The increasing penetration of renewable energy sources (RES) in the energy mix, particularly solar photovoltaic and wind power, poses significant challenges to electricity grid reliability due to their inherent stochastic variability. This study develops a stochastic framework to assess the ability of RES to balance electricity demand, with a focus on storage requirements and reliability implications. Using North Euboea, Greece, as a representative case study, normalized hourly time series of electricity demand, solar irradiance, wind speed, and temperature are analyzed to match per-capita annual energy consumption. Stochastic properties are quantified through climacograms, autocorrelation functions, cross-correlations, and estimation of the Hurst–Kolmogorov exponent, revealing strong long-term persistence in both demand and renewable generation. Results show that, despite annual energy sufficiency, demand is met only 32% of the time for photovoltaics and 44% of the time for wind power in the absence of storage. Introducing moderate storage capacity equivalent to approximately half of the average daily demand (6 kWh per capita) increases reliability to about 70-71%, yet substantial unmet demand and curtailment persist. The weak correlation between wind generation and demand, compared to a moderate correlation identified for photovoltaics, further exacerbates system imbalance. The pronounced long-range dependence of the examined processes implies clustering of deficits and surpluses, significantly increasing reliability risks. The findings demonstrate that achieving high reliability in high-RES systems requires storage and backup capacities far exceeding those implied by average energy balances. Robust energy system planning must therefore explicitly account for stochastic variability, persistence, and demand–supply misalignment when evaluating renewable-dominated power systems.
Over the past three decades, there has been a significant global shift in the energy mix toward Renewable Energy Sources (RES), primarily solar photovoltaic (PV) and wind power [1]. While the theoretical appeal of harnessing energy from the sun and wind is compelling, the inherent uncertainty and intermittent nature of solar radiation—driven mainly by variability in the atmospheric clearness index [2], and wind-speed—arising primarily from its turbulent nature [3], both treated as stochastic processes, pose substantial challenges to grid reliability [4]. Unlike dispatchable fossil-fuel or nuclear generation, solar [5] and wind [6] power output varies unpredictably with weather conditions, leading to mismatches between supply and demand.
The increasing penetration of stochastic renewable energy sources has intensified research efforts focused on the probabilistic characterization and reliability assessment of power systems under high variability conditions [7,8,9]. A substantial body of literature has demonstrated that both solar irradiance and wind speed exhibit pronounced variability across multiple temporal scales, significantly affecting the balance between electricity supply and demand [10,11,12]. Within this context, stochastic frameworks—particularly those incorporating long-term persistence, scaling behavior, and autocorrelation structures—have proven essential for capturing the intrinsic dynamics of energy-related processes [13,14,15]. Importantly, several studies have emphasized that matching annual energy production to demand is insufficient to ensure system reliability, as such approaches neglect the temporal distribution and clustering of deficits and surpluses [16,17,18]. Consequently, a range of reliability metrics and adequacy indicators has been developed to explicitly account for temporal variability and the interaction between demand and renewable generation profiles [19,20,21].
In parallel, the integration of energy storage systems and flexible balancing mechanisms has emerged as a central theme in the transition toward high-renewable energy systems [22,23,24]. Previous studies indicate that even moderate storage capacities can substantially enhance system reliability. However, they are often insufficient to fully mitigate the effects of persistent variability and prolonged low-generation periods [25,26]. The degree of correlation between electricity demand and renewable generation is also a critical factor, with low or negligible correlation—particularly in the case of wind energy—leading to increased system imbalance and infrastructure requirements [27,28,29]. Furthermore, the application of stochastic models in conjunction with high-resolution empirical data has been shown to provide robust estimates of system performance and risk under realistic operating conditions [30,31,32]. Despite these advances, significant research gaps remain, particularly regarding the joint treatment of uncertainty, cross-correlations among key variables, and the influence of climatic variability on long-term system behavior [33].
