## Abstract

In contrast to conventional power generation, wind energy is not a controllable resource because of its stochastic nature, and the cumulative energy input of several wind power plants into the electric grid may cause undesired fluctuations in the power system. To mitigate this effect, the authors propose a procedure to calculate the optimal allocation of wind power plants over an extended territory to obtain a low temporal variability without penalizing too much the overall wind energy input into the power system. The procedure has been tested over Corsica (France), the fourth largest island in the Mediterranean Basin. The regional power supply system of Corsica could be sensitive to large fluctuations in power generation like wind power swings caused by the wind intermittency. The proposed methodology is based on the analysis of wind measurements from 10 anemometric stations located along the shoreline of the island, where most of the population resides, in a reasonably even distribution. First the territory of Corsica has been preliminarily subdivided into three anemological regions through a cluster analysis of the wind data, and the optimal spatial distribution of wind power plants among these regions has been calculated. Subsequently, the 10 areas around each station have been considered independent anemological regions, and the procedure to calculate the optimal distribution of wind power plants has been further refined to evaluate the improvements related to this more resolved spatial scale of analysis.

## 1. Introduction

After the Kyoto conference on global climate change in 1997, the worldwide on- and offshore capacity of grid-connected wind power plants has increased exponentially. According to the Global Wind Energy Council Report (2006), 2006 was another record year for the wind energy market, with installations of 15 197 MW, which has brought the total installed wind energy capacity to 74 223 MW. In terms of new installed capacity in 2006, the United States continued to lead. Nevertheless, Europe still remains the market leader with 48 545 MW of installed capacity, representing 65% of the global total.

From a technical point of view, at present, wind energy is often conveniently integrated into regional electricity supply systems, but its intermittent character is not without consequences for many power systems yet. In contrast to conventional power generation, where energy input can be scheduled and regulated to be consistent with the national power supply system (PSS), wind energy is indeed not a controllable resource because of its stochastic nature.

On the local scale, the control system for a single wind power plant is usually designed just to regulate the energy output of both the overall wind farm and individual wind turbines to optimize the wind farm dynamic performance (Steinbuch et al. 1988; Chinchilla et al. 2005). At this scale, the interest is primarily focused on the evaluation of the maximum wind turbine efficiency so as to extract as much energy as possible (Mosetti et al. 1994; Milligan and Factor 2000).

On the regional or national scale, the cumulative energy input of the overall wind power plants may cause noticeable input fluctuations in the power system. Indeed, the intermittency of wind is directly transmitted into the power supply system and this dramatically reduces the marketing value of wind energy (Milligan and Porter 2005). At an operational level, the actual challenge is to develop accurate models to perform wind power forecasting to predict the overall energy input into the power system (Persaud et al. 2003). For example, the “development of a next-generation wind resource forecasting system for the large-scale integration of onshore and offshore wind farms” (ANEMOS) project (Kariniotakis et al. 2006) focuses on forecasting the wind resource available for wind power plants up to two days ahead through physical and statistical prediction models (Giebel et al. 2006; Sánchez 2006; Madsen et al. 2005). The outcome of the ANEMOS project is expected to increase the wind energy integration through an optimized management of the risk related to the intermittent nature of wind generation.

Provided that short-term wind power prediction is a primary requirement for the efficient integration of wind energy in power systems and electricity markets, it is also of particular interest to the wind power industry to develop standard procedures to find the best way to distribute wind-generating capacity among several sites, to control the stability of the overall wind energy input into the power system. For example, Pantaleo et al. (2003) noted that the high concentration of the Italian wind energy resource in few areas of southern regions could cause grid integration difficulties, like overloading and regulation problems. Archer and Jacobson (2007) suggested interconnecting wind farms as a possible solution to improve wind power reliability by reducing wind energy fluctuations on the power system. They show how by linking a certain number of wind farms together, the overall performance of interconnected systems might improve substantially when compared with that of any individual wind farm. The advantages concern both supplying base load power as well as reducing deliverable power swings caused by wind intermittency. Their idea is that while wind speed could be calm at a given location, it will be higher somewhere else, so that the wind energy production of the interconnected system is more regular and constant as the number of interconnected wind farms increases. In particular, Archer and Jacobson (2007), starting from 19 measurement sites, analyzed the performances of all the possible combinations of *k* sites (with *k* = 1, 3, 7, 11, 15, 19). In so doing, more than 130 000 different spatial wind power distributions were taken into account, as each site is considered to have a single wind turbine. For instance, when they analyzed the advantages of connecting 11 stations (*k* = 11) over 19, they studied all the possible combinations (75 582) of 11 sites among the 19 of interest. However, since they reported results averaged over all the combinations, the question about which combination performs best still remains open.

