1   Introduction

1.1   Background and motivation of the study

Significant efforts to increase the use of renewable energy (RE) for energy production have been exerted by countries around the world. Because of its abundant reserves, non-polluting characteristics, and low development costs, wind power has always been considered one of the most important sources of energy. The European Wind Energy Association (EWEA) states that the planned amount of installed capacity for wind power in Europe will reach 392 GW by 2030, driven by the robust growth of onshore and offshore wind energy (EWEA, 2015). Wind energy is gaining market share from traditional energy sources, including floating offshore wind systems (Sadorsky, 2021). In recent decades, the levelized energy cost, which continuously decreases, and the ongoing development of modern wind potential exploitation technologies, as well as the rapid growth of expertise and experience (Li et al., 2021), have accelerated the development of wind turbines (WT; Hussain et al., 2022). Correspondingly, wave energy is a form of RE that harnesses the power of waves to generate electricity. This field also includes tidal and ocean thermal energy. Appropriate technologies are used to harness this energy by capturing the movement of water caused by wind, creating waves, and generating electricity. In contrast to solar and wind power, wave energy is more predictable, as it is directly related to wind patterns (Widén et al., 2015). One advantage that will make wind and wave energy fully competitive with conventional fossil fuels as a source for electricity generation is that the WT and wave energy systems for industries are continuously improving the cost and performance of the technologies. Hence, an integrated hybrid wind–wave power generation (HWWPG) pilot system is considered in the present study. The pilot system consists of a WT and two wave energy converters (WECs) installed on an offshore platform, where electricity powers the operation of a reverse osmosis system to produce desalinated water, as well as the power grid on a northern Aegean Island in Greece. The electricity generation capacity of the WT is 800 kW, and the WECs are a total of 10 kW. However, despite the uneven distribution of installed WT and WEC systems, the application of the methodology of this research demonstrates that important conclusions can be drawn. The findings will assist in the decision-making of practitioners and researchers in the marine and offshore industries concerning the optimal HWWPG operations.

Generally, offshore systems are vulnerable to harsh weather conditions, which can increase component failure rates and severely impact maintenance processes and reliability. Considering the effects of regular and intense wind intensity variations, we propose a Markov model for the reliability assessment of the HWWPG. The following types of failure rates of WT and WEC are considered: the failure rate during regular wind speed fluctuation, the failure rate during intense wind speed fluctuation, and the lightning strike rate. In addition, considering the impact of regular and intense wind speed variations, we determine the minor maintenance (MM) and corrective maintenance (CM) times of the WT and WEC. Because of the random manner of the occurrence of failure in the system and its mechanical degradation and failure response, multiple states are employed to describe the randomness in the HWWPG system. For the multistate modeling of the system, the universal generating function (UGF) technique is used. To combine the UGF of the operational condition of the HWWPG system and that of the electric power generated due to the different wind intensity variation states into the final UGF, a composition operator of the multiplication type is introduced. To validate the efficiency of the proposed models in the study of the HWWPG system, the demand satisfaction index (DSI) that the HWWPG system can meet, the expected energy produced (EEP), the expected energy not supplied (EENS), and the total expected operational cost (TEOC) are estimated.

The WT operation and maintenance (O&M) costs can be up to 20% of the total energy costs, and this proportion increases for floating offshore wind systems (Martinez and Iglesias, 2022). Manufacturers' research and development departments, operators, and users of wind farms seek to reduce their O&M costs by understanding wind equipment failure characteristics, improving their capability to manage maintenance resources, and identifying practical measures to prevent unexpected equipment breakdowns. Compared with onshore WTs, offshore WTs have the benefits of a smaller land area use, higher regular wind speed, and additional yearly operating hours. Allal et al. (2024) investigated different machine learning techniques and algorithms in the context of renewable energy systems (RES). The identification and evaluation of existing RES technologies and the assessment of their potential for further development are some of the important issues addressed in this research. In addition, Carroll et al. (2016) provided insights into the failure rates, repair rates, and unscheduled O&M costs of offshore WTs, emphasizing the need for robust reliability assessments. A review by Hassan et al. (2023) on hybrid RES combining solar and wind resources further corroborated the reliability and stability benefits of integrating complementary energy sources.

Furthermore, these systems are associated with high maintenance costs and long repair times, emphasizing the significance of reliability assessment in the design of HWWPG systems. To optimize the electricity generation capacity, offshore hybrid RES combining WTs and wave energy systems in an integrated platform as an HWWPG concept is being developed. These systems have significant potential but are yet at the primary stage of development. Given that the European Green Deal supports the transition of the energy sector toward the decarbonization of the supply chain and includes a strategy for offshore RE technologies (Hainsch et al., 2020), these offshore hybrid systems are an important area that needs to be explored. Because of the large quantities of RE resources, RES are currently undergoing rapid installation worldwide, with the available potential of wind, wave, and tidal energy estimated to be sufficient to power one million homes (Schwartz et al., 2010; Haas et al., 2011; Chen, 2023). The performance of structural health monitoring and the maintenance of an offshore WT system involve periodic observations, analyses, and repairs of the malfunctioning parts of the system (Agbakwuru et al., 2023). This study provides a new research direction to investigate the O&M costs under different weather conditions to improve the management and reliability of the hybrid system.

A WT and a WEC are composed of numerous critical components. More specifically, a WT includes the gearbox, the systems that support its direction according to the wind, and the rotating blade systems. A wave energy system includes modular mechanical systems for transmitting the wave motion, floating or submerged oscillating body transducers that use the wave motion, the generator, and the electrical system. Each of these components is defined by distinctive failure occurrences and impacts. Significant maintenance and repair efforts are also needed.

The effect of intense weather conditions must be considered when assessing the reliability of the HWWPG system (Sannino et al., 2006; Negra et al., 2007). The wind speed determines the output of the WT and, consequently, the output of the WEC because of the height of the waves caused by wind. The output of the HWWPG system is random and dissimilar from that of conventional output production units because of the rapidly altering and unforeseeable features of wind speed and the arbitrary nature of WT and WEC failures. Studies have also evaluated the seasonal variations of wind and wave resources and their potential impact on energy farms (Onea and Rusu, 2016) and the complementarity of renewable resources in improving energy generation reliability (Costoya et al., 2023). The effect is mostly revealed in the component failure rate and repair time factors in the reliability assessment (Sannino et al., 2006; Sheng and O'Connor, 2017). The reliability inputs of the offshore systems and their components are inadequate. As a result, onshore wind farm statistical data are used in most existing reliability assessments of offshore systems (Negra et al., 2007; Moros et al., 2024). The analysis concludes that the crucial part of the precise assessment of the reliability of offshore systems, such as the HWWPG, is to ensure that the reliability parameters are adjusted to account for the effects of intense offshore weather (Scheu et al., 2012).

Waves are a promising energy resource; however, no technologies have been shown to match the performance and costs of offshore wind (Chang et al., 2018), even though several are in precommercial trials. Reliability and availability issues are associated with the large-scale deployment of various RE resources. Moreover, the resilience of power grids and WTs to extreme storm conditions has been compared with the robustness of hybrid systems (Forsberg et al., 2024). To meet the electricity demand, the reliability and availability of wind and wave power generation systems are essential and necessary (Yusuf et al., 2020). de Andrés et al. (2015) conducted a site comparison of WEC failures, operations, and O&M costs and established that failures requiring an extensive weather frame for repair increase the overall downtime, even though they occur infrequently. The availability and performance of a farm of five Pelamis WEC were evaluated by Rinaldi et al. (2016), and they determined that the outcome is highly dependent on the choice of vessel and the metocean limit at which the vessel is operated. The availability of 10 MW wave farms has been investigated by Kennedy et al. (2017), where the combination of weather frame standards, reliability levels, and available vessels are considered. Driven by the inaccessibility of offshore renewables related to onshore systems, a condition-based maintenance strategy was examined by Mérigaud and Ringwood (2016), who highlighted the expertise available from the developed offshore wind sector and other recognized offshore technologies. Rinaldi et al. (2018) presented a framework to assess dependability, availability, and maintainability and modeled the breakdowns based on the adjusted part failure rates.

