Human reliability assessment in hydrogen refuelling stations: A system dynamic approach

This study develops a methodology using a system dynamics approach to analyse human error probability (HEP) within the context of hydrogen fuelling station (HFS) maintenance, mainly focusing on the dynamic nature of performance shaping factors (PSFs) over time. The developed model offers a comprehensive understanding of how diverse human factors dynamically influence task performance, yielding critical insights for safety management strategies to study 8-hour day shifts in maintenance activity. The study explores the intricate relationship between time-dependent PSFs and human error probability, highlighting that as the number of maintenance tasks rises, so does the potential for increased fatigue levels, subsequently elevating HEP. Introducing breaks during work emerges as a promising intervention to mitigate task-related fatigue, reducing HEP. Careful break implementation is necessary to prevent shifts from extending, which could inadvertently raise HEP. This research has implications extending beyond HFS, benefiting industries where operational safety and efficiency are paramount. Future studies can build upon these findings, exploring additional interventions and diverse work scenarios to advance our understanding of human performance and error prevention strategies, ultimately fostering safer and more productive work environments. The integration of system dynamics and the insights gained contribute significantly to hydrogen safety and offer a robust foundation for future investigations in this field.


Introduction
The global pursuit of sustainability and clean energy has sparked investigations into distinct energy resources, incorporating wind, water, sunlight, biomass, and geothermal energy [1][2][3][4].Within this context, hydrogen has become a clean energy alternative [5] due to its capacity to generate negligible greenhouse gas emissions.As hydrogen is gaining popularity in the transportation industry, the development of infrastructure, particularly hydrogen fuelling stations (HFS) in urban areas, presents substantial challenges.Guaranteeing the secure operation and management of hydrogen at these stations is of utmost importance, demanding a comprehensive grasp of various operational processes and the associated risks [6].The complexities surrounding the secure installation and risk assessment of hydrogen fuelling stations have been deliberated by Ref. [7], underscoring the need for the development of risk analysis processes and implementation of efficient safety measures which is also critical to enhance public acceptance of hydrogen as a fuel [8].
Mistakes during maintenance and repair activities in hydrogen fuelling stations can potentially result in serious consequences.Existing research has emphasised that human errors constitute the predominant factor behind numerous accidents occurring during maintenance activities [9].To precisely evaluate and compute the risk at any stage of maintenance operations in these facilities, it is imperative to accurately estimate the human error probability (HEP).The analysis of HEP offers valuable insights into the efficacy of current maintenance protocols, facilitating the identification of areas necessitating enhancement to bolster the overall reliability and safety of the system [10][11][12][13].Additionally, an evaluation of how human variables (i.e.workload, level of stress, environmental conditions, and impact maintenance errors) allows for designing ergonomic solutions and creating optimized work environments that reduce the probability of situations prone to errors [14][15][16][17][18][19].The comprehensive examination of HEP in maintenance operations proactively enhances safety processes in these situations, streamlines maintenance practices, and significantly diminishes the likelihood of errors and their ensuing consequences.Several human reliability assessment techniques have also been developed to estimate the HEP.Prominent instances encompass the Cognitive Reliability and Error Analysis Method (CREAM) [20], the Technique for Human Error Rate Prediction (THERP) [21], the Success Likelihood Index Method (SLIM) [22], and the Human Error Assessment and Reduction Technique (HEART) [23].Several current techniques used to estimate the HEP are static, disregarding the variability of influential factors such as stress, workload, fatigue, and others over a given period.These influential factors exhibit temporal changes that can substantially affect HEP estimation in a particular period.Consequently, the development of advanced and dynamic approaches becomes imperative to account for the time-varying nature of these factors and enhance the accuracy of HEP estimation.
The primary objective of this paper is to develop a robust framework for human reliability analysis (HRA) during the maintenance of a system.The System dynamics approach has significant capabilities for facilitating a more precise understanding of system conditions for operational improvement and optimal decision-making [24] This method is deemed appropriate for simulating the dynamic error of humans, and its simulation allows multiple scenarios to be explored in real time.The concept of system dynamics was initially introduced by Forrester et al. [25] as an approach to numerical analysis of complex issues.The researchers contended that conventional methodologies exhibited limitations in understanding the tactical practices essential in complex systems [26].System dynamics integrates feedback loops encompassing both balancing and reinforcing loops, enabling the assessment of time delays and non-linear associations [27].System dynamics is widely applied in diverse domains such as project management, corporate growth, technology diffusion, supply chain design, and transportation policy [28].Moreover, it is employed in specific contexts including but not limited to nuclear power plant management [1]; optimising the operation and maintenance of offshore wind farms [29]; coal mining operations [27,30]; the transition of sustainable energy sources, environmental assessment, educational planning, business administration and managerial decision-making [26].Additionally, system dynamics robustly addresses petroleum exploitation [31], and the oil market [32].For various projects, constituting an ongoing process characterised by rigorous testing and the establishment of confidence in the model's reliability [33].However, despite the potential benefits, the application of system dynamics to HRA is underrepresented in the literature, with only a limited number of studies exploring the dependencies among PSFs in human reliability analysis [34] and the integration of social factors, such as safety culture, and structural features, such as safety practices, of an organisation into probabilistic risk assessment [27,[35][36][37] addressed the limitations of traditional HRA methods by employing a qualitative system dynamics model with Causal Loop Diagrams and used a systematic approach to capture the side effects of maintenance errors of a complex system.Although their approach provided a visual and analytical framework to explore the interdependencies and feedback mechanisms influencing human performance in systems, the model did not quantify the maintenance error model.To the best of the authors' knowledge, the application of system dynamics to quantify maintenance errors and consider feedback loops remains an unexplored domain.In addition, the SD model developed in this paper is helpful to consider the possibility of recovery of task after error and delay in doing the task, which is neglected by the majority of HRA methods, that may increase the probability of failure following a success.
This research aims to develop a methodology by incorporating a system dynamics approach to assess the HEP within the context of HFS while accounting for time-dependent influential variables.It is established based on the previous study on HRA in hydrogen fuelling stations [38], which developed a technique for estimating the HEP by considering the interdependencies of tasks and sub-tasks.The current investigation seeks to address the gaps in HEP estimation, particularly related to the inadequate consideration of time-dependent components and limited analysis of dynamic workflows.Specifically, the emphasis is placed on performance shaping factors (PSFs) such as fatigue, time constraints, and intensified workload during 8-hours working shifts.The application of this newly devised methodology offers valuable insights for the effective management of routine maintenance activities in hydrogen fuelling stations, thereby contributing to the reduction of risks associated with maintenance operations in these facilities.This paper is structured into several sections.Firstly, an introductory section is presented to offer an overview of the study.This is followed by a detailed description of the developed methodology, outlining the steps and tools employed in the research.Subsequently, the results and discussion section presents the findings of the computational analysis and explores their implications.Finally, the conclusion section provides conclusive remarks based on the outcomes of the study.