After stating the advantages of interconnected wind farms with respect to an individual wind farm in terms of base load power supply and reduction of the variability of wind energy production (see also Kahn 1979; Archer and Jacobson 2003; Simonsen and Stevens 2004), in the present paper we suggest a possible methodology to answer the specific question about which wind power distribution maximizes the base load and minimizes the variability. Therefore, following the idea that the spatial distribution of wind power plants on a regional scale could be optimized to guarantee the minimum temporal variability without penalizing too much the overall wind energy input into the power system, we propose a procedure to calculate the optimal allocation of fractions of wind power through the minimization of either the wind energy variability or the ratio between energy variability and energy input into the PSS. On behalf of Agence De l’Environment et de la Maîtrise de l’Energie (ADEME) and Collectivité Territorial de Corse, this procedure has been applied to Corsica (France), the fourth largest island in the Mediterranean Basin, and is based on two steps:

wind measurements at 10 m above ground level (AGL) are converted into wind power output at higher levels;

the aforementioned minimization is performed to calculate the optimal distribution of a fixed number of wind power plants among zones with different anemological regimes.

We anticipate that, in the procedure, the definition of zones similar from the wind climatology point of view is essential to calculate the final optimal spatial distribution of power plants. These zones can be characterized by individual anemological stations, if enough representative of the surrounding territory, as well as by clusters of stations. In this paper we shall analyze both cases, first by considering three large regions and then 10 smaller regions.

The present document is organized as follows: in section 2 a short description of the territory under study and of available wind measurements is reported. Results from a cluster analysis of these anemological data, performed to identify the different anemological regions of Corsica (Burlando et al. 2008) within which the total wind power should be distributed, are also briefly presented. In section 3, measurements are transformed into wind power and the procedure is verified for one station through comparison with the wind energy produced by a wind farm already installed in the area under study. Section 4 focuses on the methodology for the minimization of wind power variability onto the power system. Conclusions are drawn in section 5.

## 2. Territory, measurements, and anemology of Corsica

The present research has been developed for Corsica, a large island situated in the northwestern part of the Mediterranean Basin. When this study was undertaken, the island had an independent power supply system because of the geographic isolation from the continent: the distance from the nearest national coast, in Côte d'Azur, is indeed as far as 90 n mi and the bathymetry is as deep as about 2000 m. An interconnection with the electrical network of the adjacent Sardinia Island, that is part of Italy, has been operational since February 2006. Nevertheless, the power supply system of Corsica is still quite limited and sensitive to fluctuations in power generation.

The island does not have very relevant industrial activities, and the electricity consumption is strongly dependent on tourism, while its wind energy potential is considerable. Therefore, unexpected energy inputs into the power system can easily cause undesired fluctuations. The intermittency of wind energy is therefore particularly critical for the regional power system, and a slowly variable wind energy input would be desirable rather than a very high one.

Following these considerations, the question that arises is whether it is possible to distribute the overall wind power over the territory in such a way that the wind power plants “switch on” in turn. We suggest that the answer depends on the morphology and extension of the territory itself. Indeed, Corsica consists of a territory about 175 km wide in latitude and 80 km in longitude and it is characterized by a very complex topography, with one main mountain chain crossing the whole island from the northern to the southern edge and more than 2000-m-high peaks (see Fig. 1). In this geographical context, it is likely to expect that the topography strongly influences surface wind regimes, in that when westerly winds blow the wind power plants in western Corsica are on while the power plants in the eastern side should be almost off, and vice versa.

This scheme is in fact a rough approximation of the actual situation. Burlando et al. (2008) identified the main wind climate regimes of Corsica, showing that they can be separated into two distinct classes: thermally forced and synoptically driven wind regimes. Thermally forced regimes are usually low to medium winds corresponding to sea and land breezes, and they consist of flows that converge toward or diverge from the inland areas. Their contribution to the total wind energy budget of the island is quite relevant especially during summer. However, as the onset of these circulations is almost synchronous along the whole perimeter of the island, they could induce fluctuations on the power supply system with a typical half-day cycle. Instead, synoptically driven regimes are usually characterized by winds stronger than breezes, so that they contribute more significantly to the wind energy budget of the island. Contrary to breezes, they typically correspond to surface wind patterns with higher winds upstream of the mountains and large sheltered areas on their lee side. For example, during the northeast wind gregale, the wind mainly blows southward in the eastern part of the island and southwestward in the northern part, while the southern side is sheltered. The other synoptically driven regimes are sirocco, libeccio, and maestro, which correspond to the wind coming from southeast, southwest, and northwest, respectively. All these wind regimes contribute to lowering the wind speed correlation among their corresponding upstream and downstream areas, and consequently they contribute to make the wind energy input more regular and constant during the year (see, e.g., Kahn 1979; Archer and Jacobson 2007). Moreover, the rapid succession of different surface wind patterns is often brought about by the advection of a single weather system so that exposed and sheltered areas interchange with each other, thus contributing to maintain more regular wind energy production for a longer time. This is the case, for instance, of the typical orographic cyclogenesis in the Gulf of Genoa, just north of Corsica, induced by the interaction of a major synoptic low pressure system over central or northern Europe with the Alps (Buzzi and Tibaldi 1978; Egger 1988). During the deepening phase of the cyclone, westerly regimes (libeccio and maestro) affect Corsica, thus sheltering the eastern side of the island. When the cyclone moves away, following a typical southeastward route (Trigo et al. 1999), the opposite situation occurs: the wind blows mainly from the northeast (gregale), and the western and southern parts of Corsica are now the downstream areas.