A new method to assess offshore RE farm capacity requirements to ensure that energy can be dispatched according to local needs was presented by Gao et al. (2024). Their results showed that the combined wind and wave farm significantly reduces the need for energy storage systems and offers competitive life cycle costs compared with stand-alone wind. However, the magnitude of these benefits depends on the local resource characteristics. Additional studies have discussed the benefits of wave–wind hybrid energy systems in terms of power reliability and long-term cost (Masoumi, 2021), as well as the performance of hybrid systems combining breakwater and WEC (Zhou et al., 2023). Layout optimization methods based on wave wake preprocessing have also been proposed to accelerate wave energy development (Haces-Fernandez et al., 2021). Gao et al. (2022) investigated offshore wind and wave energy at several Australian coastal sites and their integration potential in terms of energy availability, output variability, consistency and correlation, and annual and seasonal variabilities over 7 years. Wave-dominated sites in Western and Southern Australia may be suitable for combined wind and wave energy, whereas wind-dominated sites in Eastern Australia may be unsuitable. Cipolletta et al. (2023) developed an innovative approach to wave-energy-based hybrid energy system design, including an operational strategy that meets the dispatching requirements of grid-connected generation systems.

1.2   Problem statement and contributions

A continuous-time Markov chain (CTMC) for the HWWPG system is utilized to analyze the behavior of the system in terms of the electric power generated due to the operating state of the system through the different wind speed changes, the failure rates, and the maintenance procedures. The O&M costs of the HWWPG system are influenced by various random processes, such as weather conditions, wind speed fluctuations, and component failures. A CTMC is well-suited to handle these stochastic processes, providing a robust framework for modeling the random nature of these events. A CTMC provides insights into the long-term behavior of the system by analyzing the steady-state distribution of the state probabilities, which is important in understanding how the system will perform over extended periods and is critical for planning and decision-making. Thus, the originality of this study is the evaluation of the dependability measures of an HWWPG system, given the different energy demands of the grid for 1 year that the system needs to meet for a specific island, as well as the energy demand of the desalination unit installed in the offshore structure for the production of potable water, given the environmental conditions through the different scenarios of regular and intense wind speed changes. In particular, the proposed framework is used to evaluate the potential impact of the wind speed changes on the various HWWPG system dependability measures. Because wave energy is not a fully developed energy technology or one that is converging toward a single or a few concepts, the literature does not contain any breakage curve for wave energy systems.

This study provides valuable insights into the reliability and operational costs of an HWWPG system under varying weather conditions. Using the CTMC and the UGF technique, the analysis highlights the complex relationship between wind intensity changes and the reliability of WT and WEC. Intense weather conditions notably increase the failure rates of both WT and WEC, leading to higher maintenance costs and more frequent downtimes, which, in turn, have an impact on the TEOC and EENS. Despite these challenges, this study demonstrates that the HWWPG system can still generate significant energy, although at a higher cost, underlining the importance of advanced maintenance strategies to reduce component wear, as well as the robustness of hybrid systems in harnessing RE even under adverse conditions. The DSI analysis shows that the capability of the HWWPG system to meet the energy demands decreases with higher failure rates and increased lightning strikes. However, during periods of higher energy demand, the dual reliance of the hybrid system on wind and wave energy helps mitigate some negative impacts, maintaining a relatively stable DSI even under intense weather conditions.

This research addresses several critical gaps in the current research on HWWPG systems. The study contributes to operational and environmental management by providing a detailed quantitative evaluation of the reliability and operational costs of hybrid RES. This study proposes a model that includes failure rates during different wind speed fluctuations and lightning strike rates, providing a more nuanced understanding of these dynamics. By considering regular and intense wind speed variations, we enhance the understanding of how different weather scenarios affect hybrid systems. The findings indicate that, although the HWWPG system is effective and efficient, particularly under regular weather conditions, its reliability significantly decreases under intense weather scenarios. This decrease in reliability has crucial implications for the design and maintenance of offshore RES, highlighting the need for resilient and adaptive maintenance strategies to ensure continuous and efficient energy production. In addition, the potential for integrating other renewable sources, which could optimize system reliability and efficiency, is proposed. This study also evaluates the impact of different weather conditions on maintenance costs and downtimes, highlighting the need for and providing a basis for developing robust maintenance strategies. This study not only identifies increased costs and downtimes under intense weather but also lays the groundwork for future research to explore these trade-offs in more depth.

To characterize the unpredictability of production systems, various states are used. These states consider the stochastic behavior of both production causes and the mechanical deterioration or failure behavior of the production systems. The UGF technique is utilized for the evaluation of the performance measures. A composition operator of the multiplication type is presented to amalgamate the UGF representing the electric power generated by mechanical deterioration and wind intensity variation states into the UGF for the final output production of the hybrid RES. Subsequently, the comprehensive UGF for the HWWPG system is formulated, and traditional dependability measures, such as the DSI, EEP, and EENS, are calculated based on the UGF for system generation. An illustration of the model is presented with the implementation of the HWWPG system.

This paper is structured as follows: In Section 2, the hybrid energy exploitation system and its integration with the reverse osmosis desalination system are discussed. In Section 3, the operational conditions of the HWWPG system are described in detail. In Section 4, the impact of regular and intense weather scenarios on the reliability parameters of the HWWPG system is discussed, and the main weather factor affecting the reliability of the HWWPG system is determined. In Section 5, the UGF technique is defined, and the WT and WEC are established and mathematically featured through the UGF technique. The dependability measures are presented in Section 6. The TEOC is analyzed in Section 7. The case study is presented in Section 8. The study is concluded with the main findings and research directions in Section 9.

2   Hybrid energy exploitation systems

Reducing the final cost of electricity and enhancing the availability in the long term are the major advantages of the concept of combining a WT and a WEC (Bethel, 2021). The objective is clear, i.e., to bring offshore RE resources to maturity. The idea of adding an energy harvesting device to the floating support structure of the WT is to increase the stability and dampen the movement of the structure as the wave energy is extracted, thus increasing the wind power produced as the wave energy is extracted (Meng et al., 2023). In a well-designed, modern combined approach, the wave energy device acts as a shock absorber, damping the incoming wave energy and increasing the stability of the WT pile foundation.

Lilas et al. (2022) investigated the well-designed combination of wave and wind energy devices and their advantages, including the sharing of undersea power cables, the cost of surveying and monitoring, and the sharing of support structures, foundations, mooring, and anchoring systems. In some designs, the absorption of wave energy makes the water calmer at the rear of the structure, making it easier for boats to dock, as well as making monitoring and maintenance easier and less costly. In addition, one or more WTs to a floating wave energy device will increase both steadiness and energy output. The cost of combining a WT and an offshore system is close to the cost of an onshore WT, even though, in some cases, a floating support structure is required to exploit the wave energy, as revealed by studies of the design of the Multiuse Platform MUSICA project built to be installed in the Greek insular area of the Oinousses in the North Aegean. MUSICA will provide a full suite of Blue Growth solutions for small islands, i.e., two forms of RE (wind and wave; total 810 kW), providing high RES penetration and competitively affordable electricity. The system consists of an 800 kW Enercon E53 three-blade WT, horizontal axis, direct drive gearless, variable speed, single blade adjustment, ring generator, and two 5-kW WECs of the SINN Power GmbH that are installed on an offshore platform.