Methodology
Developing a methodology that dynamically assesses human reliability analysis (HRA) in maintenance activities demands an analysis framework capable of considering inter-related and dynamic PSFs within the maintenance environment.The system thinking approach is well suited to this task, offering a means to concurrently evaluate the number of maintenance tasks pending and HEPs.This approach is considered essential for understanding and monitoring human errors within maintenance activities.The various steps involved in developing the dynamic HRA framework for the maintenance process in this paper are presented in Fig. 1.
The first step involves problem definition, understanding and articulating the problem.The next step is defining the boundary of the model and selecting the data collection process.The model specifically examines corrective maintenance operations, focusing on the reliability of human performance in executing maintenance sub-tasks.Data related to maintenance performance and human performance are collected from the literature.Maintenance data encompasses the number of tasks requiring maintenance, the minimum time required for repair, and task characterisation (whether tasks need to be performed sequentially or can be undertaken simultaneously).Task characterisation within the model is indicated by the size of the maintenance crew; a single crew member suggests sequential tasks, while two members indicate parallel tasks.Furthermore, constraints on the repair rate are determined by the minimum repair time, task characterisation, and maintenance crew productivity.
Once tasks are repaired, they can either be approved and exit the maintenance process or, if repaired incorrectly due to a human error, require further repair.The fractional rework rate, which influences the incorrect repair rate, is affected by the initial error rate and the HEP at each time step.
Data related to human performance mainly focuses on PSFs which, like human-machine interface and procedural aspects, may vary over longer timeframes (e.g., months or years) due to ageing and inherent limitations.However, during a single task that spans a few minutes or hours, these factors can be considered constant.Stress and fatigue are PSFs that accumulate over time and are considered primary factors influencing human performance [39].Though the fluctuations of the effects of training and experience are not immediately evident (in a short period), they are deemed to impact human performance and are thus considered constant in the model based on previous studies.Time availability (the time left in the schedule) is another PSF that directly influences stress [40] and fatigue [41].Time pressure, which increases the error rate, is also considered in this study and it is defined as a function of the remaining time and the perception of the time left.The impact of time pressure on decision-making and reaction time has been discussed in previous studies [42].Deficiencies in decision-making or execution can adversely affect human performance.
Following data collection, the subsequent step in the methodology entails establishing a system dynamics interface using Vensim for simulation.System dynamics modelling comprises two crucial components: visualising the problem through a causal loop diagram (CLD) and using stock-flow diagrams (SFD).The CLD describe the qualitative relationships amongst the system's elements, enabling the visualisation of the connections between maintenance tasks, HEP, and other indicators.A rudimentary CLD for this model is illustrated in Fig. 2, demonstrating the interaction between human errors, the number of maintenance tasks, and their contributory factors.
The model captures the interdependence of human errors and the number of maintenance tasks through four main feedback loops: one balancing loop and three reinforcing loops.The direction of influence between variables is indicated by arrows, with polarity marks (+ or -) outside the arrow's curve signifying the nature of a variable's impact on the dependent one.Starting with loop R3, an increase in maintenance tasks elevates the maintenance rate (in this context, the approved repair rate), which in turn increases the workload and consequently induces fatigue.This escalation in fatigue enhances the HEP, affecting the incorrect maintenance rate and amplifying the number of tasks requiring rework, thereby increasing the overall number of maintenance tasks.Loop R1 embodies the concept of "error makes error", where an increase in HEP leads to more maintenance errors, consequently, escalating fatigue and HEP, and the cycle perpetuates until interventions are implemented to balance the effects.Understanding these feedback loops can aid in devising strategies to alleviate human error and augment the maintenance process.The productivity of the maintenance crew significantly influences the maintenance process.As loop B1 indicates, escalating fatigue diminishes productivity, thereby impacting the maintenance rate.Conversely, decreased productivity tends to reduce fatigue.While time pressure can theoretically boost crew productivity by accelerating work pace, this study does not consider feedback on productivity to maintain model manageability.For this study, productivity is defined as the reciprocal of the mean time required to complete a task.After the CLD development, the subsequent phase involves translating these qualitative relationships into a quantitative framework by developing an SFD.Stocks denote accumulations after a predetermined time step, whereas flow variables describe the rate at which stocks are added or depleted.Fig. 3 represents the model based on the provided CLD.
Upon creating the SFD, the simulation model is formulated based on the dynamic problem at hand.The differential equation explained in Equation (1) represents the number of tasks awaiting maintenance, depicted as the stock variable in the model.Q is the number of tasks awaiting maintenance.Its rate of change is expressed as ( dQ(t) dt ) where t is simulation time.It is assumed that there are multiple tasks that need maintenance at the start of the simulation, which is shown with Q 0 .
A task may be repaired perfectly and subsequently exit the maintenance process.The speed at which this occurs is determined by the "approved repair rate", a me , at the rate of ) MTTR , where MTTR is the average time to repair, and C represents work capability constraints.In some instances, maintenance tasks remain in the process due to substandard maintenance at the rate of γ, indicating imperfect maintenance, requiring rework post-maintenance.The speed at which this occurs is determined by the "incorrect rate", a mi , at the rate of Tasks with inadequate maintenance may re-enter the maintenance process for rework, denoted as b im .The delay for a substandard maintenance task to return to the "tasks waiting for maintenance" state depends on the post-maintenance error detection time, D QC , to check maintenance quality.For model manageability, this study assumes immediate error detection post-maintenance.Therefore, the simplified mathematical model for the rate of change in the "Tasks waiting for maintenance" stock is shown in Eq. (1).
