1. Introduction

2. Previous work

3. Novelty of Research

4. Data and modeling

5. Results

5.1. Descriptive Statistics

As a first step, it is important to obtain an overview of the statistical properties of our reference points. The following maps present a comprehensive wind atlas of Greece accompanied by the mean daily capacity factor and the standard deviation of each individual location with a white-to-red color gradient. Areas featuring a more intense shade of red have a higher value on the corresponding variable while the white areas are neutral (values are near zero). Locations with extremely high and low values are pin-pointed on the maps to analyze the relationship among the three variables.

Locations with rich wind profiles are mainly located over the islands of the Aegean sea and Crete. This alignment is consistent with the morphological characteristics of the Aegean Basin (wind speeds are high in sites near the sea) and the presence of the Etesian winds (Meltemia) during the summer [46,47,48,49]. Etesian outbreaks tend to have higher density in August, while they present interannual variability with a negative trend over the June to September period [50]. Additionally, these outbreaks experience heightened occurrence over elevated terrains such as the Pindos mountain range and other lofty landscapes, where wind speeds are notably elevated. The volatility levels on Figure 2b, are quite the same for all the grid points. However, there is a slight change in the hue that follows the same pattern with the mean generating capacity: Coastal and mountain areas have high values both in $\sigma$ and $\mu$ and all the other areas have lower values. It is noteworthy that the geographical locations associated with the highest and lowest wind speed values on the map are in close proximity, although not identical, to those correlated with the highest and lowest capacity factors. The EPC model incorporates specific threshold values for wind speeds on both ends of the spectrum, resulting in abrupt increases or decreases in power output. This effect is particularly pronounced in coastal sites where elevated wind speeds prevail, a phenomenon amplified by the presence of the Etesian winds (Meltemia) during the summer months. At the same time, the relationship between daily generating capacity and standard deviation is not strictly linear as the five most volatile regions differ from those with the highest generating capacity. Conversely, locations characterized by the lowest generating capacity align with those of the lowest volatility levels.

The relationship between wind speed and power output can be further investigated by drawing the distributions of the grid points featuring the highest and lowest values. An area with weak winds has a right skewed distribution, the generation profile is more stable and extreme weather phenomena are rare. On the opposite case, selecting areas with strong winds generates a distribution closer to bell-shaped curve. The generation follows a similar schema, low wind speeds produce values close to zero due to the cut-in threshold. High wind speeds on the other increase the scenarios of higher generation and the distribution is shifted on the right side.

The study of Kotroni et al. [24] revealed a significant reliance of wind potential on the slope of the Greek terrain, rather than solely on the absolute elevation. This observation is somewhat reflected in the above-discussed maps and wind generation distributions. An other compelling aspect of wind generation is that it is positively correlated with the variability. This formulates a high risk-high reward relationship which is provided by the nature, as show on the left panel of Figure 4.

Figure 3. Histogram of highest and lowest daily mean wind speed and generation.

Figure 3. Histogram of highest and lowest daily mean wind speed and generation.

The addition of one unit of volatility on a single asset, increases the observed mean generating capacity. This pattern mirrors a common occurrence in financial products, such as those within the stock market, where a delicate balance between risk and return is established. Individual investments will embrace extra risk only in exchange of higher generating capacity, according to the risk appetite of the investors. This is known as a "no-free-lunch property". The presence of this phenomenon within wind generation output is indeed quite noteworthy, emphasizing the potential pitfalls associated with hasty site selection. Seeking yield aggregators or other investors may lose control of risk. Selecting optimal sites is a complex endeavor, requiring a delicate balance between risk and return aligned with defined targets. The exploration of this observed relationship in natural systems holds promise for revealing valuable insights into weather variables. Moreover, it could pave the way for the inception of innovative business ventures that leverage these insights to create novel projects with sustainable returns.

