The power grid model ensures the power balance of the energy island and the conditions for the secure operation of the system, as outlined below.

The mathematical programming model for operating energy islands must account for constraints related to both the physical limits of power grid components and operational limits to ensure safe system operation. Specifically, the proposed model includes thermal limits for AC lines (Eqs. –) and HVDC lines (Eqs. –), power limits for converters (Eq. ), AC nodal voltage bounds (Eq. ), and DC nodal voltage bounds (Eq. ). 15‖νi,t[yij(νi,t-νj,t)]∗‖≤Sijmax⁡,∀ij∈Lac,∀t∈T16‖νj,t[ykm(νj,t-νi,t)]∗‖≤Sijmax⁡,∀ij∈Lac,∀t∈T17‖ρui,tyij(ui,t-uj,t)‖≤Pijmax⁡,∀ij∈Ldc,∀t∈T18‖ρuj,tyij(uj,t-ui,t)‖≤Pijmax⁡,∀ij∈Ldc,∀t∈T19‖sc,t‖≤scmax⁡,sc,t=pc,t+jqc,t,∀t∈T20Vacmin⁡≤‖νi,t‖≤Vacmax⁡,∀i∈Nac,∀t∈T21Udcmin⁡≤ui,t≤Udcmax⁡,∀i∈Ndc,∀t∈T

Here, Eqs. () and () ensure that the apparent power flow through AC transmission lines does not exceed the thermal limits imposed by their capacity, denoted as Sijmax⁡. Similarly, Eqs. () and () enforce the maximum permissible power transfer on DC transmission lines, defined by Pijmax⁡. The converter operational limits are enforced by Eq. (), which constrains the apparent power sc,t within the converter's rated capacity scmax⁡. Voltage magnitude constraints for AC and DC nodes are imposed by Eqs. () and (), ensuring that nodal voltages remain within the prescribed operational limits of Vacmin⁡andVacmax⁡, and Udcmin⁡andUdcmax⁡, respectively.

The proposed strategy for AC/DC energy islands includes the active power dispatch of wind farms, constrained by wind power availability. Forecast wind power values are integrated into the model through Eq. (). Additionally, wind turbines can contribute to reactive power support, which is particularly relevant in offshore AC applications. The proposed approach accounts for this by incorporating reactive power dispatch. This capability is modeled by including the turbine's capability curve, which is approximated by limiting the wind turbine's apparent power to its maximum value, as defined in Eq. (). 22ℜsi,tw≤fi,t⋅Pi,nomw,∀t∈T23‖si,tw‖≤si,max⁡w,si,tw=pi,tw+jqi,tw,∀t∈T, where si,tw∈C represents the complex power generated by the wind turbine i at period t, with pi,tw∈R and qi,tw∈R denoting its active and reactive power components, respectively. The parameter fi,t∈R corresponds to the forecasted available wind power for turbine i at time t, while Pi,nomw∈R represents its nominal active power capacity. The variable si,max⁡w∈R defines the maximum apparent power output of the wind turbine. The notation ℜ(⋅) extracts the real part of a complex number, and ‖⋅‖ denotes the Euclidean norm.

Energy islands could improve flexibility by integrating energy storage systems to manage wind power intermittency, grid constraints, and curtailments. Conventional BESS, equipped with power converters, regulate both active and reactive power, minimizing curtailments while enhancing reactive power support for AC energy islands. The energy storage system is represented by Eqs. ()–(), following the linear model in . This approach determines the BESS energy state using the previous state within the prediction horizon. Charging and discharging losses are incorporated through efficiency rates ηc and ηd. To ensure continuity between time windows, the initial (E0) and final (Ef) energy states are enforced in Eq. (), ensuring that the energy state of the system is consistent and maintains a seamless transition from one time window to the next. Additionally, BESS power and storage capacity limits are imposed by Eqs. ()–(). Here, et represents the state of energy (SoE) of the BESS at period t, while et-1 corresponds to the SoE from the previous period. The variables ptc and ptd denote the charging and discharging power of the BESS at period t, respectively. The parameters emin⁡ and emax⁡ define the minimum and maximum allowable SoE, while Pc,max⁡ and Pd,max⁡ impose the upper bounds on charging and discharging power. 24et=et-1+ηcPtc-1ηdPtd,∀t∈T25etini=E026etfin=Ef27emin⁡≤et≤emax⁡,∀t∈T280≤Ptc≤Pc,max⁡,∀t∈T290≤Ptd≤Pd,max⁡,∀t∈T