The intermittency necessitates substantial energy storage or backup systems to ensure a stable electricity supply, particularly in systems aiming for high-RES penetration [34,35]. Common storage technologies include pumped hydro storage (currently the dominant form globally), batteries, compressed air energy storage, hydrogen, and emerging gravity-based storage systems [36,37]. Without adequate storage, excess renewable generation must be curtailed, while periods of low production increase reliance on fossil-fuel generation or increase the risk of blackouts [38,39,40,41,42,43].
The April 28, 2025, blackout across Spain and Portugal illustrated the risks associated with inadequate storage and limited grid flexibility in high-RES systems [44,45,46]. At the time of the incident, renewables (primarily solar) accounted for nearly 70% of the electricity supply. Still, a sudden voltage surge attributed to grid oscillations and insufficient system inertia from inverter-based resources led to cascading disconnections across the generation and transmission infrastructure. Without sufficient long-duration energy storage to buffer variability or fast-response backup resources, the system failed to recover quickly, resulting in multi-hour outages affecting millions of consumers. Although initial speculation suggested an over-reliance on intermittent renewables, official investigations ruled out excess generation as the primary cause, instead pointing to outdated grid codes, limited interconnection capacity, and insufficient voltage-control capabilities of renewable resources [47,48].
In Greece, electricity demand exhibits strong temperature dependence, with inter-peak demand driven by heating in winter and by cooling in summer, thereby exacerbating mismatches with variable RES generation [49]. National electricity consumption has remained in the range of 48-53 TWh annually, equating to approximately 4,500-4,700 kWh per capita annually in recent years [50].
In a previous study [51], it was observed that greater penetration of RES into the electricity system is linked to higher electricity prices. This observation makes it even more imperative to explore the extent to which RES can, in practice, contribute to electricity cost savings. Since electricity prices are fundamental to economic performance and societal well-being, the normalization index of the value of money [52,53] may offer useful insight into the potential benefits and possible transformations of the energy sector associated with the use of RES. However, as this perception becomes more widespread, a discussion has arisen about landscape transformation due to RES [54,55].
This paper presents a stochastic approach to sizing RES infrastructure and storage requirements, using normalized time-series data from North Euboea (Figure 1), a region located in central Greece and with climatic conditions representative of the national average, as a case study. By analyzing hourly profiles of electricity demand, solar, and wind generation, we quantify reliability metrics and show that even moderate storage capacity can significantly enhance supply security. However, substantial challenges remain for achieving high-RES penetration in energy systems.
Geographical location of Greece within Europe and the case study area of North Euboea [56,57].
Methodology and Data
Electricity demand profiles were derived from national data scaled per capita using an annual consumption of approximately 4,600 kWh per person, consistent with recent Greek averages [58].
All time series were normalized to match the mean annual energy demand per capita in Greece (≈4,567 kWh), thereby allowing direct comparison across technologies: corresponding to a requirement of ≈13.7 m2 of photovoltaic (PV) panels or a wind capacity equivalent to supplying 1,830 people per 3 MW wind turbine.
Stochastic characteristics were assessed by calculating the Hurst-Kolmogorov (HK) parameter, the mean, the coefficient of variation and skewness, as well as climacograms (variance versus scale), auto-correlograms, and cross-correlations between the time series [59,60].
To describe the operation of the energy storage system, we employ a model described by the processes defined in Equations (1) and (2) [61,62,63].
where T is time; K is the storage capacity of the system; ST is the stock in the storage energy system; xT is the inflow to the energy storage system after consumption; δT is the energy demand and RT is the actual amount of taken energy in an attempt to satisfy energy demand during the time period (t-1, t). When the storage energy system has sufficient energy, RT equals demand δT; otherwise, RT < δT.
The energy storage system was sized at approximately 6 kWh (half the average daily production per capita), simulating hourly balancing and quantifying reliability (substantial percentage of time demand is fully met without imports) and curtailment. This value is consistent with simplified storage sizing assumptions frequently used in exploratory renewable system assessments, where storage equivalent to a fraction of daily consumption is used as an indicative buffer capacity.