In the right panel of Figure 4, we present a visualization of 6 different classes of $\sigma -\mu$ points according to their altitude. Our analysis does not reveal a discernible relationship between the altitude of locations and the mean generating capacity or standard deviation within our sample. Other possible meteorological factors that could relate these variables are the logarithmic profile of the wind speed, mountain winds phenomena and the intensity of the pressure systems and fronts that cross the area of interest. A further investigation of this direction is out of the scope of this paper.

5.3. PCA and Area Selection

In the previous section we estimated the factor model and analyzed the fundamental properties of the common risk factors. A more in-depth comprehension of how each specific factor uniquely contributes to the model can be achieved through the reconstruct analysis framework. The gradual incorporation of each factor, results in an immediate transformation of the generation schema over time, thereby providing additional insights into the origins of each risk component. Given the extensive sample size, we only select 3 areas based on their loadings pattern. This selection was made to ensure that all possible combinations of exposure to the risk factors could be depicted accurately. In the following maps, the black boxes represent the 3 selected sites for the reconstruct analysis. On the first factor, only a positive sign is available, but on the second and third we can distinguish three cases: 1) Positive and positive exposure, 2) Positive and negative exposure and 3) Negative and negative exposure. The fourth factor, being of lesser significance, was not considered in the sign selection methodology outlined above, for simplicity. The chosen sites for the analysis are Chios (38.5, 25.9), Giannitsa (40.8,22.4) and Corfu (39.75,19.75). Chios has a strong positive exposure on the second factor and a weak positive exposure on the thrid factor. Giannitsa, on the other hand, demonstrate a medium negative exposure to the second factor and a medium positive exposure to the third factor. Lastly, Corfu has a moderate negative exposure to both second and third factor. The monthly averaged transformed generation of these 3 assets is anticipated to be different as each factor is added. To validate this statement, we created the monthly averaged transformed generation capacities boxplots of the designated areas. Each factor can be conceived as a building block that is added on the overall generation schema.

For the grid point of Chios, employing only one factor gives similar results with those of Figure 7. Upon introducing the second factor into the model, there is a noticeable decline in generation output during the winter and spring seasons, coupled with a peak in generation during July and August. This observation suggests that the Etesian winds (as mentioned in [50]) exert significant influence during this period, leading to a positive impact on the area’s generation output. The third factor does not add any significant alterations to the generation pattern except some minor declines on the winter months. The fourth factor enhances the production a little bit on July but it does not change the production output to a remarkable level. The grid point of Giannitsa is characterized by a different generation profile. The first factor affects the transformed capacities in a manner similar to that of all the other grid points (higher generation within winter and early spring months). When the second estimated factor is incorporated on the model, the averaged monthly transformed generation undergoes a steady reduction during spring and summer, without the production picks seen in Chios. This decrease is accompanied by low levels of volatility. The third factor contributes to greater stability in the transformed capacities by slightly increasing production during spring and summer, a phenomenon potentially explained by the positive loadings on this factor. The fourth factor does not add any considerable points. Lastly, for the grid point of Corfu, we observe the same pattern for factor 1 alike to the other areas. The addition of the second factor, imposes a decline on the transformed capacities for spring and summer accompanied by increased volatility indicated by the length of the boxes. The latter shows the variability of the wind speeds during these months, since both synoptic (e.g. front passages) and thermodynamic conditions (e.g. thunderstorms) affect the wind field. The third factor amplifies the impact of the second factor, leading to a further decline in average production. As for the fourth factor, it only generates a minor peak in average production during March, with no substantial alteration to the overall generation pattern.

Figure 7. Monthly boxplots of the 4 principal components.

Figure 7. Monthly boxplots of the 4 principal components.

Figure 8. Autocorrelation function of each one of the systematic risk factors.

Figure 8. Autocorrelation function of each one of the systematic risk factors.

Figure 9. Factor Loadings on designated areas.

Figure 9. Factor Loadings on designated areas.

