for related technicians.

Above studies have applied BN to resolve related problems, but crew influence and digital characters were rarely considered. For this purpose, based on the above research achievements, this paper proposed BNs cognitive reliability model for digital human-computer interaction. The key to the model lies in establishing a reliability analysis method considering crew influence, it can effectively analyze the cognitive reliability of digital human-computer interaction of Npps.

Cognitive reliability model considering crew influence in digital nuclear power plant

Cognitive phases

In digital human-computer interaction, human cognitive behavior will be changed. In addition to traditional cognitive ways, operators need to interact with software systems, decision support systems, navigation system, computer workstation, display, program analysis, etc.

Edwards26 proposed SHELL model. Later, this model was developed by Hawkins27. Hawkins considered the software equipment, factory conditions and the PSFs (Performance Shaping Factors, PSFs). Later, the model was improved by Caccibue. Now it should be better cognitive process model that is shown by Fig. 128:

Fig. 1
Fig. 1
Full size image

Improved SHELL mode.

Although Caccibue considered some digital factors for model, he failed to consider crew collaborative. To adapt to this situation, this paper proposes a cognitive model with crew influencing factors. Generally, cognitive process is divided into three or four phases, according to related literatures3,29, the cognitive phase in this paper was defined as three phases, the cognitive process is shown in Fig. 2.

Fig. 2
Fig. 2
Full size image

Cognitive process.

Influencing factors

It is shown in Fig. 2 that the cognitive process is divided into three stages: monitoring, decision-making, and execution. To determine influencing factors, firstly, survey and related methods (such as THERP, HCR, CREAM, IDAC) were adopted, secondly, five scholars were interviewed. In other words, initial influencing factors were obtained by survey and related research achievements, later, the evaluations were performed by five scholars specializing in human reliability analysis (HRA); each had conducted research on HRA methods for main control room operators in nuclear power plants for over five years. If the number of scholars who disapproved of an initial influencing factor as final influencing factor for a cognitive phase or crew influence reached three, the initial influencing factor would be eliminated. By analyzing initial influencing factors and scholar opinions, the influencing factors of human reliability including crew effect and three cognitive stages are shown in Table 1.

Table 1 Influence factors of cognitive process in digital system.
Full size table

Cognitive reliability model considering crew influence

As cognitive process is divided into three phases, cognitive reliability analysis should include three phases then cognitive reliability analysis method is written as follows:

$${{\text{p}}_{e\_co{g_{tas{k_k}}}}}={{\text{r}}_{crew\_tas{k_k}}}*({p_{e\_{m_{tas{k_k}}}}}+{p_{e\_d}}_{{_{{tas{k_k}}}}}+p{}_{{e\_{e_{tas{k_k}}}}})$$
(1)

Where, \({{\text{p}}_{e\_co{g_{tas{k_k}}}}}\)denotes error probability of cognitive process of the kth task; tk is the kth task; \({{\text{r}}_{crew\_{t_k}}}\)indicates crew influencing degree of the kth task; \({p_{e\_{m_{tas{k_k}}}}}\)is error probability of monitoring process of the kth task; \({p_{e\_{d_{tas{k_k}}}}}\)presents error probability of decision-making process of the kth task; \({p_{e\_{{\text{e}}_{tas{k_k}}}}}\)indicates error probability of execution process of the kth task.

To obtain human reliability of each cognitive phase, Bayesian networks (BNs) are used for calculating error probabilities of each cognitive process. Bayesian networks, also known as Bayesian Belief Network (BBN), is a directed acyclic graph (DAG). In a Bayesian network, the node that is not directed by any edge is called root node. Every node has a priori probability table or related functions. In all parent nodes, each variable has a conditional probability table that clearly shows causal dependencies of probability among variables. Let us consider a BN, U={x1,……xn}, where x1,……xn represent node. Then, by the chain rule, relation probability p{x1,……xn} is as follows:

$$P(U)=p\{ {x_1},{x_2}......{x_n}\} =\sum\limits_{{i=1}}^{n} {p({x_i}} |p{a_{{\text{ij}}}})$$
(2)

Where, xi denotes the ith task or node; p.a.ij is the jth parent nodes of xi.