The most important factors considered for the MUSICA platform foundation were the characteristics of wind energy in the area, particularly wave intensity, as it directly affects the percentage of energy produced by the platform. The wave potential (WaP) was considered because it will provide information that will further enhance the development of a combined system for energy production. Wind direction was considered, but it did not have a significant effect on wind energy, as the turbines are fitted with the "yaw, " which takes care of the movement of the turbine blade based on the wind direction. The wind speed values indicated by the meteorological data show small differences. In addition, the WaP was a particularly important factor, as it transformed into a percentage of energy produced and provided information about energy penetration from the production of electricity from the utilization of this technology. The most suitable WEC system was that with oscillating devices where the movement of the waves activates a power takeoff system.

Despite the small size and the small contribution of WEC systems to the total energy production, their installation is important. Their inclusion in the study model can generate inferences that will become a benchmark when this pilot platform is constructed on a larger scale with the use of multiple WECs. Furthermore, the utilization of WECs in offshore platform deployment has advantages because of their capacity to absorb wave energy, thereby reducing the amount of energy transmitted to the platform as they convert wave energy into electricity. The absorption of wave energy results in a reduction in the impact forces exerted on the platform, thereby limiting its movement. Consequently, the WECs function as dampers, thereby reducing the amplitude of the oscillatory movements of the platform induced by the waves. The installation of multiple WEC systems serves to enhance the overall damping effect, thereby facilitating the reduction of the vertical (uplift), horizontal (wave), and rotational (pitch, roll, and yaw) movements of the platform.

The feasibility of utilizing multiple WEC devices enhances reliability, as the failure or malfunction of one device is compensated for by the continued operation of the other devices, thereby ensuring that the overall system remains functional. Concurrently, the energy consumed by the WECs can be redistributed in a regulated manner, thereby contributing to the overall stability of the system. This redundancy enhances reliability during power generation. In addition, the system can absorb energy from a wider range of frequencies and wave directions, which not only increases the total energy harvested but also results in a more stable power generation process, thereby reducing fluctuations and improving the overall efficiency of power conversion.

In terms of platform stability, the use of multiple WECs enables each one to absorb a portion of the wave energy. The distribution of wave forces across the platform prevents the accumulation of concentrated loads that could otherwise destabilize it. The reduction in the likelihood of any single area being overstressed is a consequence of the distribution of forces. By strategically positioning the WECs, the platform can be maintained in a more balanced condition, reducing pitch and roll movements. This stability is vital for the optimal operation of WTs, which requires a relatively stable base to operate efficiently.

2.1   Spatial dynamics and design of wind–wave hybrid energy systems

The combination of wind and wave energy systems was developed because WTs and WECs in a wind farm should be spaced at a specific distance from each other, e.g., 5 times the rotor diameter. The reason for this is the wake effect and wind turbulence. For MW WTs, the WTs need to be spaced approximately half a kilometer apart. The hybrid energy production platforms, which combine wave and wind energy in one unit, can exploit the synergy between the technologies. As mentioned previously, this can increase energy production. The MUSICA floating power platform has two WECs and one WT integrated into one offshore facility. The wave energy exploitation systems can be set in a dense pattern in the same platform as WTs.

Such a combination of an offshore wind and wave energy platform results in even more power in less sea area space (Lee et al., 2018). The installation and maintenance costs are lower for wave and wind farms because the hybrid platforms can share the same:

• Bottom and surface impression

• Interconnection wiring

• Underwater cable to the network

• Vessels

• Converter on the platform

• Installation platform

• Technical support team/staff

In addition, the wind and wave devices can be combined in a single unit or can be placed close to each other to benefit from the advantages that can be achieved by the integration of wave and wind systems.

Additional benefits, such as improved dynamics and increased power generation, can be achieved by combining offshore wind and wave devices in a single unit. Some of the advantages that can be achieved by combining wave and wind energy devices in hybrid platforms are as follows:

• Reduction of weight through common platform development

• Reducing the cost of design/operation through shared cost of

      • Combined land-based facilities and other equipment

      • Overseas transportation facilities

      • Shared use of grid connections

      • Operational, monitoring, and maintenance expenditures

• Increasing the amount of RE produced per marine region

• Optimizing the utilization of marine areas

• Simultaneous use of two energy sources

• Reduction of zero energy production hours

• Significant increase in the time window for energy supply

• Increase in the value of the capacity of integrated systems

• Reduction in the variables in electricity generation

• Smoothing of the power supply

      • Improved effectiveness;

• Reduction of errors in power output forecasting

• Less transmission capacity required

2.2   Desalination unit combined with hybrid energy systems

Research on desalination using reverse osmosis in multiple-use scenarios has not been widely published. Desalination using reverse osmosis is energy intensive and requires an accompanying energy source. Traditional desalination is usually directly or indirectly powered by fossil fuels, which makes the process unecological.

Dagkinis et al. (2013) and Dagkinis et al. (2015) reported that seawater desalination can be achieved using RES operating under variable environmental conditions and proposed that optimizing their design could make the system more sustainable. In addition, desalination using reverse osmosis with RE resources will be more acceptable to society. Katsaprakakis and Christakis (2014) investigated an offshore "pumped storage" system. Their study examined the production of desalinated water during periods when there is a surplus of electricity and when the storage capacity is exceeded. They concluded that the process would have attractive financial performance without the need for subsidies. A review of the combination of wave energy and desalination was conducted by Foteinis and Tsoutsos (2017). They stated that reverse osmosis desalination plants can be directly supplied with pressurized seawater by wave energy, which can achieve significant energy savings and reduce costs and pollution.

To achieve 100% offshore wind energy utilization, He et al. (2010) used the variable state optimal control to maintain energy consumption even with the variations in wind energy. The return on investment of offshore wind farms is increased by a large offshore desalination plant powered by offshore wind. The problem of water source pollution from onshore desalination plants is solved by offshore siting. Large-scale desalination powered by offshore wind farms, gas turbines, and hydroelectric power stations was explored by Tsai et al. (2016). Water supply systems include desalination plants, water storage tanks, and reservoirs. Maxwell et al. (2022) provided an overview of the different configurations of floating WT and considered the potential impacts on marine mammals, sea birds, fish, and the benthic ecosystem. Moreover, further research and technologies, such as entanglement monitoring and deterrence technologies, should be developed to enable the deployment of floating turbines with limited ecological effects. The use of a combination of RES can improve the reliability of the system, as well as make it economical and environmentally friendly to operate (Okampo, 2021; Wang, 2024).

3   Proposed modeling approach

Wind speed is one of the most significant variables and has a direct influence on availability, maintenance activities, and energy output. To accurately measure the power output, the wind intensity at hub height for the specific site will be used together with the turbine-specific power output, usually taken from the supplier's datasheet, which indicates the capacity factor, power produced, and revenue. An appropriate weather window must be available for maintenance to be conducted (Psomas et al., 2022). A weather window is expressed as the overall time required to complete a maintenance activity and the voyage time of the maintenance support vessel.

The proposed models for the study of an HWWPG system consist of two concepts, i.e., electric power generated through the change in wind speed and electric power generated under different operational conditions of the wind–wave energy production system. The HWWPG system consists of two dissimilar sources, i.e., exterior and interior, that can interfere with its total performance. The exterior source is the wind velocity and lightning strike, and the interior source is the degradation and failure, as well as the repair behavior, of the WT and WEC.

In this study, we assume that the conditions of the WT and WEC for the HWWPG are categorized as "fully functional", "degraded", and "failed". The sojourn times at certain states of the system are assumed to be exponentially distributed. Under these assumptions, the CTMC process {X (t), t ≥ 0} can be used to model the time progress of the HWWPG system. We let E be the state space of the Markov process {X (t), t ≥ 0}, which can be partitioned into two subsets, i. e., the subset U, which contains the operational states, and the subset D, which contains the nonoperational states. Given these two subsets, the equations E = U ∪ D and U ∩ D = Ø can be derived. The longterm behavior of the system needs to be investigated based on the stationary distribution of the state probabilities of the CTMC X (t). The steady-state analysis provides insights into the long-term reliability of the system and helps in understanding how the system performs over an extended period rather than just at a specific moment in time, ensuring consistent performance and dependability. For the computation of the stationary distribution, the known linear system of the equation πQ = 0, $\sum\nolimits_{i \in E} \pi_i=1$, where Q = $\left[q_{i j}\right]_{(i, j) \in E \times E}$ is the N by N transition rate matrix, with q_ij denoting the transition rate from state i to state j and N = dim E denoting the number of states, has to be solved. The proposed operating classification is associated with three states for the WT and WEC under the following conditions: O^W representing the fully functional state, DE^W representing the degraded state, and F^W representing the failure state of the WEC, as well as O^T representing the fully functional state, D^T representing the degraded state, and F^T representing the failure state of the WT.