Figure 10. Monthly averaged transformed generation of Chios.

Figure 10. Monthly averaged transformed generation of Chios.

Figure 11. Monthly averaged transformed generation of Giannitsa.

Figure 11. Monthly averaged transformed generation of Giannitsa.

Figure 12. Monthly averaged transformed generation of Corfu.

Figure 12. Monthly averaged transformed generation of Corfu.

5.5. Portfolio Evaluation

One method for assessing the effectiveness of diversification is to compute the Risk Reduction Index, as outlined in [7]), which can be defined as follows:

$$R{R}_i=1-\frac{{\widehat{\sigma }}_p}{{\widehat{\sigma }}_i}$$

(10)

In (11), ${\widehat{\sigma }}_p$ denotes the portfolio standard deviation and ${\widehat{\sigma }}_i$ denotes the daily standard deviation of each individual asset.

In the left map of Figure 15a, the RR of each grid point is depicted. Nearly all areas have shades of red, indicating that inclusion in a portfolio is highly advantageous as it significantly reduces their risk. In the other map, we exclusively display the RR of each location of the efficient assets. Only 2 points have a blue color and 1 has a white color. These particular locations were chosen by the algorithm in the MVP solution due to their low variability in terms of risk.

An intriguing observation is that the efficient assets are distributed throughout Greece, spanning both mountainous and coastal regions across the country. Many grid points are situated within the mountain range of Pindos while others are found on the island of Crete. In particular, the eastern and southern regions of Crete are more suitable due to the complex terrain of the island which results in gap winds and wind flow acceleration as documented in [24,51]. Each of the efficient assets possesses a unique wind profile ideal for wind farm installations. For instance, in the summer season, coastal areas tend to experience higher wind speeds, while in the winter, wind speeds are elevated in the mountainous regions. This explains why both types of areas are selected by the efficient portfolios.

In addition to the Minimum Variance Portfolio (MVP) solution discussed earlier, there are other criteria that warrant consideration before finalizing the optimal allocation plan. An essential aspect to analyze is the risk exposure of the MVP portfolio to common risk factors. While diversification aims to reduce the idiosyncratic risk component by selecting areas with different generation profiles, it is possible that systematic factors may still introduce instability into the portfolio’s generation. To address this concern, we can visualize the MVP solution on a loadings map. If the locations designated by the portfolio exhibit risk factor loadings that are numerically close to zero or loadings with mixed signs, this indicates that these factors can indeed be diversified through the spatial allocation of wind generation.

In the following maps, the green-colored boxes represent the 9 MVP locations, as presented in Section 5.4. It is evident that the first risk factor cannot be diversified as all loadings are of the same sign (positive). However, the portfolio has chosen areas that, while not completely eliminating exposure to this risk component, have loadings that are not extremely high (deep red). The portfolio attempts to minimize the impact of this risk component. On the second factor, we notice that the MV portfolio areas have loadings both in the red and blue zone with the hue to varying slightly but generally not being very pronounced. This a favorable result as the portfolio can reduce or even neutralize the effect of this specific risk component by selecting areas with opposite loadings sign. The same phenomenon is also present for the third factor as 6 areas are located in the red zone while the rest are located in the blue zone with a very light hue. This factor can be as well diversified away. Finally, the last factor is diversifiable as well, given that the portfolio-designated areas feature mixed exposures to this risk component, indicated by the varied loadings.

Figure 16. Factor Loadings on designated areas.

Figure 16. Factor Loadings on designated areas.

To summarize these results, there is a 42.13% of total variance of the transformed generating capacities (as later shown in section 4.3) that cannot be diversified while the rest 36.438% is, to some extent, diversifiable.

5.6. Efficient Portfolios

The MVP solution has been found to yield a suboptimal generation output, making it impractical for implementation in a real-world scenario. Nevertheless, given the steepness of the efficient frontier, alternative portfolios warrant consideration in determining the final spatial allocation plan. To assess these alternatives, we have generated timeplots, monthly boxplots, and daily generation capacity distributions for six random portfolios located on the efficient frontier, including the MVP, EQS and MG.