The time evolution of stored energy is simulated using the storage balance equations (1) and (2), which describe the dynamic interaction between renewable generation, consumption, and storage capacity at an hourly time step. Simulations of the storage system operation were conducted over full-year periods and selected seasonal intervals (e.g., summer: mid-June to mid-July; winter: December) to highlight the effects of seasonal variability. At each hourly time step, renewable generation first satisfies electricity demand, while any surplus energy is directed to storage until the storage capacity K is reached; during deficit periods, stored energy is withdrawn to cover demand until the storage is depleted.
Meteorological data were acquired from a meteorological station located in the village of Agia Anna (latitude = 38.86 degrees, longitude = 23.40 degrees, altitude = 303 m). The station is operated by the Laboratory of Agricultural Hydraulics (University of Patras, Greece) [64,65]. Solar irradiation (Wh/m2) was derived from solar radiation (W/m2) in the study area, while wind speed (m/s) at 8 meters was used to estimate renewable generation profiles. Solar irradiance and wind speed were converted to power output using standard power curves for PV systems (per m2) [66,67] and for a typical 3 MW onshore wind turbine installed in Greece [68,69]. The temperature profile was used to identify correlations between temperature and consumption.
ResultsRelationship between Temperature and Electricity Demand
Electricity demand in Greece exhibits a strong dependence on deviations of ambient temperature from the thermal comfort zone, reflecting heating and cooling requirements. To illustrate this relationship, hourly temperature data are compared with the electricity consumption profile. In Figure 2, the hourly temperature data are plotted together with the thermal comfort zone. In the same chart, the energy consumption profile is also presented. A relationship between electricity demand and temperature deviations from the thermal comfort zone can be observed (Figure 3). When temperatures fall below or rise above the comfort range, electricity demand increases due to heating and cooling requirements. This relationship explains the seasonal peaks observed in the demand time series and motivates the use of temperature deviations as explanatory variables.
The energy consumption profile per capita and the temperature profile of the same year (daily step). The thermal comfort zone is also depicted.
The energy consumption profile per capita and the deviation of temperatures from the thermal comfort zone (daily step).
Renewable Generation and Demand Matching
To evaluate the ability of renewable sources to satisfy electricity demand, photovoltaic and wind generation profiles were scaled to match the annual per-capita consumption. In the case study considered in this work, it is assumed that the installed photovoltaic capacity is sized to meet the total annual energy demand, complemented by a proportional share of the annual energy output of a 3 MW wind turbine. This share corresponds to the energy needs of a single individual, equal to 1/1830 of the turbine’s total annual output. The respective annual production and consumption profiles are presented in Figure 4. A more detailed view is provided in Figure 5 for the period from June 15 to July 15, which includes the longest day of the year with maximum solar availability (June 21: the summer solstice), and in Figure 6 for the period from December 1 to December 31, which includes the shortest day of the year with minimum solar availability (December 21: the winter solstice).
Hourly per-capita electricity demand compared with renewable energy production profiles (photovoltaic and wind) scaled to match annual energy demand.
The energy profile of per capita energy consumption in relation to the absolute values of the renewable sources profile (wind and PV) that cover the energy needs (June 15-July 15) (hourly step).
The energy profile of per capita energy consumption in relation to the absolute values of the renewable sources profile (wind and PV) that cover the energy needs (December 1-December 31) (hourly step).
We observe that, even when assuming the existence of infrastructure (wind turbines and photovoltaic systems) capable of meeting energy requirements in absolute terms, their stochastic nature allows demand to be met only 32% of the total time for an energy mix consisting solely of photovoltaics, and 44% of the total time for an energy mix consisting solely of wind turbines. The unused energy amounts to 2.583 kWh for photovoltaics and 1.913 kWh for wind turbines.
Stochastic Properties of the Time Series
The stochastic characteristics of the examined time series were analyzed to quantify variability and long-term persistence in both electricity demand and renewable generation [70,71,72,73]. To quantify the uncertainties arising from the analysis of these time series, Table 1 presents their stochastic characteristics, Figure 7 shows the corresponding climacograms, and Figure 8 illustrates the autocorrelation functions of the time series, the advantages and limitations of which are discussed in [74]. Climacograms indicate strong long-term persistence in temperature and consumption, weaker persistence in wind generation, and pronounced daily periodicity in PV, with minima at 12 and 24 hours. The autocorrelation function further confirms a diurnal periodicity across all examined time series.