3.1   Markov model of the wind speed variation

In this study, the variation of the wind velocity over 1 year will be considered. The National Meteorological Service provided the meteorological data for 1 year. These data are from the Northeast Aegean are Greece (position: N 38°30′05.23″, E 26°15′03.67″). According to these data, four wind speed scales and the corresponding wave height scales are created, from which different values of electric power are generated. Every state of the diagram shown in Figure 1 comprises a cumulative amount of electric power generated by the wind–wave energy production system. Each amount can be expressed as $g_{\mathrm{WH}_i}^{\mathrm{WS}_i}$, i = 1, 2, 3, 4, where WS is the wind speed, and WH is the wave height, which determines the different cumulative amounts of the electric power generated due to wind speed and wave height for each of the four states of the model based on the different wind speed fluctuations. The state space of the wind veloc ity model is E₁ = $\left\{g_{\mathrm{WH}_1}^{\mathrm{WS}_1}, g_{\mathrm{WH}_2}^{\mathrm{WS}_2}, g_{\mathrm{WH}_3}^{\mathrm{WS}_3}, g_{\mathrm{WH}_4}^{\mathrm{WS}_4}\right\}$, and the sojourn times at certain states of the wind speed model are assumed to be exponentially distributed. Under these assumptions, the CTMC {Z (t), t ≥ 0} can be used to model the time progress of the wind speed intensity. For the computation of the stationary distribution, the known linear system of equations π'Q' = 0, $\sum\nolimits_{i \in E_1} \pi^{\prime}=1$ has to be solved. In the equation, $\boldsymbol{Q}^{\prime}=\left[q_{i j}^{\prime}\right]_{(i, j) \in E_1 \times E_1}$ is the N by N transition rate matrix, with q'_ij denoting the transition rate from state i to state j and N = dim E₁ denoting the number of states for the wind speed Markov model. The model for all of the feasible transitions among the four states of the electric power generated is shown in Figure 1.

Figure  1  Four-state transition diagram for varying electric power generated based on the wind intensity

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Each transition rate is an indication of the number of times the wind speed has changed in a year. The transition rates $\lambda_{1, 2}^{\mathrm{WS}}, \lambda_{1, 3}^{\mathrm{WS}}, \lambda_{1, 4}^{\mathrm{WS}}, \lambda_{2, 3}^{\mathrm{WS}}, \lambda_{2, 4}^{\mathrm{WS}}$, and $\lambda_{3, 4}^{\text {WS }}$ show how wind speed increases, whereas $\lambda_{2, 1}^{\mathrm{WS}}, \lambda_{3, 1}^{\mathrm{WS}}, \lambda_{3, 2}^{\mathrm{WS}}, \lambda_{4, 1}^{\mathrm{WS}}, \lambda_{4, 2}^{\mathrm{WS}}$, and $\lambda_{4, 3}^{\mathrm{WS}}$ show how wind speed decreases. When the wind intensity changes, a transition from one state to another state occurs.

Both the WT and the WEC cannot produce 100% of their nominal rated output. The generator receives the rotational forces and uses electromagnetic induction to produce electricity. Then, the energy generated is transmitted through a cable system running down the turbine. The energy is distributed to supply electricity to homes or buildings after connection to the grid, where some voltage adjustments may be made. WTs and WECs are significantly influenced by crucial factors, such as wind intensity, wave height, weather conditions, and electrical losses, within the system and during grid transmission. These factors limit the energy production of WTs and WECs to a fraction of their nominal rated capacity.

Another vital aspect affecting WT performance is the power coefficient, which varies based on the design of the turbine and depends on several variables, including wind speed, angle of wind attack, turbulence intensity, the shape and material of the turbine blades, and the speed of the rotor. The power coefficient (C_p) serves as a critical performance metric for WTs, indicating their efficiency in converting kinetic energy into electrical energy. The theoretical maximum for this coefficient, known as the Betz limit, is approximately 59.3%. This limit, derived from the work of German physicist Albert Betz in 1919, asserts that no WT can convert more than 59.3% of the kinetic energy of wind into mechanical energy. For the specific WT used in this case study, the maximum power coefficient is 49%.

Thus, according to Li et al. (2023), Ahamed et al. (2020), Diaz et al. (2007), and Mansouri et al. (2023), the percentage of electrical losses for the WT and WEC is approximately 1.7% and 9%, respectively. Ultimately, the percentage of electrical losses of transmission to the grid is 7%. Table 1 shows the electric power generated at the specific wind speed and wave height variation by the HWWPG system, and Table 2 shows the electric power generated by the HWWPG system based on the operational status of the WT and WEC. The operational rate indicates how often the HWWPG system operates at its full or reduced capacity, and this rate is integrated into the maximum power coefficient and all of the aforementioned corresponding losses or not at all.

3.2   Markov model of the HWWPG system

MM and CM activities are performed on the wind–wave energy production system in the proposed model shown in Figure 2. MM activity is planned to be implemented to prevent the complete failure of the HWWPG system and lengthen its useful operational life. When MM activity is performed, the HWWPG system is reverted to its previous operational state. CM activity is implemented after a failure has occurred and is designed to revert the HWWPG system to the original state that enables it to perform the required functions (Koutras et al., 2017). In the occurrence of failure, the HWWPG system will be out of service until the necessary repairs have been made.

Figure  2  State transition diagram for varying electric power generated based on the wind intensity

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More specifically, we consider that the HWWPG system is in state (O^T, O^W), where O^T is the fully operational condition of the WT and O^W is the fully operational condition of the WEC. Degradation increases with the operating time of the system. In the degradation level, the system enters the state (D^T, O^W) with degradation rate λ^T, where D^T is the degradation state of the WT and O^W is the fully operational condition of the WEC. Moreover, the system enters the deterioration state (O^T, D^W) with degradation rate λ^W, where the WT remains in the perfect functioning state O^T and the WEC enters the degradation state D^W. Furthermore, the system can transition from the previous states (D^T, O^W) and (O^T, D^W) in the state (D^T, D^W) with degradation rates λ^W and λ^T, respectively, where both the WT and WEC are in the degraded state. In addition, from the latter state, the system might enter the states (D^T, F^W) and (F^T, D^W) with failure rates λ^W and λ^T, respectively, where, in the former state, the WT is in the degraded state D^T and the WEC has failed F^W and, in the latter state, the WT has failed F^T and the WEC is in the degraded state D^W. The system may reach total failure (F^T, F^W) from the state (D^T, F^W) with the failure rate λ^T and from the state (F^T, D^W) with the failure rate λ^W. Furthermore, in this study, we assume that the degradation rates of the WT and WEC from the operational states to the degraded states are identical to the failure rates of the WT and WEC from the degraded states to the total failure states.

Finally, there is the possibility of a sudden breakdown of the system, mostly caused by exterior causes, such as lightning strikes. In such a case, the system from the fully operational state (O^T, O^W) and the states (D^T, O^W), (O^T, D^W), and (D^T, D^W) can transition to the complete failure state (F^T, F^W) with failure rate λ_l and from the state (O^T, F^W) where the WT is in the perfect functioning state O^T and the WEC has failed F^W along with the state (F^T, O^W) where the WT has failed F^T and the WEC is in the fully operational condition O^W can transition to the complete failure state (F^T, F^W) with failure rates λ_l^T and λ_l^W, respectively.