Figure 17. Histogram of mean daily generating capacities of 6 portfolios of the efficient frontier.

Figure 17. Histogram of mean daily generating capacities of 6 portfolios of the efficient frontier.

The histograms of relative frequencies reveal an interesting trend along the efficient frontier. In the first 20-30 portfolios a right kurtosis is observable in the distribution, particularly pronounced in the MVP. In contrast, in the final 5 portfolios, there is some degree of left kurtosis, especially in the MG portfolio. As we progress towards the middle of the efficient frontier curve, specifically in portfolios within the range of 30-40, the distribution of daily capacities gradually tends to approximate a Gaussian distribution. This phenomenon can be attributed to the inherent objectives of the optimization algorithm. To attain a significantly low level of risk, the algorithm efficiently selects locations where extreme weather conditions or very high wind speeds are infrequent. However, when the goal is to boost generation output, the algorithm must include areas with more diverse and robust wind profiles, even if this leads to increased variability.

The implementation of the MVP solution may lead to a high number of zero production days, rendering the wind farms unproductive. On the other hand, Portfolio 50 (MG) is not a viable option due to its high variability and uncertain generation output. Portfolios within the range of 16-35 present a more acceptable compromise, as they help mitigate excessive variability while increasing wind energy production. To make a well-informed decision, it’s advisable to examine these portfolios in detail, taking into account factors such as the number of selected assets, ease of wind farm installation, alignment with the surrounding environment, and micro-climate conditions. By considering these aspects, an investor can identify the portfolio that offers the best balance between risk and generation output while being conducive to practical implementation.

Figure 18. Mean daily generating capacity of 6 portfolios of the efficient frontier.

Figure 18. Mean daily generating capacity of 6 portfolios of the efficient frontier.

The mean daily generation of the portfolio can be computed by utilizing the optimal portfolio weights obtained from the algorithm in previous step. As we examine the time plots, it becomes clear that both the mean and variance of generating capacity have significant variations along the efficient frontier. For instance, the MVP portfolio displays multiple clusters of near-zero production days, resulting in a lower risk level. Conversely, portfolio 50 demonstrates the opposite trend with higher generation but also greater risk.

Wind speeds and other climatology variables are highly affected from seasonality. To capture the seasonal patterns in the mean daily generation of the portfolios, monthly boxplots are utilized. These boxplots reveal that mean daily generation tends to peak during the summer season for all portfolios, with an increasing trend from the MVP to the MG solution. Conversely, the winter season also experiences high wind speeds, while in spring and autumn, production decreases. This sequential pattern reflects the four seasons’ influence on generation. Moreover, the variability in production differs across seasons. Winter, marked by extreme weather conditions, exhibits significantly higher wind speed variability. In contrast, during the summer, the range of the boxes is narrower, indicating lower variability. An intriguing finding is that even in the MVP boxplot, a notable amount of variability remains, underscoring that generation risk is not entirely eliminated. In the subsequent section, factor analysis will be conducted to delve into the sources of variability in the efficient portfolios, dissect wind generation risk, and validate the results of the optimization algorithm.

An alternative approach that can be easily applied to facilitate the selection of the final spatial allocation plan is to quantify the incremental risk, denoted as ${\sigma }_i$ and the supplemented generation output, denoted as ${\mu }_i$ that is added in the sequence of the efficient portfolios along the frontier. In addition, the Coefficient of Variation (CV) can be calculated to identify the relationship between generation output and volumetric risk for each portfolio. Typically expressed as a percentage, the CV can be integrated into the same graph alongside the other two measures. This standardized relative risk measure enables a direct comparison of relative variability among different datasets, further aiding the decision-making process. If the Coefficient of Variation (CV) for an individual portfolio surpasses 100%, it signifies that the standard deviation ($\sigma$) exceeds the mean ($\mu$), indicating an unfavorable scenario. The lower the CV, the more favorable the relationship between $\sigma$ and $\mu$ becomes. In fact, the optimization algorithm could be adjusted to target the minimum CV ratio.