γ, the fraction of the rate of incorrectly maintained tasks depends on the initial error rate and occurs when human error occurs in the process.In this study, γ is formulated using a sigmoid function, , where α is the prior error rate in the completion of corrective maintenance tasks, which is defined as the initial error rate.To calculate the initial error rate a conditional probability (by applying the Bayesian network) has been used by considering the effect of PSFs on the completion of maintenance.In this study, the PSFs that are not considered in the SD model have been used in the BN.The HEP calculated by BN at the start of the simulation has been shown as α in the model.
The HEP, which ranges between a minimum value (H min ) and a maximum value of 1, is defined as a function of the moving average of several PSFs that influence human error, as expressed in Equation ( 2).
where H 0 is the reference human error probability, H i , and H j are the effect of those PSFs that increase the HEP (e.g.fatigue, stress, and time pressure) and the effect of those PSFs that decrease the HEP (e.g.experience and training), respectively; and e i and e j are the weight of PSFs affected in HEP.H 0 is considered at the system level, necessitating a focus on the order and dependency of subtasks.It has been widely recognized that tasks with dependencies result in higher HEPs, and neglecting these dependencies leads to inaccuracies in results [43].Therefore, in simulations, sequential subtasks are critically considered to ensure that the determined nominal HEP realistically reflects operational conditions.H 0 is calculated as , where R i (t) is the probability of any human action succeeding in time t.In this study, (Ri(t)) are calculated based on the generic nominal human unreliability (GTT) for each subtask provided in the HEART method [23].Although several HRA methods consider nominal human unreliability, HEART's straightforward and industry-agnostic approach has led to its effective implementation across various sectors.
The fatigue level, based on task duration, is modeled based on previous literature [44,45] with modifications to fit the dynamic model.Equation ( 3) shows the fatigue level at time t.
Here, λ represents the fatigue growth parameter when a job or action is completed during time T w .
In addition, to calculating HEP, the model includes three novel features.Firstly, the model accommodated both serial and parallel scheduling of multiple tasks during the maintenance process.Secondly, it captures the impact of human errors leading to significant schedule overruns due to potential reworks.Thirdly, the time at which the simulation ends is not constant for all situations.In other words, the time of simulation ends when all sub-tasks are done.Although the main objective of the model is calculating the HEP in a scheduled time for maintaining tasks (for example, if there are 24 tasks that need repair and each task takes 10 min to be repaired, the scheduled time for these 24 tasks is 4 hours), the time to repair all tasks may increase due to error and rework.So, in this model, the final time of simulation is formulated based on the time needed to maintain all the tasks defined.That is why fatigue also can be increased due to the exceeding shift time of the day (typically set at 8 hours).
To assess the contribution of shift duration to fatigue, a defined shift length is considered.A regular 8-hours shift doesn't contribute to the fatigue score.The contribution of shifts longer than 8 hours is represented by a nonlinear function.In this study, a table function has been employed to show this index, depicted in the model by the Fatigue Value Table .The calculation of this factor accounts for homeostatic factors and the nature of the task and is estimated based on research conducted by Ref. [46].
One of the most frequent error-contributing factors is the ratio of available time to the time required to complete a task [47], and it is one of the most impactful PSFs in HRA [48].The ratio of the remained scheduled time and perception of time left is used in this model to determine the time pressure in each time step.Furthermore, the pressure produced by the deficit of time reserved for performing in a time unit is represented in the system as a "mental workload".The mental workload gets activated when the perception of time left is higher than the scheduled time left and affect the attention, working memory and executive function cognitive factor of human [49].
After the model is formulated, the next step involves calculating HEP based on real-time data and simulating the model for base run simulation.Then, the model can be applied to explore potential scenarios, test its implementation suitability, and examine policies.
One major possible mechanism is included in the model to represent organisational reactions to growing HEP.Fig. 4 shows the mechanism (highlighted in red), and other PSFs affecting HEP in the model, which is not shown in Fig. 3.The model developed is instrumental in understanding system behaviour and recognising strategies that coincide with an organization's objectives, offering an optimal response to stimuli.
In this study, two distinct interventions are examined, physical performance intervention and holistic performance intervention.Physical performance intervention primarily focuses on the impact of rest breaks on the physical performance of crew members.Here, the concept is that breaks primarily counteract physical fatigue, such as that resulting from repairing equipment.Grounded on recommendations from HSE [50] the model restricts the duration at a single workstation to 2 hours, followed by a 15-min rest break.Holistic performance intervention expands the scope to consider the potential benefits of breaks on both physical and cognitive or psychological performance.It acknowledges the multi-faceted benefits of breaks, from reducing physical fatigue to enhancing cognitive function and overall psychological well-being.It integrates the assertion by Di Pasquale et al. [51] that the human reliability curve, post-break, reverts to a previous, enhanced state.For both interventions, the primary objective is to evaluate the efficiency and effectiveness of breaks.The model simulation, considering the potential impacts on both physical and cognitive aspects, uses the Wright learning curve.Assumptions, such as a post-break decrease in HEP, are made based on prior studies.The parameters, as highlighted by Di Pasquale et al. [51] describe the recovery factor as an exponential function, with both the break duration and a coefficient (set at 6.91) as the function's variables.Through this modelling, this paper aims to reduce long, uninterrupted work durations and evaluate the potential benefits of varying break frequencies and lengths.
The next step is updating the model if needed and formulating a new strategy or intervention.The capability of the model to dynamically update the human errors when new evidence becomes available is very important especially when many operational and environmental factors are fluctuating while they do not have a static nature.Since new evidence becomes available, the developed model can present a revised HEP and simulate and analyse the behaviour of the system and formulation strategy or policy.However, this study only investigates the different strategies for the rest-break approach of maintenance crew in HFS.