To catch the alternation of contributions of different areas to the wind energy production, we have based our analyses on measurements from 10 anemometric stations located along the shoreline of Corsica in a reasonably even distribution along the whole perimeter of the island. All stations belong to the Météo France network and record the horizontal wind intensity and direction at 10 m AGL. They are numbered counterclockwise from 1 (Solenzara) to 10 (Figari) in Fig. 1, while Oletta, used in the previous work by Burlando et al. (2008), is not explicitly considered in the present paper for the reasons discussed in the following. All wind measurements are averages over the last 10 min of the hour. The datasets, available for a 3-yr period from 1 October 1996 to 30 September 1999, have a sampling rate every hour (24 times per day from 0000 UTC to 2300 UTC), apart from the dataset of Cap Corse, which records every 3 h (8 times per day from 0000 to 2100 UTC). A summary of the main characteristics of the stations and of the corresponding datasets is reported in Table 1: columns 3–5 report geographical coordinates and elevation of the stations; average wind speeds at 10 m AGL and wind calm percentages are shown in columns 6 and 7; the shape and scale parameters of the corresponding nondirectional Weibull probability density function (Weibull 1951), evaluated from the wind speed time series at 10 m AGL, are reported in columns 8 and 9; finally, column 10 provides average wind speeds at 50 m AGL calculated as discussed in section 3a. As suggested by ADEME, inland stations were not taken into account, since such areas are scarcely populated with respect to the coast and not very suitable for the installation of wind power plants because of the very complex topography as well as the presence of several national parks.

Recently, Burlando et al. (2008) proposed a classification of Corsica into distinct anemological regions based on a cluster analysis of the aforementioned wind measurements. As explained in that paper, the hierarchical cluster analysis requires using synchronous datasets (see also Kaufmann and Whiteman 1999), so that the analysis was performed only on measurements simultaneously collected at all stations, namely for a maximum of *N* = 8760 (3 years × 365 days per year × 8 measurements per day). However, the number of available measurements per station was *N*′ = 7271, which is somewhat lower than *N* because of recording interruptions for maintenance or damage, so that the time series of contemporaneous measurements actually consisted of about 83% of *N*.

The anemological regions were defined through the comparison of 15 different clustering techniques resulting from the combination of three distance measures and five agglomerative methods. The results of this analysis identified the following three wind climate regions: the eastern region (ER), the northwestern region (NWR), and the southwestern region (SWR). In particular, ER includes the cluster comprising stations 1, 2, and 3, NWR includes the cluster comprising stations 4, 5, and 6, and the cluster containing stations 8, 9, and 10 identifies SWR. Station 7 (Ajaccio) showed to be a transitional station between NWR and SWR, whereas Oletta (not numbered in Fig. 1) results were unreliable so that it was not taken into account both in that clustering procedure and in the present paper.

In the framework of the present paper, the concept of anemological regions will be used to calculate the optimal distribution of wind power among ER, NWR, and SWR (section 4a). Subsequently we have applied the minimization procedure also to single stations, as a generalization to the case of clusters with just one element each (section 4b). Therefore, we have calculated the minimization in both cases so as to verify whether our classification into three anemological regions is sufficient to define an optimal spatial distribution of wind power plants consistently or if a further subdivision in 10 regions is recommended.

## 3. Conversion of wind measurements into wind energy output

Wind speed measurements are usually available at the international standard reference height of 10 m AGL, while the power curve of a wind turbine refers to the height at which the turbine operates. A common problem then arising is how to evaluate the wind speed at the turbine hub height. Since in our study only turbines with hub height of *h* = 50 m have been taken into account, we had first to convert, for each anemometric station *j* (*j* = 1, . . . , 10), the wind speed data at 10 m, *υ*^{10}_{j}, into the corresponding wind speeds at 50 m, *υ*^{50}_{j}. The procedure adopted to perform this conversion is described in the following subsection.

### a. Calculation of wind speed data at the hub height

Simple approaches to evaluate the wind speed at the turbine hub height are, for instance, the logarithmic and the power-law method (Gipe 1995), which both require an estimation of the surface roughness. Archer and Jacobson (2003) developed a more sophisticated methodology to evaluate the wind speeds at 80 m AGL from sounding and surface data.