From the 50 efficient portfolios, the minimum value of CV is found on portfolio 33, where $CV=33.729\%$. However, even if on that node the relevant $\sigma$ is as low as possible for the corresponding $\mu$, an investor should still has to accept a quite high risk. On the other hand, on portfolio with $id=8$, the standard deviation is lower and the % increase of daily mean generation for the following portfolios is much higher. Specifically for the 8 first portfolios of the frontier, the % increase of $\mu$ is greater than the % increase of $\sigma$. Thus, portfolio 8 could be a usefull alternative to overcome the low wind generation of the MVP solution. The spatial allocation plan of this portfolio is depicted on Figure 21. In addition to this, the portfolio that has the same mean generation output with the EQ portfolio but with lower volatility is presented on the right of the figure.

Figure 19. Monthly boxplot for the daily generation capacities of 6 portfolios of the efficient frontier.

Figure 19. Monthly boxplot for the daily generation capacities of 6 portfolios of the efficient frontier.

Figure 20. Percentage change on $\sigma$ and $\mu$ of 50 portfolios and CV plot.

Figure 20. Percentage change on $\sigma$ and $\mu$ of 50 portfolios and CV plot.

Figure 21. Alternative portfolios for spatial allocation of wind resources in Greece.

Figure 21. Alternative portfolios for spatial allocation of wind resources in Greece.

The two portfolios depicted in the maps share common assets with the MVP solution but include additional locations. The Efficient Equally Weighted portfolio comprises a total of 17 areas and portfolio with $id=8$ has 16 assets included. These 2 portfolios have nearly identical selections of locations, differing by only about 2-3 areas.

5.7. Risk Decomposition on the Efficient Portfolios

As a final step in the implementation of an optimal allocation plan, we apply the PCA method to all individual portfolios. However, when conducting PCA on portfolio factor analysis, we cannot employ the same approach as we did for the transformed generating capacities. This is because we must also consider the portfolio weights, which are now a factor influencing the left-hand side of the factor model equation.

Since we have already conducted the PCA analysis for the entire dataset and estimated the factor scores, we can now apply an Ordinary Least Squares (OLS) model. The factor scores will now serve as the independent variables, while the portfolio generation will be the dependent variable. The unknown coefficients in this model are the portfolio loadings, which are valuable for assessing the exposure of the optimal portfolio to the common factors. We can perform a risk decomposition, similar to the one conducted for the entire dataset, specifically for the portfolio case. This allows us to investigate the sources of variability by calculating the total variance explained by the systematic and idiosyncratic components (denoted as ${\widehat{s}}_t$ and ${\widehat{e}}_t$ correspondingly). The estimated model and the variance decomposition of the portfolio mean generating capacity can be described by the following equations:

$$\begin{array}{ccc}\hfill {\tilde{y}}_t=c_0+\sum _{k=1}^K{\widehat{\beta }}_k{F}_{kt}+{\widehat{e}}_t& \hfill & \hfill k=1,2,...\mathit{K};t=1,2,...,\mathit{T}\end{array}$$

(11)

$$\begin{array}{c}\hfill \widehat{Var}\left({\tilde{y}}_t\right)=\widehat{Var}\left({\widehat{s}}_t\right)+\widehat{Var}\left({\widehat{e}}_t\right)={\widehat{\beta }}_1^2+{\widehat{\beta }}_2^2+\dots +{\widehat{\beta }}_K^2+\widehat{Var}\left({\widehat{e}}_t\right)\end{array}$$

(12)

The slope coefficients (${\widehat{\beta }}_1,{\widehat{\beta }}_2,\dots ,{\widehat{\beta }}_k$) are the portfolio loadings and $F_{tk}$ is the score of factor k at time t, already computed through PCA in section 5.2. The sign and magnitude of portfolio loadings describe the influence of each systematic risk factor on the portfolio’s generation output. The systematic variance can be computed as the sum of squares of the loadings while the remaining percentage of the total variance accounts for the idiosyncratic variance of the portfolio. The outcomes of the loadings estimation and the risk decomposition are summarized in the following table:

Table 2. Estimated efficient portfolio loadings of each factor.