Methodology application
The Australian government plans to prepare for a zero-emissions transport future in Australia with substantial infrastructure investment, exemplified by the hydrogen initiative.Presently, Australia has two publicly accessible hydrogen fuelling stations, one managed by Toyota in Altona, Victoria, and the other by ActewAGL in Canberra, mainly serving government or council fleets of passenger cars.This study focuses on the HFS tailored to the environment of a humid sub-tropical city and the operational conditions of an urban area with a population density exceeding 1170 persons per square kilometre in the same subtropical location.Employing the developed approach, the investigation centres on assessing the probabilities of human errors occurring during task performance within the HFS.

System boundaries, variables, and data collection
This research focuses on the maintenance procedure applied to the pressure relief valve, an emergency venting system located within an HFS storage unit.The study evaluates the HEP associated with various maintenance steps concerning the pressure relief valve through safety analysis.
The maintenance process entails seven primary tasks, commencing with informing operators about the maintenance activities to ensure safe execution.Subsequently, the steps include isolating the storage unit from the supply line, draining hydrogen from the storage unit, conducting pressure relief valve testing using inert gas, draining the inert gas, re-filling the storage unit with hydrogen, and documenting all procedures covered and relevant data.These seven tasks are further broken down into twenty-three sub-activities, and a comprehensive account of these activities can be found in Table 1 [38].
Table 2 lists the parameter values used for this study.Some of the sources of data are presented in detail in previous sections, and the remaining are discussed herein.These values are mostly consistent with the literature such as Chauhan et al. [38] and Jaber et al. [45] while some are selected to illustrate the effect of varying conditions in simulation settings or assumed due to data unavailability.For instance, number of subtasks are based on Table 1.

Simulation scenarios
The simulation experiments include a base run simulation that explores the change of HEP relevant to collected data for the maintenance of HFS, considering the impact of interventions, such as applying breaks during work.The dynamic changes in increased subtasks and application of interventions are tested as listed in Table 3.The table also details how each test is implemented in the analysis.In this study, the dynamic simulations have been conducted using the Vensim software.The total simulation duration is dynamic and depends on the duration of work.The unit for time is an hour with a time step of 0.0078125 hours for precise resolution of system behaviour.The time step size was carefully chosen to ensure accurate capturing of the system dynamics while maintaining computational efficiency.The simulation thus captures the dynamics of HEP in the maintenance of HFS over an extended period.

Results
Table 3 comprehensively illustrates all the scenarios under consideration.The primary scenario, denoted as the base case, involves 23 initial subtasks that must be completed during the maintenance job, assuming a minimum repair time of 0.2 hours.In this case, a single crew is considered, implying a sequential execution of tasks, and an initial error rate of 0.49 is provided.

Table 1
Considered tasks and sub-tasks for the maintenance of HFS.

A1
Inform other operators when maintenance activities have started and ensure proper safe distance.

SA1
Warn all operators that the storage line maintenance procedure is beginning.

A2
Isolate the storage unit from the supply line.