Similar to wind turbines, BESS can provide reactive power support through the Q-control of their converters, enabling additional grid services. This capability is modeled as a control variable in the optimization framework to meet the reactive power demands of the AC grid. To this end, the converter's capability curve is incorporated by limiting its apparent power to the maximum allowable value, as shown in the following equations: 30stb=ptd-ptc+jqtb,∀t∈T31‖stb‖≤sbmax⁡,∀t∈T, where the complex power injected or absorbed by the BESS at period t is denoted as stb, which consists of an active power component (ptd-ptc) and a reactive power component qtb. Additionally, Eq. () ensures that the apparent power of the BESS does not exceed its nominal capacity, denoted as sbmax⁡.

Hydrogen production systems exhibit inherently nonlinear behavior, particularly in electrolyzers, where efficiency depends on factors such as voltage, current density, temperature, and degradation effects. This nonlinearity primarily stems from the electrochemical relationship between electrolyzer voltage and current density, which influences power input and hydrogen output, as explained in the work by . Common modeling approaches include constant-efficiency, polynomial, piecewise linear, and convex approximations, as discussed by and . Although constant-efficient models are widely used, such as the works by and , they have limitations in capturing these nonlinear dynamics. To enhance accuracy, the proposed approach incorporates a linear model, formulated through the following constraints: 32Mt=Mt-1+ht,∀t∈T33ht=bhPte+ch,∀t∈T34Mti=M¯,∀t∈T35Mtf=M‾,∀t∈T360≤Mti≤M¯,∀t∈T, where hi,t represents the hydrogen produced at node i during period t. In addition, the model considers the initial cumulative hydrogen state Mi,ti and its final state Mi,tf, which correspond to the hydrogen demand for the analysis period. Finally, Pi,te represents the demanded power of the electrolyzer, and bih and cih denote the parameters of the linear model for the green hydrogen production system.

The objective function of the proposed framework is designed to maximize revenue from offshore wind power generation, reflecting the primary economic driver of hybrid AC/DC energy islands. This focus ensures optimal utilization of renewable energy resources while indirectly minimizing system power losses through efficient active and reactive power dispatch. By optimizing resource allocation within the nonlinear AC/DC power flow model, the framework achieves a reduction in power losses, as demonstrated in Sect. 5.2. Additionally, the model accounts for curtailments through network constraints, with BESS and hydrogen production mitigating potential curtailments by storing or converting excess energy, as detailed in Sect. 5.3. This approach enhances operational efficiency and economic performance while maintaining computational tractability for multi-period and seasonal analyses. The mathematical expression for the revenue maximization objective is as follows: 37max⁡z=∑t∈T∑i∈CCi,tPi,tm, where Ci,t represents the power price at country i during period t and Pi,tm is the optimization (decision) variable representing the active power delivered to country i in period t within the time window T.

Accurate forecasting of wind power generation and electricity market prices is critical for the optimal operation of AC/DC energy islands. However, forecast errors are inevitable due to the inherent variability of wind and the stochastic nature of electricity markets. Ignoring these uncertainties can lead to suboptimal decisions and potential economic losses. Therefore, it is essential to evaluate the robustness of the proposed optimization framework under realistic conditions that account for forecast inaccuracies.

To this end, a probabilistic power flow (PPF) analysis was performed to assess the impact of uncertainty on the system. A Monte Carlo simulation approach was applied, where multiple realizations of the uncertain input variables – namely, wind power generation and energy prices – were generated according to their probabilistic distributions and used as inputs to the nonlinear optimization model. This methodology enables a systematic evaluation of how variability in forecasts affects the operational performance and economic outcomes of the energy island. The detailed methodology and results of this analysis are presented in Sect. .

The Princess Elisabeth Energy Island is planned for construction by the Belgian Transmission System Operator (TSO) Elia in the Belgian sector of the North Sea, approximately 45 km offshore, as described in , , and , and illustrated in Fig. . The energy island is designed to integrate three future offshore wind power plants: a 700 MW installation (PE-I), expected to be operational by 2028, and two additional wind power plants (PE-II and PE-III), each with a capacity of up to 1400 MW, planned for 2029, as shown in Fig. a.