We can see from TABLE I that each cognitive process has several influencing factors, (knows as parent nodes). Assuming multi-parent nodes are remarked by M1, M2 … MK respectively, then the Mj node has some states, namely,\({S_{{M_{1,j}}}},{S_{{M_{2,j}}}},......{S_{{M_{i,j}}}}\)(j = 1……k, j is state level). For each task node, it is great difficult to calculate the conditional probability. But, according to the theory33, when a node has two parent nodes in Bayesian networks, namely, B and C, the probability can be calculated by the Eq. (3):

$$P(A|B,C)=\lambda P(A|B)P(A|C)$$
(3)

λ is the standardization correction factor, it often is used to determine \(\prod\nolimits_{{a \in A}} {P(A|B,C)}\);P(A|B, C) denotes error probability of a task known as A influenced by B and C.

Equation (3) can be further extended to a general form:

$$\begin{gathered} P(A|{X_1},{X_2},......{X_n}) \hfill \\ \quad =\lambda p(A|{X_1})P(A|{X_2})......P(A|{X_n}) \hfill \\ \end{gathered}$$
(4)

As shown in Eq. (4), a node conditional probability with multi-father nodes can be obtained by conditional probability of a single father34. A node condition probability with multi-father nodes can combine with the theory of BN probability advanced by thand Sankaran Mahadevan35, then the node condition probability with Multi-father nodes can be obtained:

$$P(t_{k} |F_{1} = S_{{F_{1},v}},F_{2} = S_{{F_{2},v}},......F_{J} = S_{{F_{J},v}} ) = \lambda \prod\limits_{{j = 1}}^{{j = n}} {P(t_{k} |F_{j} = S_{{F_{j},v}} )}$$
(5)

Where Tk is the kth task; Fj denotes the jth influencing factor; v represents the vth level of a influencing factor;\({S_{{F_j},v}}\)indicates the status of the vth level of the jth influencing factor.

Based on Eqs. (4), (5) can be abbreviated as:

$$\begin{gathered} {\text{p}}({t_k}|{F_1},{F_2},......{F_j})=\lambda \prod\nolimits_{{j=1}}^{{j=n}} {P({t_k}} |{F_j}={S_{{F_{{\text{j}},v}}}}) \hfill \\ \quad =\lambda p({t_k}|{F_1}={S_{{F_{1,v}}}})*P({{\text{t}}_k})|{F_2}={S_{{F_{2,v}}}})*......p({t_k}|{F_j}=={S_{{F_{j,v}}}}) \hfill \\ \quad =\lambda p({t_k}|{F_1})*P({{\text{t}}_k})|{F_2})*......p({t_k}|{F_j}) \hfill \\ \end{gathered}$$
(6)

According to related achievements36, reliability demonstrates an exponential distribution, such as system/ product reliability, further, several research results indicated human error was exponential distribution2,3,22,29,37, then, based on Eq. (6),the calculation method about \({p_{e\_{m_{tas{k_k}}}}}\)in Eq. (3) is:

$$\begin{gathered} {{\text{p}}_{e\_{m_{tas{k_k}}}}}{\text{=p}}({t_k}|{z_1},{z_2},......{z_j})={\lambda _m}\prod\nolimits_{{j=1}}^{{j=n}} {P({t_k}} |{z_j}={S_{{z_{{\text{j}},v}}}}) \hfill \\ \quad ={\lambda _m}p({t_k}{^{{}}_{}}|{z_1}={S_{{z_{1,v}}}})*P({{\text{t}}_k}|{z_2}={S_{{z_{2,v}}}})*......p({t_k}|{z_j}=={S_{{z_{j,v}}}}) \hfill \\ \quad ={\lambda _m}[p({t_k}|{z_1})*P({{\text{t}}_k})|{z_2})*......p({t_k}|{z_j})] \hfill \\ \quad ={\lambda _m}*[{{\text{e}}^{ - v\_{z_1}*w\_{z_1}}}*{{\text{e}}^{ - v\_{z_2}*w\_{z_2}}}......*{{\text{e}}^{ - v\_{z_j}*w\_{z_j}}}] \hfill \\ \quad ={\lambda _m}*{e^{ - (v\_{z_1}*w\_{z_1}+v\_{z_2}*w\_{z_2}+......v\_{z_{\text{j}}}*w\_{z_j})}} \hfill \\ \end{gathered}$$
(7)

Where, zj is the jth influencing factor of monitoring process; λm indicates an error correction coefficient of monitoring process;v_zj presents a value of the jth influencing factor of the kth task; w_zj is a weight value of the jth influencing factor.