MM activity has only a slight effect on the improvement of the system, and if the system is in the states (D^T, O^W) and (O^T, D^W), then an MM activity is implemented, and the WT and WEC may transition with the rates μ_M^T and μ_M^W, respectively, to the perfect functioning state (O^T, O^W). MM activity is also conducted when the WT and WEC are both in the degraded state (D^T, D^W), and the system may transition with a rate μ_M^W either to the state (D^T, O^W), where D^T is the degradation state of the WT and O^W is the fully operational condition of the WEC, or the state (O^T, D^W) with the rate μ_M^T, where the WT is in the perfect functioning state O^T and the WEC remains in the degradation condition D^W.

CM activity is implemented when the system is in the states (D^T, F^W) and (F^T, D^W). With the rate μ^W_CM, the HWW-PG system enters the state where the WT remains at the degraded state and the WEC becomes fully operational (D^T, O^W) or, with the rate μ^T_CM, the HWWPG system enters the state where the WT becomes completely operational and the WEC remains in the degraded state.

Moreover, if the WT is in the failure state and the WEC works perfectly (F^T, O^W), or if the WT is in the fully operational condition and the WEC is in the failure state (O^T, F^W), then with the CM activity, the system returns with the rates μ^T_CM and μ^W_CM, respectively, to the state where both the WT and WEC are in the fully operational condition (O^T, O^W). Finally, in the case where the WT and WEC are both in the failure state, then a CM activity is performed, and with the rate μ^{W, T}_CM, they transition to the perfect functioning state (O^T, O^W).

4   Impact of regular and intense wind speed changes on the reliability parameters of the HWWPG system

We need to determine and understand the offshore WT and WEC failure rates and repair resource requirements to model and reduce the O&M and energy costs. However, a global challenge for the industry is the reduced data volume on the failure of offshore RES and the absence of research on this subject. Reliability is important in the development of wave energy, as the consequences of faults often entail expensive and complex involvements that must be performed even at the prototype stage under inherently difficult conditions (Zhukovskiy et al., 2021). Given the consequences of failure in structures found in the marine environment involve costly repairs, the improvement of the reliability analysis for oceanic systems is to provide a stimulating and essential challenge. Therefore, the prediction of failure rates and system capacity needs to be as accurate as possible to enable further improvements and enhancements (Abramovich et al., 2020).

In contrast to other RES, WECs operate in a severe environment and require expensive procedures at sea. Therefore, additional challenges arise in the design of oceanic RES. As a result, flexible tools for the evaluation of dissimilar technologies with different features need to be developed. These tools include the assessment of reliability, availability, and maintainability, which can reduce the considerable costs involved in the application of marine RES at the planning, maintenance, and operational planning stages (Li et al., 2015).

Research has shown that lightning strikes are responsible for 7% to 10% of WT blade damage and economic losses due to blade repair, replacement, and turbine downtime are the most serious of all lightning incidents (Yokoyama et al., 2014). The blades, the tower itself, the hub, the nacelle frame, and the lightning receptors on the nacelle are the most vulnerable parts of the turbine to direct lightning strikes. In accordance with the International Electrotechnical Commission (IEC) 62305-1: 2010 standard, the four types of lightning strikes that influence a turbine are direct lightning strikes on the turbine, lightning strikes near the turbine, direct lightning strikes on the lines linked to the turbine, and lightning strikes close to the lines linked to the turbine.

To reduce the O&M and energy costs, we need to determine and understand the failure rates and repair resource requirements of offshore WTs (Belsky et al., 2019). Although there has been little published data on the failure rates of offshore WTs in the past, there is even less detail available in the public domain on the requirements for repair resources (Carroll et al., 2016). Approximately 30% of the reliability of an offshore WT and the resources required to maintain it can be attributed to the total energy cost (Dinwoodie et al., 2013). Generally, higher failure rates and repair resource requirements (i.e., material and labor costs) lead to higher energy costs (Zhukovskiy et al., 2022).

In terms of maintenance, the persistence of weather conditions is particularly important. All offshore works and repairs require a certain amount of time to complete. The effect of intense wind speed changes on the reliability of offshore WT parts is mostly revealed in two features, i.e., the increase in component failure rates and the increase in CM time after failure.

The main components of offshore WTs include the gearbox, generator, rotor blades, and yaw system. Because of the tallness of WT and its wind-operated feature, the effect of intense wind speed changes on the reliability of WT is noticeable and significant (Lin and Quemener, 2016). Previous studies have stated (Chou and Tu, 2011; Ribrant and Bertling, 2007) that mechanical parts, such as gearboxes and blades, account for the majority of WT failures, with over 70% of blade failures due to lightning strikes or extreme wind gusts.

Thus, this study considers the consequences of intense wind speed changes, as well as regular wind speed changes, that the HWWPG system normally operates on the failure rate of WT and the maintenance rate. The wind speed variation data for both regular and intense weather conditions are derived from historical records provided by the Hellenic National Meteorological Service. These records include data specific to the Northeast Aegean area covering 1 year. Using the proposed model, we investigate the impact of robust and sudden wind speed changes. Table 3 shows the wind intensity transitions under regular and intense weather conditions.

5   UGF technique description

To fit multistate models to the aforementioned systems and ensure enhanced precision in reliability evaluation, this study introduces a particular method. The approach comprises the integration of the UGF and random processes. The UGF was initially introduced by Ushakov (1986) to simplify the computational complexity of multistate systems (MSS). The mathematical foundation of this method was further expanded by Gnedenko and Ushakov (1995) and Ushakov (2000). Detailed and modern explanations of the UGF technique with various technical applications were provided by Lisnianski and Levitin (2003) and Levitin (2005).

The UGF technique enables the derivation of the complete steady-state performance sharing (PS) of the system through a rapid algebraic method based on the steady-state PS of its subsystems or components. This approach enables researchers to apply the same recursive procedures to systems with different performance characteristics and varying types of component interactions. Ushakov's (1986) introduction of the UGF has proven highly efficient for assessing the reliability of various types of MSS, as noted by Lisnianski and Levitin (2003). The u function extends the well-known ordinary moment-generating function, with the main difference being that the UGF can evaluate probabilistic distributions for the overall performance of systems with diverse topologies, interactions, and physical performance characteristics. This analytical tool is highly effective for describing multistate components and constructing comprehensive models for complex MSS.

The u functions representing individual generation units within the distributed generation system are merged to formulate the model for the entire distributed production system. Subsequently, this consolidated model is solved to compute the reliability measures, accounting for the uncertainty in load demand, which is also denoted by the u function.

5.1   UGF technique based on stochastic methods

In the general case, the power output of the HWWPG system is modeled as a discrete-state, continuous-time stochastic process X (t). The continuous-time stochastic process X (t), which has K distinct performance levels {x₁, x₂, …, x_K}, is a discrete random variable G accompanied by the sharing of state probabilities{P₁ (t), P₂ (t), …, P_K (t)}.

At any given time t, the u function of an independent discrete random X (t) is expressed as a polynomial, as follows:

where the stochastic process X (t) has K possible values and P_n(t) is the probability that X (t) is equivalent to x_n.

Asymptotically, Eq. (1) can be written as follows:

The function u(z) signifies the various potential states of a component or subsystem, establishing a connection between the probabilities associated with every state and the performance of the component or subsystem in that particular state.