The infrastructure will incorporate both AC and DC technologies, where AC cables will be used for wind farm collection, while high-voltage direct current (HVDC) connections will facilitate interconnections. The HVDC links with Denmark and Great Britain are scheduled for commissioning in the future. While some elements, including the full 3.5 GW capacity and the hybrid interconnector with the UK, may be postponed or scaled back , the core construction of the island and HVAC infrastructure for the initial 2.1 GW offshore wind connection continues uninterrupted. Given that these wind farms are yet to be developed, this study defines and simulates their turbine position layout within the designated areas, employing commercially available tools from . Assuming full capacity utilization, the layout is designed using the IEA-22 MW reference wind turbine defined in , applying a staggered grid arrangement with optimized row orientation and spacing for the local wind rose to minimize wake losses, as shown in Fig. a. Wind power plant PE-I contains 32 turbines, while wind power plants PE-II and PE-III contain 64 turbines.

The system includes a green hydrogen production unit and a battery energy storage system to utilize surplus wind energy for industrial decarbonization. Hydrogen is expected to play a critical role in decarbonizing hard-to-abate sectors, such as steelmaking, ammonia production, and heavy transport, due to its high energy density and versatility, as highlighted by the and the . Given the current absence of a mature hydrogen market, a fixed daily hydrogen production quota is assumed to represent anticipated demand, reflecting policy-driven or contractual obligations to supply industrial end-users. For downstream logistics, hydrogen is transported offshore via marine carriers rather than pipelines, motivated by: (i) geographical flexibility, enabling delivery to multiple European ports, aligning with the cross-border hydrogen economy vision ; (ii) operational flexibility, allowing production during high wind generation periods to reduce curtailment; and (iii) infrastructure scalability, avoiding large upfront investments in dedicated pipelines while offering a modular solution for future hydrogen value chains.

To support the analysis, Fig. b presents the single-line diagram of the test system, conceptually based on the Princess Elisabeth Energy Island. The diagram illustrates the topological structure, including offshore wind farms, internal AC collection systems, HVDC converters, storage options, and export transmission links to multiple regions. This configuration serves as the foundation for simulation scenarios used to verify the proposed optimization framework. The main technical parameters are summarized in Table , including the parameters of the BESS, which are based on the study by , and the parameters of the green hydrogen system, calculated as explained in Sect.  and .

The hydrogen production system in this study was modeled in more detail and implemented in MATLAB Simulink, from which measurements were obtained to identify the parameters of a linear hydrogen production model, considering both the electrical and downstream components of an electrolyzer. The electrical model is based on a voltage–current characteristic, using an empirical formula with approximated parameters for alkaline electrolyzers from . The downstream model links the electrolyzer current to the hydrogen production rate via chemical coefficients and Faraday efficiency, following the equations and parameters from . The electrolyzer stack is modeled as a series connection of cells, introducing nonlinearity between consumed power and hydrogen production due to the voltage–current characteristic and Faraday efficiency effects at low current levels. Additionally, the measurements are taken on the DC side, excluding power supply losses. Finally, the parameters used for the measurements are detailed in Table .

A significant challenge in modeling hydrogen production systems is accurately estimating the parameters of the model presented in the Sect. . This section presents a methodology for estimating the parameters of the green hydrogen production model through an optimization approach. Accordingly, the following optimization problem is proposed: 38argminbh,ch∑k=1MEks.t.Ek=‖hk-h̃k‖,hk=bhPke+ch, where h̃k is the measurement of the hydrogen produced with the power Pke, hk is the hydrogen production estimation based on the linear model presented in Eqs. ()–(), and bh and ch are the parameters estimated by the optimization problem presented in Eq. (). Similarly, the parameter of the constant-efficiency model is estimated using the problem model presented in Eq. (), with ch=0.

The estimated parameters of the linear model for the hydrogen system are presented in Table . Figure a illustrates the estimation performance of the hydrogen production system using both constant-efficiency and linear models, identified against the measured data. The associated modeling errors are presented in Fig. b. The linear model achieves a maximum error of approximately 12.91 %, with an average and median error of 1.20 % and 0.69 %, respectively. In comparison, the constant-efficiency model exhibits a higher maximum error of 14.95 %, along with an average and median error of 3.98 % and 2.08 %. These results clearly indicate the improved accuracy of the linear approach. In particular, the significantly lower mean and median errors of the linear model reflect its enhanced robustness and sensitivity in capturing variations in the input power to the hydrogen electrolyzer, making it more suitable for dynamic operational conditions.