Similarly,\({p_{e\_{d_{tas{k_k}}}}}\),\({p_{e\_{{\text{e}}_{tas{k_k}}}}}\).\({{\text{p}}_{crew\_{t_k}}}\)in Eq. (1) can be respectively written as:

$${{\text{p}}_{e\_{d_{tas{k_k}}}}}={\lambda _d}*{e^{ - (v\_{d_1}*w\_{d_1}+v\_{d_2}*w\_{d_2}+......v\_{d_{\text{j}}}*w\_{d_j})}}$$
(8)
$${{\text{p}}_{e\_{e_{tas{k_k}}}}}={\lambda _e}*{e^{ - (v\_{a_1}*w\_{a_1}+v\_{a_2}*w\_{a_2}+......v\_{a_{\text{j}}}*w\_{a_j})}}$$
(9)
$${{\text{r}}_{{\text{cr}}ew\_{t_k}}}={\lambda _{crew}}*{e^{ - (v\_{g_1}*w\_{g_1}+v\_{g_2}*w\_{g_2}+......v\_{g_j}*w\_{g_j})}}$$
(10)

Similarly, for decision-making process: λd indicates an error correction coefficient of decision-making process; v_dj presents a value of the dth influencing factor about the kth task; w_dj is a weight value of the jth influencing factor.

For execution process, λe indicates an error correction coefficient of execution process; v_aj presents a value of the jth influencing factor about the kth task; w_aj is a weight value of the jth influencing factor.

For crew influencing degree, λcrew indicates a correction factor of crew influence degree; v_gj presents a value of the jth influencing factor about the kth task; w_gj is a weight value of the jth influencing factor.

The values about v_z j, v_d j, v_a j, v_g j in Eqs. (7)–(10)

General, influencing factors are divided into 3–9 levels, the levels of influencing factors including crew, monitoring, decision-making and execution in this paper were divided into four different levels (excellent, good, moderate, poor), the value ranges of each level are [0.75,1], [0.6,0.74], [0.4,0.59] and (0,0.4), respectively3.The steps obtaining each influencing factor value are as follows: (1) based on a task scenario, a level of each influencing factor is determined by operators or experts; (2) the better the performance of influencing factors is, the larger the values range of which is; (3) based on predefined ranges for each influencing factor, quantitative ratings were independently assigned by subject matter experts—either certified nuclear power plant main control room operators with ≥ 10 years of operational experience, and HRA researchers with ≥ 5 years of dedicated methodological expertise in human reliability analysis for main control room operators.

The weights calculating method of influencing factors (w_z j,w_d j,, w_a j, w_g j)

Up to now, there are some methods to obtain influencing weights, such as subjective experience method, AHP (Analytic Hierarchy Process), DEMAETL (Decision Making Trial and Evaluation Laboratory). The weights of influencing factors of each cognitive stage and crew in this paper are obtained using the Entropy value method, the main steps of which are as follows38:

Step1: Standardization of influencing factors observation values.

Standardization of influencing factors values adopts extremely large processing method:

$$v\_x_{{ij}}^{\prime }=\frac{{v\_{x_{ij}} - \hbox{min} (v\_{x_{ij}})}}{{\hbox{max} (v\_{x_{ij}}) - \hbox{min} (v\_{x_{ij}})}}$$
(11)

V_xij is an observation value of the ith evaluation object on the jth influencing factor; max(v_xij) denotes the maximum value of v_xij; min(v_xij) denotes the minimum value of v_xij.

Step2: Calculating feature proportion of xij.

$$f{e_{ij}}=\frac{{v\_{x_{ij}}^{\prime }}}{{\sum\nolimits_{{i=1}}^{n} {v\_{x_{ij}}^{\prime }} }}$$
(12)

Step3: Calculating entropy value of the jth influencing factor.

$$e{n_j}= - t\sum\limits_{{i=1}}^{n} {f{e_{ij}}} *In({p_{ij}})$$
(13)

Where, n is number of evaluation objects; t = 1/In(n), t > 0,enj>0.

Step4: Calculating the coefficient of difference.

$$c{o_j}=1 - e{n_j}$$
(14)

Step5: Determining the weights.

$${w_j}=\frac{{c{o_j}}}{{\sum\nolimits_{{i=1}}^{n} {c{o_j}} }}$$
(15)

Where, w indicates the standardization weigh of the jth influencing factor.

Determining the weights of influencing factors (w_z j,w_d j,, w_a j, w_g j)

The weights of influencing factors including crew and three cognitive phases were obtained by questionnaires and entropy value method mentioned above. A total of 10 questionnaires were distributed, 10 of them were valid. All experts or scholars have good experience on digital Npps.

Because there were several cognitive phases, this paper only used monitoring process as an example of acquiring the weights of influencing factors, the following steps were used:

(1) Initial value.

Initial values of each influencing factor were given by experts and scholars, a given value was based on the importance of each influencing factor. The range of values is defined as between 1 and 10, The more important the influencing factor is, the larger the given value will be. Based on the questionnaire, initial values of influencing factors in monitoring process were listed in Table 2.