The stochastic characteristics of the distributions.
HK
Average (kW)
Average (daily kWh)
Coefficient of variation
Skewness
Energy consumption profile
0.93
0.521
12.5
0.015
0.956
Energy from solar panels
0.90
0.582
1.315
Energy from a wind turbine
0.82
0.256
0.656
HK
Average (°C)
Coefficient of variation
Skewness
Temperature (°C)
0.95
15.5
63.6
0.3
Distance from thermal comfort zone (°C)
0.94
5.1
25.2
0.7
Climacograms of the time series of energy production from solar panels, energy production from wind turbines, the consumption profile, temperatures, and the deviation of temperatures from the thermal comfort zone, where γk represents the variance of the aggregated time series at time scale k.
Auto-correlograms of the time series of energy production from solar panels, energy production from wind turbines, the consumption profile, temperatures, and the deviation of temperatures from the thermal comfort zone.
Since electricity generation from RES is equal, in absolute terms, to electricity consumption, the mean values of average power and daily energy production coincide with the respective consumption profile for the considered time series.
The long-term behavior of the examined time series was characterized by estimating the Hurst-Kolmogorov exponent (HK), a measure of long-range dependence widely applied in geophysical and hydrological processes. Values significantly greater than 0.5 indicate persistent behavior, whereby trends and fluctuations tend to cluster over extended periods rather than alternating randomly.
All series exhibited strong persistence (Table 1). Exceptionally high values for temperature and comfort-zone deviation reflect prolonged spells of extreme weather, resulting in extended periods of elevated demand (H = 0.93). The renewable generation series also displays marked persistence, with solar displaying slightly higher persistence than wind, implying that both cloudy/calm conditions and clear/windy spells tend to cluster over multiple days to weeks.
These findings underscore that the variability of renewable energy sources (RES) cannot be characterized as white noise but instead exhibits pronounced Hurst–Kolmogorov dynamics, with HK exponents ranging from 0.82 to 0.95 across the examined series. This strong long-range persistence implies enhanced clustering of production deficits and surpluses, resulting in increased uncertainty and prolonged periods of stress for the energy system. Consequently, achieving a reliable supply in high-RES-penetration scenarios necessitates substantially more robust system design, including larger storage capacities, significant overbuilding of generation infrastructure, or the retention of firm dispatchable resources.
Demand–Renewable Correlation
An important factor affecting the reliability of renewable systems is the degree of temporal alignment between electricity demand and renewable energy generation. The correlation coefficient was calculated between hourly per-capita electricity consumption and the normalized time series of renewable generation.
The correlation between electricity demand and PV production was 0.43, indicating a moderate positive linear relationship. This suggests that solar output tends to increase during periods of higher demand, primarily because peak summertime cooling loads coincide with maximum solar irradiance. Such alignment partially mitigates the intermittency challenge for solar systems, as higher generation occurs during periods of heightened demand.
In contrast, the correlation between demand and wind generation was only 0.09, essentially negligible. This near-zero value reveals that wind power production shows little or no systematic correspondence with consumption patterns: strong winds may occur during low-demand periods (e.g., nighttime or mild seasons), while calm conditions may coincide with peak loads. Consequently, wind energy provides substantially less natural load-following capability than solar generation in the Greek context.
Storage System Performance
The performance of the renewable system was further evaluated by introducing an energy storage component designed to buffer short-term mismatches between generation and demand. To stabilize energy, an energy storage system with a capacity of approximately half the expected average daily production (≈6 kWh per capita) was introduced according to the balance equations presented in Section 2. The objective is not to determine an optimal storage size, but rather to examine the improvement in system reliability achieved with a moderate level of energy storage.