As observed in Fig. b, the largest discrepancies in hydrogen production estimation occur when the electrolyzer operates at low power levels. However, as shown in Fig.  in Sect. , the hydrogen production system operates most of the time close to its nominal capacity across different seasons. Consequently, the practical impact of estimation errors at low power levels is limited, as the system predominantly operates at levels where the linear model demonstrates high accuracy.

This section analyzes the wind power profiles of the three offshore wind power plants integrated into the Princess Elisabeth Energy Island. The objective is to characterize both seasonal and spatial patterns in the available power, based on hourly production data over a full year. The production data were generated by a simulation of the wind parks with wake losses in the Youwind platform with the Niels Otto Jensen wake model (), using wind speed and direction time series provided by the , which are based on LiDAR measurements at the PE sites. These patterns are used to identify representative daily profiles for each season, which are later used to define typical operating conditions in the optimization framework. The analysis includes the statistical distribution of daily average power, representative daily curves, and turbine-level power output for a representative day.

Figure  shows the seasonal distribution of average daily power for each of the three wind power plants. The boxplots summarize the statistical variability of the available power across the four seasons, based on daily average values. The central line indicates the median, the box spans the interquartile range (25th to 75th percentiles), and whiskers extend to 1.5 times the interquartile range. Autumn shows the highest median daily power across all wind power plants, while summer consistently presents the lowest. Spring and winter exhibit intermediate values, with winter showing greater variability. Rather than using a full year of data, we demonstrate the optimization on representative days for each season, to keep the calculation time for the study limited, while still capturing the key seasonal characteristics of offshore wind variability. Based on these analyses, we obtained turbine-level power forecasts, which constitute one of the key inputs for implementing the proposed optimization approach.

Figure  shows the representative daily time series of hourly wind power generation for each wind power plant: (a) PE-I, (b) PE-II, and (c) PE-III. For each season, a representative day was selected by identifying the daily profile whose average total power was closest to the seasonal median. The curves reflect typical intraday generation behavior under seasonal wind conditions. These profiles provide seasonally realistic input scenarios for the operation of the hybrid energy island.

Finally, Fig.  shows the hourly wind speed and power output per turbine in PE-I for the representative winter day. The plot illustrates the spatial variability in turbine performance over the course of a day, influenced by wake effects and wind direction, and highlights the importance of considering spatial resolution in wind power modeling.

In this subsection, we evaluate the performance of the optimal operation strategies proposed for the Princess Elisabeth Energy Island, focusing on representative days that capture typical seasonal variations in wind power availability and electricity prices. The nonlinear mathematical programming model, developed in Sect. , is implemented using the Python-based optimization modeling library Pyomo, as described by and solved with the IPOPT solver, developed by . It employs a primal-dual interior-point algorithm with a filter line-search method to efficiently handle the nonlinearities inherent in the system.

However, before proceeding with the multi-period and seasonal analyses, a preliminary verification of the proposed optimization approach was conducted to ensure its reliability under nominal operating conditions. Specifically, a cross-check was performed using the Python-based power flow tool pyflow_acdc, developed by . pyflow_acdc is an open-source library designed for power flow and optimal power flow (OPF) studies in hybrid AC/DC networks. However, it does not support the modeling of energy storage systems or green hydrogen production units, which are essential for the comprehensive operation of energy islands. Therefore, the verification was limited to a simplified case excluding BESS and hydrogen production. The comparison shows that both approaches yield very similar results, with the proposed method achieving a 1.5 % improvement in the objective function value compared to pyflow_acdc, thereby confirming the accuracy and robustness of the proposed optimization framework.

After verifying the proposed approach, the analysis focuses on the representative days selected to characterize the seasonal operation of the energy island. Figure  illustrates the seasonal representative daily profiles of hourly power generation for the three wind power plants (PE-I, PE-II, and PE-III) integrated into the energy island, with each plant's profile displayed in dedicated panels. These profiles provide a detailed representation of wind power availability across different seasons, serving as a fundamental input for the optimization model. Complementing this, Fig.  shows the day-ahead electricity price profiles for the representative seasonal day in the Great Britain, Belgium, and Denmark, which are pivotal for the economic optimization of the energy island's operation. The electricity price data for Belgium and Denmark were obtained from , while the Great Britain prices were sourced from . Leveraging these wind power and electricity price profiles, the nonlinear model optimizes the power dispatch from the wind turbines, the charging and discharging schedules of the battery energy storage systems, and the production rates of green hydrogen, while accounting for the detailed characteristics of the energy island's AC/DC infrastructure. The results demonstrate the effectiveness of the proposed optimization framework in coordinating the energy island's resources, ensuring both operational efficiency and system reliability across diverse conditions.