To derive the u function for a system comprising two subsystems or components, composition operators are utilized. To define the u function for two subsystems connected in series, basic algebraic operations are performed on the individual u functions of each subsystem. The composition operators are expressed as follows:

The derived u function establishes a connection between the steady-state probability of every state in a system (calculated as the product of the steady-state probabilities of states for its components or subsystems) and the efficiency ratio of the system in that particular state. The function ser (·) within composition operators articulate the overall power rate of the system, comprising two components or subsystems connected in series. In the case of a series connection, the performance rate is denoted by the minimum of the performance rates of the individual subsystems. The composition operator ⊗_min is employed when calculating the overall performance rate of a system composed of two subsystems connected in series. In such a configuration, the system's total output is determined by the minimum of the individual performance rates of the subsystems. Hence, the composition operator ⊗_ser expressed for connecting a pair of components or subsystems in series correspondingly takes the following form:

5.2   Structural functions of the performance rates of the HWWPG system and the wind speed model

Let G_(i)^HWWPGS and G_(j)^WS be the variables representing the electric power generated for the different states of the HWWPG system for i = 1, …, 9 and the electric power generated due to wind speed variations for the different states of the wind speed model for j = 1, …, 4, respectively. We assume that G_(i)^HWWPGS and G_(j)^WS are independent of each other and downscaled into E_HWWPGS = {G₍₁₎^HWWPGS, G₍₂₎^HWWPGS, G₍₃₎^HWWPGS, …, G₍₉₎^HWWPGS} and E_WS = {G₍₁₎^WS, G₍₂₎^WS, G₍₃₎^WS, G₍₄₎^WS}, respectively. Let π^HWWPGS ={π₍₁₎^HWWPGS, π₍₂₎^HWWPGS, π₍₃₎^HWWPGS, …, π₍₉₎^HWWPGS} and π^WS = {π₍₁₎^WS, π₍₂₎^WS, π₍₃₎^WS, π₍₄₎^WS}, denote the steadystate probabilities of the HWWPG system and WS, respectively.

The u function of the HWWPG system is expressed as follows:

The u function of the WS is expressed as follows:

The complete u function of the HWWPG system and WS can be derived as follows:

6   Dependability assessment measures

6.1   Demand satisfaction index

An assessment of the generation capacity that is available in the electrical system is essential. In our case, this includes both wind and wave power. The capability of the system to meet consumer needs is determined by the adequacy of generation capacity relative to demand. To assess the level of satisfaction or fulfillment of consumer demand, the DSI is given as the probability of the electrical system operating normally.

In the modeling of MSS, DSI is the probability that the MSS is in states where the output is equal to or greater than the demand W (Zio et al., 2007). Through the operator δ_DSI, we can determine the availability DSI using the u function, as follows (Levitin et al., 1998):

where the operator δ_A is the sum of all of the probabilities of the system states meeting the requirement F(G_i, W) ≥ 0. Furthermore, K is the number of states of the MSS, G_i is the power output of the MSS at state i, W is the demand, and 1_{{Gi ≥ W}} is the indicator function.

The indicator function ensures that only states where the power output meets or exceeds the demand contribute to the sum. Asymptotically, Eq. (8) can be rewritten as follows:

6.2   Expected energy produced

The EEP is a prediction of the overall output performance of the HWWPG system and the estimated amount of electrical energy a WT and a WEC are expected to produce over a given period. The maximum potential output under ideal operational conditions and resource availability is the primary concern of EEP and is considered the sum of the performance rate of the total system and its related state probability, and it can be expressed as follows:

where the operator δ_EEP is defined as the sum of the total performance rate at state i, G_i, and its corresponding state probability P_i. Asymptotically, Eq. (10) can be rewritten as follows:

6.3   Expected energy not supplied

EENS is the expected quantity of energy that will not be delivered by the system to consumers during a certain period because of the lack of system power or an unexpected major shutdown (Platis et al., 1996). Thus, another crucial and necessary dependability measure, i. e., EENS, can be obtained. Therefore, we define a subset B that contains all of the operational states i for which the generated power G_i is less than the demand W. The subset is defined as follows: B = {i ∈ U |W > G_i}.

To describe the energy not supplied (ENS), let:

where 1_{{X (t) = i}} is the indicator function that is 1 if X (t) = i and 0 otherwise, G_i is the power output in state i, and W is the demand.

Thus, the EENS can be expressed as follows:

where P_i (t) is the state probability distribution at time t.

Because we are interested in the long-term behavior of the described system, we need to calculate the EENS asymptotically. Using Eq. (13), we derive the following expression:

where π_i is the asymptotic probability of the system being in state i.

Then, the EENS in a time interval of T time units can be written as follows:

7   Total expected operational cost

By incorporating the maintenance and fixed costs associated with the crew transfer vessel into the model, this research aimed to provide an accurate assessment of the TEOC of the HWWPG system over 1 year, for which recorded data were available. The reliability and total operational cost per unit of time of the system were particularly scrutinized under different weather scenarios to understand their impact on the maintenance procedures and overall system performance.

In our approach, where maintenance activities occur under various weather scenarios, the overall operational cost per unit of time is highly important (Koutras et al., 2017). Thus, in this study, the overall operational cost per unit of time is used to measure the performance of the system, as it is critical. Initially, the activities that result in the operational cost for the system need to be delineated. These activities include the MM and CM, as well as the fixed cost associated with the crew transfer vessel. To define the wind–wave energy production total expected system operational cost per unit of time, let r(i) = h_iC_S be the reward function, where:

In addition, C_S is the cost function incorporating all of the possible costs that may occur for the HWWPG system. The cost function contains the cost c_TV that occurs when a crew transfer vessel is used to transfer technicians and smaller parts. Moreover, the cost c_M occurs when an MM is implemented in the states (DE^T, O^W), (O^T, DE^W), and (DE^T, DE^W). C_CM is the cost of a CM process after a complete failure, which is implemented when the system is in the states (F^T, F^W), (DE^T, F^W), (F^T, DE^W), (F^T, O^W), and (O^T, F^W). Because of the nature of the MM, this type of maintenance has a lower cost than CM, i.e., c_M < C_CM. To describe the TEOC per unit of time, let

be the cost rate at time t, then 1_{{X (t) = i}} is the indicator function and X (t) is the state of the system at time t. In this way, the TEOC at time t for the proposed model of the system can be obtained as the expectation of the cost rate l(X (t)), as follows (Koutras et al., 2017):

where π_i(t), i ∈ E is the state probability distribution at time t.

However, our main objective is to examine the effects of the maintenance operations under different weather scenarios on the asymptotic efficiency of the proposed model. Therefore, we express the asymptotic TEOC per unit of time and the asymptotic probability distribution of the process X (t) as follows:

8   Results

8.1   Input parameters for the different weather scenarios: the north aegean case

To ensure the accuracy and reliability of the CTMC model used for the HWWPG system, a rigorous validation process was undertaken. The validation involved comparing the outputs of the model with historical operational data collected over 1 year. These data encompassed various operational states and weather conditions, enabling a comprehensive assessment of the performance of the model.

The failure, MM, and CM rates of the HWWPG system under regular and intense weather conditions are recorded in Tables 4 and 5, respectively. The data shown in Tables 4 and 5 are derived from historical records, empirical studies, and industry reports that provide statistics on the performance of WTs and WECs under typical weather conditions. In addition, the information is based on the data obtained from more extreme weather events and their impact on RES, as well as from meteorological data from national weather services that provide insights into the frequency and severity of intense weather conditions.

The case study of the HWWPG system that we examined in the area of the Oinousses Island in the North Aegean refers to an 800 kW Enercon E53 three-blade WT, horizontal axis, direct drive gearless, variable speed, single blade adjustment, ring generator. The turbine is manufactured by Enercon GmbH based in Germany. The WT initially works at a wind velocity of 3.0 m/s, and the cutoff point is at a wind velocity of 34.0 m/s. The rotor diameter is 52.9 m, and the rotor surface is 2 198.0 m². The WT is set with a set of three rotor blades. The generator is set to synchronous operation with a maximum speed of 28.3 r/min. The voltage is 690.0 V, and the grid frequency is 50.0 Hz. The reverse osmosis desalination system used in this work, which requires 5 000 kWh/day of electric power to produce 1 000 m3/day of fresh water, is powered by the grid and by a 10-kW WEC, an 800-kW WT, or by both.