Figures 9-14 illustrate the temporal behavior of the storage system under photovoltaic and wind generation scenarios. Figure 9 and Figure 10 present the annual evolution of stored energy relative to the maximum storage capacity. Figures 11-14 provide detailed views of representative seasonal periods, highlighting the impact of summer and winter conditions on system reliability.
Photovoltaic panels. Maximum storage capacity, available stored energy, and energy imports into the system after consumption.
Wind turbines. Maximum storage capacity, available stored energy, and energy imports into the system after consumption.
Photovoltaic panels. Available energy; energy imports into the system after consumption; consumption profile (June 15-July 15).
Photovoltaic panels. Available energy; energy imports into the system after consumption; consumption profile (December 1-December 31).
Wind turbines. Available energy; energy imports into the system after consumption; consumption profile (June 15-July 15).
Wind turbines. Available energy; energy imports into the system after consumption; consumption profile (December 1-December 31).
In Figure 9 and Figure 10 energy imports into the system following consumption are depicted in gray, the maximum storage capacity is indicated by a red dashed line, and the available stored energy is shown in orange, over a one-year time frame.
In Figures 11-14, energy imports into the system after consumption are illustrated in gray, the available stored energy is represented by an orange line, and the consumption profile is shown in blue. These figures provide temporal zooms into the data presented in Figure 9 and Figure 10, focusing on specific periods (summer: mid-June to mid-July; winter: December) that encompass the summer solstice (21 June, the longest day of the year) and the winter solstice (typically 21-22 December, the shortest day of the year), while depicting the same underlying data as Figure 9 and Figure 10. This alternative visualization approach more clearly highlights seasonal variations in the time series and their effects on consumption. Instances in which the consumption profile falls to zero indicate that demand cannot be satisfied, resulting in a blackout.
Reliability is defined here as the percentage of time steps in which electricity demand is fully satisfied without external energy imports. In the configuration considered, the reliability ratio reaches 71% for photovoltaics (PV) and 70% for wind turbines. The unused energy amounts to 1,036 kWh for PV and 1,127 kWh for wind turbines.
It is worth noting that the aforementioned values are conservative, since accounting for cross-dependencies among solar radiation, wind speed, air temperature and other processes of the hydrological cycle influencing the energy demand [75,76,77,78,79] within water-energy nexus management, would likely further reduce RES reliability due to the additional uncertainty arising from the combination of non-negligible cross-correlations and strong long-term persistence behaviors [80,81,82].
DiscussionPhotovoltaic Generation Reliability
Although solar photovoltaic (PV) generation can theoretically meet annual per-capita electricity demand, its stochastic variability significantly limits system reliability in the absence of storage. In the examined case study, a PV-only configuration satisfies demand during only about one-third of the examined time steps, while also producing substantial surplus energy that must be curtailed.
The introduction of moderate storage capacity—equivalent to approximately half of the average daily production (about 6 kWh per person)—substantially improves system performance, nearly doubling the proportion of time during which demand is satisfied. Nevertheless, a considerable probability of unmet demand remains. This outcome indicates that storage capacity estimated from average daily balances underestimates the actual requirements for a reliable electricity supply, as it does during extended periods of low solar generation.
Wind Generation Reliability
In the examined system, wind-only generation meets demand more frequently than PV in the absence of storage; however, this apparent advantage reflects the higher short-term variability of wind production rather than a systematic alignment with demand patterns. Strong wind events may coincide with periods of low electricity consumption, while calm conditions may occur during peak demand.
This behavior is consistent with the weak correlation identified between wind generation and electricity demand. As a result, wind energy provides limited natural load-following capability in the Greek climatic context. The pronounced temperature-driven seasonality of electricity demand further amplifies these mismatches, particularly during summer cooling peaks and winter heating periods.
Implications for Storage and System Design
The stochastic analysis, including climacograms and autocorrelograms, reveals strong long-term persistence in both demand and renewable generation. Elevated Hurst exponents indicate clustering of deficit and surplus periods, meaning that low-generation conditions can persist for extended durations. Such clustering increases the stress on energy systems and cannot be adequately captured by traditional statistical indicators based solely on averages or short-term variability.