SA2
Close the two flow valves, FV1 and FV2, the generation unit to the 3-way valva-1 SA3 Close flow valve, FV3, on the line from the 3-way valva-2 to the compressing unit SA4 Close flow valve, FV4, on the line from the 3-way valva-2 to the dispensing unit SA5 Close flow valve, FV5, on the line from the compressing unit to the storage unit A3 Drains hydrogen from the storage unit

SA6
Open flow valve, FV6, on the line from the storage unit to the dispensing unit to empty the lines and the storage unit through the dispensing unit SA7 Check the pressure value of the pressure sensor on the storage unit to ensure the storage unit is empty SA8 Close flow valve, FV6 A4 Test the pressure relief valve on the storage unit using inert gas SA9 Prepare an external source of inert gas and a line with a connector suitable to connect to the flow valve, FV5, on the line from the compressing unit to the storage unit SA10 Connect the inert gas line connector to the flow valve, FV5, between the compressing unit to the storage unit SA11 Fill the storage unit with inert gas until the pressure relief valve on the storage unit is activated SA12 Check the pressure sensor between flow valve FV5, and the storage unit to ensure the correct value of the pressure relief valve on the storage unit venting pressure A5 Drains the inert gas from the storage unit SA13 Disconnect the inert gas line connector from the flow valve FV5 SA14 Ensure proper fitting and close flow valve FV5 SA15 Open flow valve, FV6, on the line from the storage unit to the dispensing unit to empty the inert gas from the lines and the storage unit through the dispensing unit SA16 Check the pressure value of the pressure sensor on the storage unit to ensure the storage unit is empty A6 Re-filling of hydrogen into the storage unit

SA17
Close flow valve, FV6 SA18 Open flow valve FV5 SA19 Open the flow valve FV3 on the line from the 3-way valva-2 to the compressing unit SA20 Open the flow valve FV1 and FV2 on the line from the generation unit to 3-way valva-1 SA21 Fill the storage unit with compressed hydrogen SA22 Check the pressure sensor between flow valve FV5, and the storage un A7 Record keeping of all the tasks and relevant data on the SA23 Record all the tasks and relevant data on the checklists to

Scenario 1: base case
The particular scenario showcasing trends in the HEP is illustrated in Fig. 5a.Over the initial 2 hours, there is a linear surge in HEP, which then begins to plateau, culminating in a peak value of approximately 0.49 at 4.45 hours.Following this peak, a slight decline of 0.43 at 6.6 hours was retained till the end.This scenario exemplifies how HEP can increase, decrease, or remain constant, contingent upon the effects of the PSFs.The HEP is notably influenced by PSFs such as time pressure, fatigue resulting from time constraints, fatigue from prolonged shifts, and fatigue induced by task execution.Consequently, higher values of these PSFs correspond to elevated levels of HEP.
The progression of fatigue resulting from schedule pressure is depicted in Fig. 5d.Initially, the fatigue is minimal at the commencement of the project; however, as time elapses, it gradually intensifies until it reaches its peak value of unity.The manifestation of this PSF relies on the accuracy of the remaining time perception.If the perception of time remaining is erroneous, the fatigue induced by time pressure is correspondingly higher: 0.969 at 6.6 hours.
In Fig. 5c, the fatigue arising from the execution of tasks during maintenance is presented.There is a gradual increase; in the end, it tends to stabilise at the value of 0.40 after 6 hours.This type of fatigue depends upon three essential factors: the fatigue growth parameter, determined from a prior research paper.The approved repair rate is influenced by the capabilities of the personnel and the incorrect rate arising from errors encountered during the execution of maintenance tasks.An increase in the approved repair and incorrect rates increases fatigue due to the task performance.The relationship between time pressure and time enlightened in Fig. 5b shows a gradual increment in stress levels until it reaches a peak value of 0.89 at approximately 3 hours, then decreases.These findings substantiate that time pressure intensifies as tasks are executed and time elapses.However, as the tasks are completed and the workload diminishes, the time pressure commences downward, indicative of stress reduction.It should be noted that the total time considered in this scenario falls below the standard 7-hours shift duration.Consequently, the fatigue induced by the length of the overtime remains at zero.Fig. 6 shows the effect of increasing HEP on the number of maintenance errors.As shown in Fig. 6a, the number of subtasks that need rework reaches the peak at 5 hours and hence the increased average number of rework (Fig. 6b) increases the number of tasks waiting for maintenance and consequently increases fatigue from doing tasks.
These parameters presented herein are related to the crucial aspects of the base case.The accompanying illustration of PSFs in conjunction with HEP serves to elucidate the degree of dependence of HEP on PSFs.Upon examination of the graphs above, it becomes evident that, within a time period of less than 8 hours, HEP is influenced by time pressure, fatigue resulting from time pressure, and fatigue incurred from task execution within the allocated time frame.However, as the shift length increases due to error rate and rework, fatigue due to shift length is also added to the PSFs of HEP.
This scenario highlights the significance of human error in maintenance operations and its consequential impact on system downtime.Through a focus on HEP and its influence on task durations, the simulation reveals that elevated HEP corresponds to a rise in the proportion of subtasks needing rework, thereby augmenting system downtime.The dynamic trajectory of HEP illuminates the sway of PSFs, particularly time pressure and fatigue.Such fluctuations in HEP accentuate the urgency for robust mitigation measures to curtail human error and enhance maintenance efficiency.The study of [47] supports the importance of managing time pressure and fatigue in maintenance tasks to ensure peak performance and diminish incident rates.By targeting the root causes of human error, organisations are better positioned to bolster system reliability and curtail downtime.