Table 2 Questionnaire data of monitoring process.
Full size table

(2) Feature proportion results of influencing factors in monitoring process.

According to Table 2 and Eqs. (11) and (12), feature proportions of influencing factors in monitoring process see Table 3.

Table 3 Feature proportion result of monitoring process.
Full size table

(3) Entropy value.

Based on Table 3 and Eq. (13), the entropy value results were listed in Table 4.

Table 4 Entropy value of monitoring process.
Full size table

(4) Coefficient of difference.

Based on Table 4 and Eq. (15), the coefficient of differences were listed in Table 5.

Table 5 Entropy value of monitoring process.
Full size table

(5) Weights of influencing factors.

Based on Table 5 and Eq. (15), the weights of influencing factors in monitoring process were listed in Table 6.

Table 6 Weights of influencing factors in monitoring process.
Full size table

Similarly, the weight values including decision-making process, execution process, and crew influence see Tables 7, 8 and 9, respectively.

Table 7 Weights of influencing factors in decision-making process.
Full size table
Table 8 Weights of influencing factors in the execution process.
Full size table
Table 9 Weights of crew influencing factors.
Full size table

Determining the values of parameters including λm,λd,λe,λcrew

Obviously, In Eqs. (7), (8) and (9), except for λm, λd or λe, the order of magnitude of calculation result is 10− 1. Generally, order of magnitude about mean error probability is 10− 2 in digital system39,40,41,42. in order to reach 10− 2, for the sake of conservatism, the values of λm, λd and λe are 0.1. Crew influencing degree is as a correction coefficient of cognitive process, it is not an error probability, then Eq. (10) do not consider correction factor, so λcrew is defined as 1.

Results and discussions

Expert judgment

Literature review and expert judgment are the main research method in this study43. Expert-based methods play a vital role in Human Reliability Analysis (HRA), despite being a subject of debate44,45. A variety of HRA techniques, such as SLIM46and A Technique for Human Error Analysis (ATHEANA)47, depend significantly on expert judgment. Expert judgment is indispensable in HRA, particularly for its qualitative components48. As emphasized in SPAR-H, enhancing the qualitative evaluation of Performance Shaping Factors (PSFs) for particular issues is essential49. The expert-based method is particularly effective for developing the qualitative aspects of PSFs. Its rationale lies in the collective intelligence of groups, where individuals, under suitable conditions, can arrive at accurate assessments and predictions50,51.

Influencing factors

The determined influencing factors went through two steps: (a) initial influencing factors were obtained by related research achievements and surveys; (b) each initial influencing factor was judged by five scholars.

Initial influencing factors were based on classic research achievements and considered digital background. Further, each scholar has related experiences of digital Npps for over five years. Obviously, the process obtaining initial influencing factors is sufficient and reasonable. Later, each initial influencing factor was evaluated according to five scholars’ opinions, only when the number of scholars who agree reaches 3, can the initial influencing factor be retained. The judgment criteria are reasonable, then, to a certain extent, the obtained final influencing factors is convincing.

Weights of influencing factors

Currently, there are various methods to obtain weight of influencing factors, such as AHP, DEMATEL, Subjective evaluation method, factor analysis method, Entropy value method, etc. The objects analyzed by AHP method have characteristics such as diverse attributes and complex structures, etc. DEMATAL method is used to solve complex and practical problems, discussing the impact of one factor on all other factors. For subjective evaluation method, the weights of influencing factors were obtained through expert evaluation and judgment, which has a certain degree of subjectivity. Factor analysis method can condense multiple influencing factors into several comprehensive influencing factors and use variance interpretation rate to obtain the weights of each influencing factor. To minimize complexity and subjectivity as much as possible, Entropy value method was used for obtaining the weights of influencing factors. So, the chosen method in this paper is relatively reasonable. The questionnaire was filled out by experts and scholars with experience in digital Npps, which improved the accuracy of the data, so the obtained weights of each cognitive phase and crew influence are reasonable.

Performance analysis of cognitive reliability model

To describe variation trend of error probability about cognitive process, the changing trends were shown in Figs. 3 and 4.

Fig. 3
Fig. 3
Full size image

Variation trends of error probability.

Fig. 4
Fig. 4
Full size image

Error probability variation trends when crew influencing degrees are in the best and worst cases.