The HWWPG system has two connected WECs obtained from SINN Power GmbH. The wave energy technology is a floating buoyancy chamber with a single-point absorber with a piston. The WEC consists of a floating buoy, a rod attached to it, a gearbox, a generator unit, and a voltage source inverter (VSI). The floating buoy of the WEC floats on the water and moves up and down with every wave that passes, and the pile comes in contact with the buoy and moves with it. In its upward and downward motion, the rod acts on drive rollers and rolls the set of multiple linear generator units. A gearbox, a permanent magnet synchronous motor (PMSM) with coils, and a VSI are the main components of the linear generator. The current that passes through the PMSM is controlled by the VSI so that the torque of the generator cylinders is under control. In this way, by breaking the movement of the rods, the mechanical energy can be converted into electrical energy. The linear generator units of each WEC produce 5 kW, in total 10 kW peak for the platform. The WEC is designed for modular power arrangement, which enables easy adjustments to the overall power plant layout for consumption requirements and local wave climates.

The lightning strike rates of various weather scenarios are also considered an important factor. Each system has a different likelihood of being struck by lightning. Therefore, we assume that lightning has an 82% chance of hitting the turbine because of its height, a 12% chance of hitting the shaft system, and a 6% chance of hitting both systems at the same time and causing damage. Notably, and as previously mentioned in Section 3, the lightning strike rate is higher under intense weather conditions.

To assess the performance of the offshore wind and wave power generation system under different weather scenarios, a more detailed analysis of the data for 1 year was conducted for both the WT and WEC. We assume that the failure rates of both the WT and WEC increase under intense weather conditions, resulting in 35% more damage to both systems. Furthermore, as the failure rate increases, the time necessary to perform CM on both the turbine and wave energy production system also increases.

8.2   Reliability evaluation and economic assessment

The analysis of the model data provided insightful observations regarding the operational cost and dependability of the HWWPG system. The CTMC model, combined with the UGF technique, facilitated a detailed investigation of the performance measures of the system, such as the TEOC and DSI.

The system investigated in this study is not the sole provider of electricity for the island. The Public Power Corporation (PPC) also plays a significant role in supplying electricity. The HWWPG system not only contributes to the RE capacity of the island but also operates in conjunction with DEI's conventional power grid infrastructure. This integrated approach ensures a reliable and continuous supply of electricity, enhances the overall energy resilience and sustainability of the island, and leverages both renewable sources and traditional power generation methods.

The impact of different weather scenarios on the reliability and TEOC of the HWWPG system is assessed. To comprehend the availability of a system based on the wind intensity variations, weather conditions, and MM and CM times and evaluate the performance of the system under different conditions, it is crucial to analyze not only the parameters that are related to the weather conditions but also to have reliable models and input values. All of these inputs are difficult to acquire for an immature technology, for which there is only a small amount of data available to the public. The approach adopted in this study is using available inputs and generating other experimental inputs from experts' knowledge related to the field and data of other related schemes. Using the economical input values that are listed in Table 6, the TEOC has been computed according to Eq. (19). The cost data presented in Table 6 were derived from detailed financial documentation from similar offshore wind and wave energy projects outlining the maintenance costs. Furthermore, cost estimates and maintenance guidelines were provided by equipment manufacturers like those for WTs and WECs. The demand W for electricity that needs to be met by the HWWPG piloting system is 1 280 274 kWh and concerns Oinousses Island with 1 006 residents. However, this demand changes during the periods in which the population of the area increases because of visitors, who increase the number of inhabitants by an average of 60%. As a result, the total energy demand increases, and the annual demand reaches 2 034 422.40 kWh.

Having defined the different weather scenarios and various parameters, the DSI, EEP, EENS, and TEOC can now be assessed using Eqs. (9), (11), (15), and (19), respectively. Indicative figures presenting the DSI, EEP, EENS, and TEOC with respect to the energy demand W under different weather scenarios are provided (Figures 3‒7).

Figure  3  Asymptotic demand satisfaction index with the annual energy demand W under different weather scenarios

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Figure  4  Expected energy not supplied (EENS) with the annual energy demand W under different weather scenarios

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Figure  5  Expected energy produced (EEP) and total expected operational cost (TEOC) under different weather scenarios

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Figure  6  Remaining EEP and EENS under different weather scenarios

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Figure  7  Remaining EEP with the annual energy demand of the desalination unit under different weather scenarios

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Figure 3 illustrates the change in DSI under the annual energy demand W. This figure specifies the impact of various weather scenarios on the DSI of the HWWPG system. Notably, the DSI for W = 1 280 274 kWh is lower under intense weather conditions than under regular weather conditions because, in harsher weather conditions, the failure rate, as well as the probability of an external environmental factor, such as a lightning strike, increases.

Furthermore, Figure 3 shows that when the energy demand of the island is higher (W = 1 859 222 kWh), the DSI of the system decreases. The HWWPG system is not operating at its maximum capacity all of the time because of wind speed variations, MM, CM, and electrical grid constraints, which lead to reduced DSI when the demand is high. In addition, as the demand for electricity increases, the HWWPG is insufficient to deliver the electricity generated because of the electrical losses of the WT and WEC and the electrical losses during transmission to the grid, which result in reduced DSI. Thus, the DSI of the HWWPG system during the periods when the demand is higher is expected to decrease.

However, Figure 3 illustrates a remarkable observation. Because it is a hybrid system, the HWWPG relies on both the WT and WEC to meet the energy needs of the island as demand increases. Hence, along with the higher wind speeds, the DSI of the HWWPG system is higher under intense weather conditions than under regular weather conditions. As a result, the increase in the failure rates of the WT and WEC, as well as the higher rate of lightning strikes, does not affect the DSI. Consequently, the reliance of the hybrid system on both wind and wave energy helps maintain a stable DSI. Higher wind speeds contribute more kinetic energy, enabling the WT to generate more electricity despite the adverse conditions. Moreover, the MM activity is planned to avoid complete failure and is being implemented more frequently during intense weather conditions. The DSI of the HWWPG system is higher during intense weather for the given demand because of the increased energy production from higher wind speeds, which offsets the negative impact of increased failure rates and lightning strikes. The hybrid nature of the system, utilizing both wind and wave energy, enhances its capability to maintain a stable energy supply even under adverse conditions.

Figure 4 illustrates the increase in the EENS with respect to the energy demand under regular and intense weather conditions. Intense weather conditions, such as significantly higher wind speeds, higher wave heights, and longer periods of high wind intensity, can lead to overloading and mechanical stress on the components of WTs and WECs. To ensure structural integrity, WTs and WECs are designed with specific operating limits. Exceeding these limits can result in mechanical damage or failure, which, in turn, increases the quantity of ENS. As a result, during periods of severe weather, the HWWPG system has a higher failure rate and a higher rate of lightning strikes, which, in turn, increases the ENS.

Furthermore, as the demand W (kWh) of the residents of the island increases, so does the ENS, as shown in Figure 4, which can be attributed both to the capability of the particular hybrid system under study to gratify a large proportion of the resultant requirement and to the final performance of the HWWPG system. In addition, if the electricity demand suddenly increases and the wind is not blowing at a sufficient speed, then the output of the WT may not be able to meet the increase in demand, thereby increasing the amount of energy that is not supplied.

Plotting the EEP against the TEOC helps visualize the trade-off between the energy produced and the cost incurred to produce that energy, which is crucial for economic assessment and optimization of operational strategies. By analyzing how the TEOC changes with varying EEP, stakeholders can identify operational points where the system operates most cost-effectively. Understanding the relationship between EEP and TEOC aids in financial forecasting and helps in projecting the economic viability of the system under different operational scenarios and weather conditions. The EEP by the system is higher under intense weather conditions than under mild weather conditions, as shown in Figure 5, because, at higher wind speeds, the WT is exposed to more kinetic energy in the wind, which enables it to produce more electricity.