Sensitivity analysis
The model is grounded in reliable data, however, constraints in accessing operational HRS data and the scarcity of specific research have necessitated assumptions for certain parameters.Most parameter values Fig. 5. Base run simulation results (HEP and PSFs).
A. Chauhan et al. for our simulations align with existing literature, while some are chosen to reflect variations in different settings.The values for "weight of PSFs affecting HEP" are selected based on expert opinion and their relevance to the chosen application.The Monte-Carlo analysis is employed to assess the model's robustness to these parameters' variations.The HEP sensitivity is tested to variations of up to double and half the base run value, conducting 2000 simulation runs for each test.The results, as depicted in Fig. 7, demonstrate qualitative robustness with variability maintained within reasonable bounds.

Scenario 2: change in number of subtasks
The primary purpose of this analysis is to evaluate how the application of SD in maintenance activities can impact the associated risk of HEP.To achieve this, three additional scenarios were chosen after considering the baseline case, denoted as S1.For the present scenario (S2), the first test (S2-T1) involved the completion of 30 tasks within a 7hours timeframe, the second test (S2-T2) considered 48 tasks over 8 hours, and the third test (S2-T3) encompassed the completion of 36 tasks within 6 hours.All other parameters remained constant throughout the second scenario.These cases were then examined to assess changes in HEP-dependent factors and the time required for completing maintenance tasks.
The first scenario considered 23 distinct subtasks with an MTTR of 10 min.Under optimal conditions, without the influence of human errors, the entirety of these tasks should take less than 4 hours to complete.However, as HEP increases, a portion of these tasks are not executed correctly, leading to the necessity of rework.This additional work, resulting from the initial inaccuracies, extends the overall repair time and, consequently, increases downtime.The second scenario reveals a notable negative impact of human errors on maintenance performance.However, there is limited research quantifying the impact of human error on maintenance performance and downtime of the system [52].The tests of this scenario focus on how maintenance errors increase the planned downtime of the system and concentrate on the interplay between maintenance scheduling and HEP.The HEP exhibits distinct variations across different cases.Fig. 8a presents a comparative analysis between scenario 2 cases and S1.In S2-T1, the initial number of tasks is increased by 30 %, from the base value of 23-30, resulting in an extended completion time from 6.6 to 8.4 hours and an increase in the HEP from 42 % to 45 % at the end of the simulation.The peak of HEP in S2-T1 is 51.5 % after 5 hours of starting the work and it decreases as the time pressure declines because the perception of time remaining decreases.In S2-T2, a further increase in the number of initial tasks to 48 for an 8-hours shift leads to a prolonged completion time of 14 hours and a substantially higher HEP, approaching its maximum value of 77.5 % without decreasing the HEP during the work.It happens because as soon as the perception of the remaining time decreases, human error starts to rise because of the fatigue due to shift length.While, in S2-T3, 36 tasks are undertaken for a 6-hours shift, taking 9.8 hours to complete, with a maximum HEP of 53 %.The aforementioned cases demonstrate the significance of both the initial number of tasks and the length of the shift in maintenance tasks, as they directly impact HEP.The consideration of fatigue resulting from task performance is of paramount importance, as it progressively intensifies with the execution of tasks over time, as evident in Fig. 8e.The fatigue from task execution is directly influenced by the approved repair and incorrect rates, both of which escalate with human errors.The incorrect rate arises from human error in maintenance tasks, while the approved repair rate reflects the increased maintenance frequency resulting from human errors.Higher values of these rates correlate with elevated fatigue from task execution.Furthermore, all three tests within scenario 2 initially demonstrate a pattern resembling that of the base case.However, this pattern unfolds over a prolonged period due to the higher number of tasks involved.In case S2-T1 increase in the number of tasks (30) results in an increase in fatigue and time.In S2-T2, with 48 tasks, the fatigue value reaches a maximum fatigue level of 0.7 in 14 hours, while in S2-T3, with 36 tasks, the time consumed is less than 10 hours, accompanied by a maximum fatigue value from doing tasks of 0.5.Fig. 8c provides a comparative analysis between scenarios 1 and 2 in relation to the base case, particularly focusing on the impact of schedule pressure-induced fatigue.The observed trend in time pressure reveals a gradual escalation until it reaches its peak in terms of fatigue levels.This observed pattern signifies that as time passes, pending tasks accumulate, and stress levels elevate, a critical juncture is reached wherein the potential for human errors is significantly amplified.For instance, S2-T1 reaches maximum level at 7th hour, S2-T2 reaches maximum fatigue at 11th hour, and S2-T3 took 9 hours to reach maximum fatigue value.Fig. 8d depicts the time pressure, defined as the ratio between the remaining time and the perception of time left.The trends observed in the figures demonstrate a common pattern.As soon as it becomes evident that the actual remaining time is sufficient to complete the remaining tasks, the time pressure diminishes.An increase in the measure of tasks leads to a corresponding elevation in time pressure, demanding a greater duration for the completion of maintenance activities.This, in turn, causes a heightened probability of human errors.Markedly S2-T1 maximum time pressure of 0.9, S2-T2; 0.92 and S2-T3; 0.94 all occur between 2nd and 3rd hour time.Fig. 8f illustrates fatigue resulting from the length of the shift.The nomenclature shows that this fatigue occurs when the time exceeds the standard shift hours of 7. Thus, no fatigue is observed if the tests are completed within the 7-hours shift.However, if the duration exceeds 7 hours, fatigue due to the extension of shift hours ensues and an increase in this factor is associated with an increment in downtime.
The scenarios mentioned above were examined across varying task quantities and completion times.Particularly, as the number of tasks increased, the potential for human errors leading to rework also rose.Furthermore, prolonged maintenance task durations were linked to heightened stress and fatigue levels, resulting in an increased probability of human errors.In the subsequent scenario, specific measures were implemented to mitigate these errors.