Consequently, reliable operation of systems dominated by variable renewable energy sources may require significantly larger storage capacities, substantial overbuilding of generation infrastructure, enhanced interconnection capacity, or the retention of firm dispatchable resources. Short-duration storage alone may not be sufficient to mitigate prolonged low-generation periods associated with persistent stochastic variability. Even in small regional systems, there seems to be a need for additional energy infrastructure to handle peaks and provide a stable base energy. In Greece, commonly used resources include hydropower, coal, and natural gas. The potential of energy transfers has also been discussed in this regard. In the case of solar energy, its contribution is not expected to be important (due to common nighttime hours in areas with meaningful proximity for energy purposes). At the same time, wind should be analyzed separately at a larger spatial scale, accounting for associated losses.
The storage model adopted in this work is intentionally simplified and does not explicitly incorporate charge–discharge efficiencies or operational optimization strategies. The purpose of the analysis is to evaluate the stochastic mismatch between renewable generation and electricity demand rather than to perform a detailed engineering design of storage technologies. A more comprehensive techno-economic evaluation of storage sizing and system costs represents an important direction for future research.
Conclusions
Renewable energy sources offer substantial environmental and resource advantages; however, achieving high penetration levels requires system designs that extend well beyond simple annual energy balancing. The stochastic nature of renewable generation, combined with the pronounced long-term persistence identified in both demand and supply time series, creates extended periods of deficit and surplus that cannot be effectively mitigated by short-duration storage alone. As a result, reliable operation of renewable-dominated power systems may require substantial overbuilding of generation capacity, long-duration energy storage, enhanced interconnection capacity, or the retention of dispatchable backup resources. In the Greek context, where electricity demand exhibits strong seasonal variability and per-capita consumption remains moderate by European standards, hybrid solar–wind configurations and complementary flexibility options deserve further investigation.
The analysis demonstrates that although solar and wind resources can, theoretically, satisfy annual electricity demand, their inherent variability significantly constrains reliability in the absence of adequate system flexibility. Moderate storage capacity improves system performance but does not eliminate supply deficits stemming from clustering of low-generation periods driven by long-range dependence. These findings highlight the importance of explicitly accounting for stochastic variability and persistence when planning renewable-based energy systems. The stochastic framework employed here provides a practical approach for assessing reliability risks. It may support more robust planning of renewable energy transitions in regions with similar climatic and demand characteristics.
Acknowledgments
The authors would like to thank Dr. Nikolaos Malamos who provided the meteorological data from the weather station of Agia Anna.
Author Contributions
Conceptualization, G.-F.S; Methodology, G.-F.S; Software G.-F.S, N.T; Validation, G.-F.S, P.D, DK; Formal Analysis, G.-F.S; Investigation, G.-F.S; Resources, G.-F.S; Data Curation, G.-F.S; Writing – Original Draft Preparation, G.-F.S; Writing – Review & Editing, G.-F.S; N.P.; I. B.; N. K.; P. D.; N. T.; R. I.; I. A.; M-A. A.; D. S.; G.-D.L.; M.M.; O.I.A.; D.M.; A.A.; N.M.; D.K.; Visualization, G.-F.S; Supervision, n/a.; Project Administration, n/a; Funding Acquisition, n/a.
Funding
This research received no external funding but was motivated by the scientific curiosity of the authors.
Competing Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data Availability Statement
Dr. Nikolaos Malamos provided the meteorological data from the weather station of Agia Anna.
AI-Assisted Technologies Statement
Artificial intelligence (AI) tools were used solely for basic grammar correction and language refinement in the preparation of this manuscript. Specifically, OpenAI’s ChatGPT was employed to improve the readability and linguistic clarity of the English text. All scientific content, data interpretation, and conclusions were developed independently by the author. The authors have thoroughly reviewed and edited the AI-assisted text to ensure its accuracy and accept full responsibility for the content of the manuscript.
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