As shown Figs. 3 and 4, the error probability includes monitoring, decision-making and execution process decreases as the influencing factor input value increases, it is reasonable. A larger value indicates a higher level of influence factors, a higher level of influencing factor indicates a fewer error rates; on the other hand, the greater influencing degree of crew is, the less the error probability is, which is reasonable, because, when the better the level of crew influencing factors is, the greater the influencing degree of crew is according to Eq. (11), further, from a realistic perspective, when the level of influencing factors is better, the error probability is lower.

Case analysis

  1. (1)

    Determining an analyzed task or scenario.

    To describe the analysis process of proposed method, this paper used a steam generator tube ruptures (SGTR) accident of an NPP as an example. Further, a safety injection (SI) scenario that is a part of SGTR accident process was used for an analyzed object.

  2. (2)

    Determining cognitive process.

    According to the author’s simulator observation, the operation process of SI should include three cognitive phases, so the error probability of each cognitive process including monitoring, decision-making and execution process need to be calculated.

  3. (3)

    Determining the values of influencing factors for related cognitive phases.

    The purpose of this section is to illustrate the application process of the cognitive reliability model, so the input values of each influencing factor corresponding to a level were not obtained through operator surveys. For scenario data of SI, the value of each influencing factor at a certain level mainly came from similar data collected by the author before. If the previous data collected did not contain the data of influencing factors in this paper, they were given by operators after simulator training. Table 10 shows the values of influencing factors including monitoring process, decision-making process, execution process and crew.

  4. (4)

    Calculating error probabilities of each cognitive phase and crew influence degree for SI.

    1. (a)

      Crew influence degree.

      Based on Eq. (10), Tables 9 and 10, influencing degree of crew is as follows:

      $${r_{{\text{cr}}ew\_{t_k}}}=1*{e^{ - (0.82*0.191+0.85*0.173+0.8*0.079+0.92*0.384+0.85*0.173)}}=0.42$$
    2. (b)

      Error probability of monitoring process.

      Based on Eq. (7), Tables 6 and 10, error probability of monitoring process is as follows:

      $${{\text{p}}_{e\_{m_{tas{k_k}}}}}=0.1*{e^{ - (0.45*0.124+0.85*0.119+0.6*0.337+0.65*0.113+0.8*0.128+0.45*0.179)}}=0.0541$$
    3. (c)

      Error probability of decision-making process.

      Based on Eq. (8), Tables 7 and 10, error probability of decision-making process is as follows:

      $${{\text{p}}_{e\_{d_{tas{k_k}}}}}=0.1*{e^{ - (0.75*0.248+0.75*0.125+0.8*0.074+0.95*0.095+0.65*0.312+0.8*0.146)}}=0.0482$$
    4. (d)

      Error probability of execution process.

      Based on Eq. (9), Tables 8 and 10, error probability of execution process is as follows:

      $${{\text{p}}_{e\_{e_{tas{k_k}}}}}=0.1*{e^{ - (0.45*0.285+0.6*0.193+0.5*0.164+0.7*0.142+0.5*0.104+0.75*0.112)}}=0.0571$$
    5. (e)

      Calculating the cognitive error probability of SI.

      Based on Eq. (1) and above calculation results of error probabilities, cognitive error probability is written as:

      $$\begin{aligned} {{\text{p}}_{e\_co{g_{tas{k_k}}}}}&={{\text{r}}_{crew\_tas{k_k}}}*({p_{e\_{m_{tas{k_k}}}}}+{p_{e\_d}}_{{_{{tas{k_k}}}}}+p{}_{{e\_{e_{tas{k_k}}}}}) \hfill \\ &=0.42*(0.0541+0.0482+0.0571)=0.0669 \hfill \\ \end{aligned}$$

      By analyzing the reliability of SI, the error probability of the SI task is 0.0669, its result range of order of magnitude is reasonable, because, generally, order of magnitude about error probability is between 10− 3 and 10− 2 in digital system interaction process.

Table 10 The values of influencing factors.
Full size table

Conclusions

To provide a cognitive reliability analysis method for digital nuclear power plants, this paper presented a cognitive reliability analysis model. Through this research, main achievements of this study included the following: (1) cognitive phases were determined; (2) the influencing factors of each cognitive phase were defined; (3) a cognitive reliability model was proposed; (4) a calculation method of error probability was presented based on Bayesian networks; (5) the weights of influencing factors were determined; and (6) the proposed method was analyzed to be a better performance. On the other hand, the proposed model also has some shortcoming, for instance, the influencing factors considered in this paper may not be very comprehensive, the value of the correction coefficient is somewhat conservative. In future work, the authors will further extend the model, improve influencing factors, amend correction coefficient by collecting data, and further consider its application in some specific areas.