Moreover, Figure 5 shows that the TEOC increases under intense weather conditions because of the increased wear and damage on the turbine components that result from higher wind speeds and increased operational activity. Although energy production may increase, MM and CM costs may also increase because of more frequent maintenance and the need to address wear-related issues. The overall operational costs may increase because of these additional maintenance costs. The longevity of the turbine components may be affected by continuous operation at high wind speeds and subjected to higher levels of stress, potentially resulting in a shorter service life, which could contribute to increased operational costs if it results in more frequent replacements or major overhauls.

Figure 6 presents the difference between the remaining EEP that the HWWPG system can produce annually after it has satisfied the demand of the residents of the island and the EENS, considering the various weather intensity scenarios and the demand of the Oinousses Island. Higher EEP with lower EENS indicates a reliable system, whereas higher EENS relative to EEP indicates reliability issues that need addressing. Figure 6 also shows that when the population of the island increases, which, in turn, results in an increase in the demand (W = 1 859 222 kWh), and when mild weather conditions prevail, the EENS is more than the remaining EEP of the system. From the point of view of the HWWPG system, and for a specific weather scenario, both wind and wave energy have an inherent variability that depends on the weather. Under intense weather conditions, the EEP is still significant because of higher wind speeds, and the EENS increases due to higher failure rates that reduce the overall system uptime, increased maintenance activities leading to more frequent downtimes, and more frequent lightning strikes. If the wind and wave patterns are not right, then the amount of energy produced could be less than anticipated. Fluctuating wind speeds and wave heights can affect the overall energy produced. Consequently, under regular weather conditions, the reliability of the HWWPG system is higher, with the remaining EEP surpassing the EENS, indicating a wellfunctioning system capable of meeting the demand effectively. By contrast, intense weather conditions challenge the reliability of the system, leading to higher EENS despite increased energy production. The higher failure rates, frequent maintenance, and increased lightning strikes during intense weather significantly impact the overall performance and operational costs of the system. This comparative analysis underscores the importance of robust maintenance strategies and system resilience to optimize performance under varying weather conditions.

Figure 7 presents the production of energy under different weather intensity scenarios for the different demands of the desalination unit. Figure 7 also shows the remaining expected energy that the HWWPG system can produce annually after accounting for the demand of the residents, as well as the remaining EEP after the EENS. Notably, the cases where the HWWPG system can meet the annual needs of the desalination unit for the production of fresh water are when the HWWPG system produces a total of 604 577.73 kWh and 721 418.44 kWh. However, for the specific data used, regarding the nominal power of the WT and WEC, as well as the wind speed, two of the four cases cannot meet the demand. Inefficiencies and losses can occur in the process of converting energy from wind and wave sources into electricity. As mentioned previously, the overall energy output of the system can be reduced by several factors, such as transmission losses, conversion losses, and mechanical inefficiencies in the WT or WEC.

Plotting the remaining EEP (after accounting for the demand of the residents) against the demand of the desalination unit shows whether the system can supply additional energy requirements, ensuring that secondary applications (e.g., desalination) are adequately powered. This plot helps in deciding how to allocate the generated energy between different needs and ensures that the primary energy demand (i.e., the needs of the residents) is met first; then, any surplus can be directed to secondary needs, such as desalination. This plot also provides insights into how effectively the system is being utilized. If the remaining EEP frequently exceeds the secondary demand, then it indicates the potential for further utilization or expansion of the system.

Table 7 presents the expected energy that the investigated system can produce annually, the EENS due to degradation, and the failure of both systems under different weather scenarios. The EEP depends only on the wind intensity variations; thus, it is constant for all of the cases that we investigated. By contrast, EENS depends on the demand; as a result, the remaining EEP of the HWWPG system has satisfied the demand of the residents of the island, as shown in Table 7. In addition, the table shows the production of energy after a certain amount is lost and the amounts of energy that can meet the demand of the desalination unit.

9   Conclusions

9.1   Main findings

The study presents a model that uses a mathematical approach to evaluate the operational condition of an HWWPG piloting system and a wind speed model that represents the electric power generated at different wind velocity variations and wave heights. The reliability indices of the HWWPG system were evaluated considering two different weather scenarios. In addition, the UGF technique is developed for the calculation of the performance measures of the HWWPG system. To acquire the final system model, which will enable the straightforward computation of the reliability measures, their states are combined using a novel, properly introduced composition operator. The regular and intense wind speed variation data have been analyzed, and the susceptibility of the hybrid HWWPG system has been assumed.

By applying the model to a real case scenario, the study showcases its capability to accurately reflect the performance and reliability of the HWWPG system under varied weather conditions. Leveraging historical and empirical data ensures that the inputs of the model are grounded in real-world observations, enhancing the reliability and accuracy of the predictions. Furthermore, the CTMC approach is well-suited for reliability analysis and provides a dynamic method to simulate the performance of the system over time, thereby validating its capability to capture the stochastic nature of system failures and repairs. The established mathematical foundation of the UGF technique and its capability to handle complex systems provide a robust framework for validating the outputs of the model. By benchmarking against established reliability data from onshore systems, the study also provides a reference point that supports the credibility of the modeled performance of the offshore system.

Key findings indicate that the failure rates and maintenance costs of both WT and WEC increase under harsh weather conditions, leading to higher TEOC and EENS. Despite these challenges, the systems still show potential for substantial energy output, although at increased costs. The DSI analysis further underscores the capacity of the hybrid system to meet the energy demands, even during periods of increased need, because of its dual reliance on wind and wave energy. These findings contribute significantly to the field of environmental management by providing a detailed evaluation of the dependability measures of HWWPG systems. The methodology and results of the study provide practical insights into optimizing maintenance strategies and designing more resilient RES.

In addition, as the demand increases, the HWWPG system is dependent on both the WT and WEC to meet the energy requirements of the island. Therefore, together with the higher wind speeds, the DSI of the HWWPG system is greater under harsher weather conditions than under mild weather conditions. Consequently, the increased failure rates and lightning strikes do not affect the DSI. Furthermore, the effect of intense weather on the HWWPG system is depicted by the TEOC of the HWWPG system. In addition to the wind sources in the maritime region, the reliability of the HWWPG system is dependent on the weather circumstances, e.g., the greater wave heights. The proposed model is expected to increase the precision of the assessment results of the reliability of the HWWPG system and provide some useful guidelines for the operator. Moreover, the EENS and TEOC increase under harsh weather conditions because of the mechanical stress on the components of WTs and WECs and the longer periods of high wind intensity. If energy production increases, then the MM and CM costs also increase. The total operational costs may increase because of the additional maintenance costs. The data analysis underscored the significant influence of weather on system performance. Wind speed variations and harsh weather events were major factors affecting the failure rates of both WT and WEC. The need for robust maintenance protocols was emphasized to mitigate the impact of severe weather, ensuring continuous energy production and system reliability.

In conclusion, the validation and data analysis of the CTMC model for the offshore HWWPG system demonstrated its accuracy in reflecting real-world operations. The model provided valuable insights into the operational costs and dependability of the system, highlighting the importance of adaptative maintenance strategies to manage the challenges posed by varying weather conditions.

9.2   Future research directions

For future work, the model can be advanced to the integration of other kinds of harsher weather beyond the effects of higher wind speeds, lightning, and waves that we considered in this study, such as icing or higher temperature in critical components due to lightning strikes. Although we have only investigated the variation of the transition rates between the wind intensity states by assuming that the failure and repair rates are constant, we intend to extend our study by considering the evolution of wind intensity over time in a cyclic manner, where the transition probabilities between different wind speed states vary with time. This kind of behavior can be described as cyclical, and for this reason, we can use a cyclic nonhomogeneous Markov chain.

It would also be interesting to add more wind speed categories, although this would make the model more complex, to obtain more accurate results on the different performance measures of our model. The HWWPG system design phase can also incorporate the reliability assessment model combined with other RES, such as solar panels, to further optimize its configuration reliably.