To facilitate a structured analysis, the evaluation has been divided into two stages. First, we present a detailed operational analysis of a representative day for the autumn season. This focused assessment allows for an in-depth examination of the system's performance throughout a typical day under a specific seasonal scenario. Subsequently, we extend the analysis by presenting the representative daily profiles for all four seasons. This broader assessment demonstrates the capability of the proposed approach to adapt the optimal operational strategy to the varying conditions across the year.

Figure  presents the nodal voltages and turbine-level reactive power injections for the representative autumn scenario, clearly evidencing the correlation between reactive power dispatch and voltage magnitudes. This relationship is fundamental in offshore AC networks, where voltage regulation plays a critical role in ensuring system stability – particularly during periods of high wind generation or significant power export. Capturing this interaction requires a modeling framework capable of representing both voltage magnitudes and reactive power flows, which is not possible in simplified DC power flow approximations where reactive power is neglected and voltages are assumed constant. To address this, the proposed nonlinear AC power flow formulation explicitly models and optimally dispatches reactive power from both the wind turbines and the battery energy storage system (BESS), enabling coordinated voltage support and a reduction in active power losses. In our case study, dispatching reactive power from the wind turbines and the BESS reduced total system losses by approximately 1 % compared to an equivalent scenario without such reactive power support. Furthermore, the high sparsity of the network admittance matrix ensures competitive computational performance despite the increased modeling detail: for example, the autumn representative day simulation was completed in approximately 27 s on a standard laptop (Intel Core i5-1235U, 16 GB RAM, Python 3.11). This demonstrates that the AC-based nonlinear formulation not only delivers a more accurate and operationally meaningful representation of the energy island but also remains computationally tractable for extended multi-period and seasonal analyses.

Figure  presents the representative autumn daily profiles of electricity prices and dispatched power for Belgium (BE), Great Britain (GB), and Denmark (DK). The top panels show that electricity prices peak around EUR 199 MWh⁻¹ at hour 10 in BE, EUR 191 MWh⁻¹ at hour 6 in GB, and EUR 199 MWh⁻¹ at hour 10 in DK. In response, the bottom panels illustrate the dispatched power profiles optimized according to these electricity price profiles. In BE, substantial exports occur between hours 8 and 22, with a maximum dispatch of approximately 3.43 GW at hours 19–20, coinciding with the higher electricity prices observed during these periods. In GB, exports are concentrated around hours 4–7 and 15–18, following the increases in electricity prices during the early morning and late afternoon. Despite similar price levels between BE and DK, dispatched power to DK remains lower, with peaks around 656 MW at hour 19, primarily due to the greater distance and associated transmission losses between the energy island and the Danish grid. The operation of the BESS, as illustrated in Fig. , further enhances the dispatch strategy: the BESS charges predominantly during lower-price periods around hours 15–18 and 22–24, with charging powers up to 1155 MW, and discharges during high-price intervals, notably at hours 10 and 19–20 with discharging peaks above 1140 MW. The hydrogen production system dynamically adjusts its operation, reducing electrolyzer power to 22.5 MW at hours 10 and 19–20 to prioritize electricity exports during price peaks, while otherwise absorbing surplus renewable generation, thus guaranteeing the fulfillment of the daily hydrogen production demand, which constitutes a critical operational constraint for the energy island.

Figures , , and illustrate the seasonal operation of the energy island, covering the hourly electricity prices, the dispatched power to the onshore grids, and the performance of the battery and hydrogen production systems. The results clearly reflect the seasonal variability in both offshore wind generation and market prices. Despite these fluctuations, the proposed optimization framework dynamically adjusts energy dispatch, BESS management, and hydrogen production to maximize economic revenues while ensuring the secure operation of the energy island and the power system. In the representative days analyzed, the dispatched energy varies from 15 506 MWh in summer to 51 246 MWh in winter, with intermediate values of 23 018 MWh in spring and 43 971 MWh in autumn. Similarly, the revenues range from approximately EUR 5.31 million in spring to EUR 10.14 million in winter, with EUR 6.27 million in summer and EUR 7.26 million in autumn.