Scenario 3: rest break intervention
In previous cases, we investigated downtime time due to some critical scenarios.In S2-T1, S2-T2 and S3-T3, the time exceeds 8 hours of day shift.This raises the motivation to explore why an 8-hours shift is applied, its pros and cons, and how it is applied.
Incorporated in the model is a recommendation from HSE [50] to limit the time at a single workstation to 2 hours, followed by a structured 15-min rest break.This rest period, grounded in research, suggests that the level of fatigue from doing task approach zero post-break, given the preventive role of breaks in musculoskeletal issues.The main focus here is to use the model as a tool to evaluate varying policies and strategies associated with maintenance processes across different organisations.In evaluating the effectiveness of interventions, especially the incorporation of breaks during task execution, the findings from the simulations yield insightful implications.For the scenario S3-T1, which covers 23 tasks, the introduction of breaks leads to an extended completion time, from 6.6 hours in S1 to 7 hours.However, HEP, a vital metric denoting the potential for mistakes, experiences a notable reduction.In the S1 scenario without breaks, the HEP stands at 0.43, which drops significantly to 0.35 with the introduction of breaks.Impressively, the peak value of HEP, originally at 0.5 in S1, diminishes to 0.43 when breaks are incorporated.The S3-T2 scenario is complex and requires thoughtful consideration.When tasked with completing 36 tasks within a 6-hours window, the introduction of breaks extends the completion time significantly to 10.4 hours.However, similar to the previous test, there is a tangible reduction in the HEP, from 0.48 compared to the S2-T3 scenario to 0.4 with breaks.Furthermore, the maximum observed HEP drops from 0.54 to less than 0.47 upon introducing breaks.
Further granularity is provided in Fig. 8.With the inclusion of breaks, the average number of subtasks that require repairs in the S3-T1 scenario decreases from 0.9 per time interval to 0.75.In the S3-T2 setup, this metric declines even further to 0.9 from an initial 1.2.This reduction strengthens the argument that, aside from enhancing physical performance, breaks also play a pivotal role in mitigating maintenance errors.Nonetheless, it is crucial to weigh these benefits against potential detriments.While breaks do curtail errors, they can inadvertently introduce time losses due to task interruptions.
Taken together, these findings underscore the inherent trade-offs in the introduction of breaks.Although task completion times may witness an increase, the potential for human error, a parameter of maximum significance in precision-demanding contexts, sees a substantial decrease.Such results emphasize the need to refine break durations and frequencies to optimize worker performance without unduly prolonging completion times.
As established, breaks do have potential drawbacks.They signify a work pause, and when faced with tight deadlines, even brief interruptions can cumulatively lead to significant time loss.An excessive increase in the number of breaks may not yield productive results [45].presented that adequate recovery breaks ease accumulated fatigue, but excessively long recovery times extend lead times and degrade production performance due to forgetting in the learning-forgetting-fatigue-recovery model.Similarly, in the latest study by Nam and Lee [53] the working time and break time of tasks by applying the learning-forgetting model in the equipment operating process illustrates that with the increase in break period, the knowledge gained during the work is elapsed.As shown in S3-T1 and S3-T2, interruptions can have repercussions, including lost time, task disengagement, procrastination, and diminished productivity.This is evident when a rest break, although alleviating immediate fatigue, extends the task completion time, thus indirectly increasing fatigue because of the prolonged shift length.
On the contrary [54], emphasised the importance of timing rest breaks appropriately.Furthermore, Tucker [55] argued that incorporating breaks can reduce the total working hours within a shift, which can be effective to a certain degree.Trougakos et al. [56]noted that breaks involving respite activities, as opposed to mundane tasks, are more conducive to a positive mood post-break.Further studies highlight a clear association between consistent breaks, cognitive function retention, error rate reduction, and an uptick in productivity [57].Even if these studies do not directly address their findings in the context of schedule-induced fatigue or time constraints, the results remain relevant.The consensus is that rest periods not only improve a worker's physical performance but also their mental faculties.
The next series of tests (tests 3 through 6) were designed to delve deeper into the impacts of break time and frequency while taking into account the potential improvements in both physical and cognitive human performance post-break.The metric for such improvements is referred to as the recovery rate.
In S3-T3, the break duration is maintained at 15 min per 2 hours, consistent with previous tests.Fig. 9 depicts there is a notable decline in HEP to 0.34 by the task's culmination, clocking a completion time of 6.2 hours.In relation to the base case scenario, S1, this presents an appreciable improvement.The peak HEP also saw a decrease to 0.42.Test S3-T4 considers the Pomodoro Technique, a time management method that involves working for 30 min straight and then taking a 5-min break.This method, backed by some empirical evidence, suggests that these short, regular breaks can boost productivity and mental agility, which might be beneficial under schedule pressure.Fig. 9 depicts S3-T4 yields a completion time of 6.6 hours, identical to S1.However, the HEP saw a considerable reduction to 0.29 by the end of the task and a peak value of 0.37.With S3-T5, by introducing a 20-min break every hour as per [51] recommendation, the task completion time slightly increased to 6.7 hours.Yet, the HEP metrics indicated an improvement with a peak value of 0.3.In S3-T6, with 30-min breaks scheduled every 2 hours for maximum recovery, the work completion matched that of S3-T3 at 6.2 hours.Impressively, the HEP at the end of the task was the lowest among all scenarios at 0.24, even though its peak HEP was slightly lower at 0.38.
When analysing the average number of subtasks requiring rework, S3-T3 and S3-T4 saw a decrease to 0.63 and 0.61, respectively, compared to S1's 0.9.Furthermore, S3-T5 and S3-T6 demonstrated substantial improvements with values at 0.33 and 0.45, indicating enhanced efficiency and reduced error propensity when these break strategies were applied.The results from the conducted tests emphasize the importance of well-timed rest periods in enhancing both physical and cognitive performance.The data reveals a clear trend: shorter, more frequent breaks lead to a noticeable reduction in HEP.This is especially evident when considering approaches like the Pomodoro Technique and the recommendations from Di Pasquale et al. [51].
The decrease in the number of subtasks requiring rework further supports this observation.It suggests that proper rest, achieved through strategic break intervals, can lead to more accurate task performance with fewer errors.However, while breaks are beneficial, it is crucial to implement them thoughtfully.The balance between the number and length of breaks is essential.Over-elongating the overall task duration due to numerous or long breaks might inadvertently increase fatigue, negating the benefits of the breaks.Future studies should further investigate the optimal balance of break frequency and duration, especially in settings with tight schedules.
The SD model presents numerous advantages over traditional methods.Unlike conventional models that rely on static inputs, under  the assumption that relationships and weights among factors remain unchanged, the proposed model is adaptive, permitting alterations based on updated or real-time data.This adaptability proves invaluable in the domain of human error, characterized by inherent uncertainty and dynamism.It is important to emphasize that the weight attributed to factors like fatigue must be periodically reassessed and adjusted based on emerging data, research outcomes, or changing work dynamics.In the current model, the weights for fatigue from task execution, fatigue from schedule pressure, fatigue from shift duration, and time pressure are grounded in extant literature and team consensus, standing at 0.5, 0.3, 0.5, and 0.7 respectively.Nonetheless, considering the unpredictable nature of fatigue and time pressure, which may vary across contexts and individuals, these weights can be re-evaluated.For instance, if subsequent studies or differing work contexts suggest revised weights of 0.8, 0.5, 0.8, and 1, the model can be promptly updated, paving the way for strategic and intervention recalibrations.Fig. 10 offers a comparative analysis between HEP and PSFs based on updated data across the initial six scenarios.Notably, S2-T2, which commenced with the highest number of tasks, records a HEP of 1, alongside elevated values for all fatigue-related PSFs.S3, which implements 15-min breaks every 2 hours, presents an intriguing case.S3-T1 and S3-T2, despite their differing initial tasks, both highlight the beneficial effects of breaks on reducing fatigue from task execution.However, there is an increase in fatigue from extended shift lengths.This underlines the necessity of judiciously scheduling breaks, ensuring they do not prolong shift hours, which, while diminishing one form of fatigue, could inadvertently amplify another, thus elevating HEP.