Table 2. Estimated efficient portfolio loadings of each factor.

Portfolio id	F1	F2	F3	F4
MV Portfolio	0.9025	0.3054	0.6056	-0.3945
Portfolio 8	0.3539	0.1266	0.0674	-0.1938
EQS Portfolio	0.3479	0.1275	0.0507	-0.1881
Portfolio 35	0.4350	0.1706	0.0487	-0.2567
Portfolio 42	0.4574	0.2993	0.0521	-0.3976
MG Portfolio	0.3043	0.7813	0.4889	-0.7969

Table 3. Risk decomposition of Efficient Portfolios.

Table 3. Risk decomposition of Efficient Portfolios.

Portfolio id	F1	F2	F3	F4	Sys.Var	Idio.Var	$\mathit{P}\mathit{o}\mathit{r}\mathit{t}.$$\mathit{\mu }$	$\mathit{P}\mathit{o}\mathit{r}\mathit{t}.$$\mathit{\sigma }$
MV	36.13%	4.14%	16.26%	6.90%	63.46%	36.54%	1.32%	2.87%
Portfolio 8	46.96%	6.01%	1.70%	14.08%	69.63%	30.37%	9.17%	4.16%
EQS Portfolio	48.13%	6.47%	1.02%	14.07%	70.59%	29.41%	18.14%	6.63%
Portfolio 35	47.86%	7.36%	0.60%	16.67%	73.53%	26.47%	39.45%	13.33%
Portfolio 42	33.24%	14.23%	0.43%	25.12%	74.19%	25.81%	47.30%	16.98%
MG Portfolio	5.92%	23.51%	7.90%	28.11%	68.95%	31.05%	57.39%	29.13%

All of the efficient portfolios have a positive loading on the first factor with a moderate value (greater than zero but not very high). This outcome aligns with the loading maps for individual assets in the previous section. The MV portfolio, possesses the highest loading coefficient value for the first factor. The loading coefficients of factors 2 and 3 are also positive for all the portfolios and are higher for MG and MV Portfolio. Finally, all of the portfolios are negatively exposed on the last factor with Portfolio MG to have the higher value. The loading coefficients multiplied with the factor scores determine the direction of the portfolio generation (increase or decrease in each month of the year).

The variance decomposition analysis reveals that the MV Portfolio has the lowest exposure to the systematic component of variance and the highest exposure to the idiosyncratic part, which is not the most favorable scenario. Diversification primarily aims to reduce the idiosyncratic variance by strategically allocating capacity to areas with diverse production profiles. Portfolio 42 has the lowest idiosyncratic variance and the highest level of systematic variance. These findings can be combined with the average daily generating capacity and risk levels to support the decision of a specific spatial allocation plan. For example, the EQS portfolio, previously discussed as an alternative solution, demonstrates lower idiosyncratic risk compared to the MV portfolio, a quite higher generating capacity and moderate levels of volatility.

It is important to emphasize that there is no single definitive answer when it comes to selecting a specific portfolio model, even after employing all the methodologies described in this section. The optimal choice of allocation plan ultimately depends on the risk profile and utility function of the individual investors. Each of the techniques presented here can, however, play a significant role in the decision-making process, providing valuable insights and aiding investors in making informed choices that align with their risk tolerance and objectives.

6. Conclusions and future research

Abbreviations

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