This section presents a case study to evaluate the effectiveness of the proposed approach in mitigating curtailments under constrained transmission conditions. To simulate a contingency scenario, the capacity of all HVDC transmission interconnections is reduced to 33 % of their nominal value, representing a severe limitation in power export capability, such as might occur during maintenance or unexpected outages. This case scenario analysis tests the robustness of the proposed nonlinear optimization framework, described in Sect. , by assessing its performance under atypical operating conditions. The optimization objective, focused on maximizing revenue from offshore wind generation, indirectly minimizes curtailments by prioritizing efficient resource allocation, including BESS charging/discharging and hydrogen production scheduling. The results demonstrate how the BESS stores excess energy during periods of high wind generation and constrained transmission, while the hydrogen production system absorbs surplus power to meet daily production targets, thereby reducing curtailments and enhancing economic performance.

Figure  illustrates the dispatched power profiles to the onshore grids of Belgium (BE), Great Britain (GB), and Denmark (DK), alongside the operational profiles of the BESS and hydrogen production systems for the representative day under constrained HVDC capacity. During hours 4–8, the optimization framework prioritizes power dispatch to GB, as shown in Fig. , driven by peak electricity prices in GB during this period, as observed in Fig. . This strategic allocation maximizes revenue by capitalizing on high market prices, aligning with the optimization objective.

From hour 8 onward, the dispatched power to Belgium reaches the maximum allowable capacity under the constrained HVDC limits, as depicted in Fig. . This preference for Belgium is primarily due to its proximity to the Princess Elisabeth Energy Island, which minimizes transmission losses compared to GB and DK. Additionally, electricity prices in Belgium during these periods are generally comparable to or higher than those in GB and DK, as shown in Fig. , further incentivizing exports to Belgium. In contrast, the dispatched power to GB exhibits lower peaks around hours 8 and 12, as seen in Fig. , reflecting the influence of lower electricity prices in GB during these hours, consistent with Fig. .

Beyond hour 8, the dispatched power to all three countries frequently reaches the maximum constrained HVDC capacity, as observed in Fig. . This behavior is driven by a significant increase in available wind power from the wind power plants (WPPs), particularly WPP PE-I around hour 12 and WPP PE-III from hour 16 onward, as indicated by the wind profiles in Fig. . To mitigate curtailments under these conditions, the optimization framework prioritizes hydrogen production from hour 8, as shown in Fig. , where the electrolyzer power Pe(t) increases to absorb surplus wind generation, ensuring that daily hydrogen production targets are met while minimizing unutilized renewable energy.

The BESS complements the hydrogen production system by dynamically adjusting its operation to balance the constrained transmission capacity and variable wind generation. As illustrated in Fig. , the BESS discharges during the early hours when wind power availability is low, supporting power exports to the onshore grids. During periods of high wind generation, particularly around hour 12 for wind power plant PE-I and from hour 16 for wind power plant PE-III, the BESS charges to store excess energy, as shown in Fig. , thereby preventing curtailments.

This section presents a probabilistic assessment of the energy island's operation under uncertainties in wind power and electricity prices, using Monte Carlo simulations to evaluate variability in key performance indicators, as detailed in Sect. . In this study, wind power generation and electricity market prices were modeled with Gaussian-distributed uncertainty, using the forecasted value as the mean and a standard deviation of 10 %, consistent with the approaches in and .

A Monte Carlo simulation approach was applied to evaluate the impact of these uncertainties on the operational performance of the energy island. Multiple realizations of the uncertain input variables were generated and used as inputs to the nonlinear optimization model. A total of 1000 simulation runs were performed per scenario, and the statistical distribution of the resulting economic benefits was analyzed to quantify the impact of forecast uncertainty. Finally, a statistical analysis of the output values is performed, studying key indexes to evaluate and visualize the probability distribution of the resulting effect of the uncertainty of energy price and wind power forecasting on the operation of the energy island. Three scenarios were considered: (i) uncertainty in wind power forecasts, (ii) uncertainty in electricity price forecasts, and (iii) combined uncertainty in both wind power and prices. Table  summarizes the key statistical indicators of the economic profit obtained for the simulations of each scenario.

The following points highlight the primary findings from the Monte Carlo analysis of wind power and price uncertainty impacts on the energy island's economic performance:

Wind power uncertainty alone has a minor effect on economic profits, as indicated by the very low coefficient of variation (0.14 %) and narrow percentile range. The mean and median are almost identical, highlighting the symmetry of the distribution. Relative to the deterministic case, this scenario reduces the mean profit by about 2.20 %, with risk metrics like the conditional value at risk (CVaR) showing a slightly larger drop of 2.48 %.