Conclusion
Integrating system dynamics in this research has presented a novel approach in the HFS context.The adoption of system dynamics has facilitated a comprehensive exploration of HEP, considering the dynamics of time-dependent PSFs.The model developed in this study has provided a deeper understanding of how human factors dynamically influence task performance over time, offering valuable insights for safety management strategies.The investigation provided valuable insights into the impact of PSFs on HEP, shedding light on the complexities of human performance in challenging work environments.The analysis of various scenarios, including those with interventions, revealed important safety and operational efficiency implications.As the number of tasks increased, so did the time taken for their completion, resulting in higher fatigue levels.Furthermore, the application of breaks demonstrated a potential to alleviate fatigue from task execution, contributing to reducing the HEP.However, careful consideration is required to ensure that the introduction of breaks does not inadvertently extend the shift hours, as this may lead to increased fatigue from shift length and consequently raise HEP.
Overall, the research contributes to a better understanding of human factors in maintenance tasks and offers practical insights for designing interventions to improve safety, efficiency, and performance.The implications of this study extend beyond the HFS context, with potential applications in various industries to enhance operational safety and efficacy.Further investigations in this area could explore additional interventions and test their effectiveness in different work scenarios, thus advancing the understanding of human performance and error prevention strategies.Ultimately, by proactively addressing human factors, organisations can foster a safer and more productive work environment for the benefit of workers and stakeholders.The integration of system dynamics and the comprehensive insights gained from this research contribute substantially to hydrogen safety.Future investigations may leverage these findings to explore additional interventions and diverse work scenarios, further advancing the understanding of HEP in hydrogen safety.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 4 .
Fig. 4. The representation of the model and application of policy to decrease fatigue by ensuring that workers have and take adequate and regular breaks during a shift.

Fig. 10 .
Fig. 10.Human error probability and performance shaping factors based on updated data across the initial six scenarios.(HEP = Human error probability; FFDT = fatigue from doing tasks; FFSP = fatigue from schedule pressure; FFSL = Fatigue from shift length; TP = Time pressure).

Table 2
Model parameters and their selected values for the case study.

Table 3
Scenarios considered for the case study.
A.Chauhan et al.