In the present paper, we have adopted a methodology that makes use of wind speed measurements at 10 m AGL and numerical simulations of three-dimensional wind fields. These simulations were performed with three different numerical models. The wind fields over the northwestern region, where Ile Rousse and Calvi are placed, and over the southwestern region, which comprises Ajaccio, Pila-Canale, and Sartene (ARIA Technologies 2002), were simulated by means of the model “MINERVE” (Finardi et al. 2001). The wind fields over the southern area of Figari (Ratto et al. 2000) were simulated by the wind-field interpolation by nondivergent schemes (WINDS; Ratto et al. 1990; Burlando et al. 2007). Both MINERVE and WINDS are mass-consistent models (Ratto et al. 1994). The wind fields over eastern Corsica, which Solenzara, Alistro, and Bastia belong to (OptiFlow 2002a), and over the northern part of the island, where Cap Corse is found (OptiFlow 2002b), were simulated by means of a more sophisticated code (Ferziger and Perić 2002) [i.e., a Reynolds averaged numerical simulation (RANS)]. Unfortunately, we only had available the mean wind fields resulting from the simulations mentioned above, apart from those concerning the southern area, on which we directly performed the simulations (Burlando et al. 2002).

Therefore, the transformation of wind speed measurements at 10 m AGL into wind speeds at higher levels above ground level was performed through the subsequent steps:

the 10-yearly mean wind speeds,

^{h}_{j}, at the hub height of the turbine,*h*, are obtained from the cited wind flow simulations (see column 10 in Table 1);the 10 long-term time averages at 10 m AGL,

^{10}_{j}, are calculated from measurements;a scale factor

*γ*_{j}is defined as the ratio between^{h}_{j}and the mean wind speed at 10 m AGL,^{10}_{j}, namely*γ*_{j}=^{h}_{j}/^{10}_{j};the 10 time series of the wind speed at the level of interest,

*υ*^{h}_{ij}, that is, the time series of wind speed at 50 m AGL, are obtained by multiplying the wind measurements at 10 m,*υ*^{10}_{ij}, by the factor*γ*_{j}as*υ*^{50}_{ij}=*γ*_{j}*υ*^{10}_{ij}, being (*i*= 1, . . . ,*I*= 8317) the time index.

For example, the nondirectional frequency distribution of the wind speed at 50 m AGL calculated at Calvi and obtained through this procedure is shown in the left panel of Fig. 2. The last column of Table 1, moreover, reports the average wind speed values at 50 m AGL corresponding to the considered 10 stations.

If we had had at our disposal the individual simulated wind fields, we could have defined the scale factor in more refined ways, for example by taking into account the dependence on wind direction, speed, and, possibly, atmospheric stability. This would have explicitly taken into account the local effects of topography. In the authors’ opinion, however, this drawback is not that relevant in the present context, since we are interested in presenting the methodology rather than obtaining the most accurate evaluation of the wind potential. On the contrary, more sophisticated procedures would be recommended whenever possible.

In comparison with Archer and Jacobson’s paper (2007), the time series used in the present work have a larger sampling time, that is, 3 h instead of 1 h. The reasons to use this time step have been explained in section 2. This means that we are filtering out the wind speed fluctuations with a shorter period. However, on the one hand these datasets are expected to reproduce properly the most important atmospheric phenomena, from local thermal circulations to mesoscale and synoptic motions, which occur with larger periodicities (Van der Hoven 1957); on the other hand, from a statistical point of view, the frequency distributions of 3- and 1-h sampled datasets should present similar shapes, at least when time series are sufficiently long.

### b. Comparison between measured and calculated wind speeds

We have tested our procedure with wind data obtained from the Punta Aja wind farm, near Calvi. More precisely, we have calculated the wind speeds at 50 m AGL in Calvi and compared them with measurements from an anemometer placed within the wind farm at approximately the same height above ground.

The Punta Aja wind farm became operative in the end of 2003, so that a direct comparison with the wind data described in section 2, relative to the period 1996–99, was not possible. Therefore, the validation of the methodology described in the previous subsection has been applied to two new datasets, both ranging from 1 January to 31 December 2004:

the database of wind speed measurements collected at 10 m AGL at the Calvi station,

*υ*^{10}_{i6}(being for this dataset*i*= 1, . . . , 2920), with a sampling rate every 3 h (Δ*t*= 3 h);the database of daily wind speed measurements (minimum, mean, and maximum) at 48 m AGL taken on the mast of an anemometric station placed at Punta Aja, within the area of the wind farm, at a distance of about 10 km from Calvi.

Since the second database has a daily frequency while the first one is 3-hourly, some kind of elaboration is required to make them comparable. Indeed, after having converted the eight wind speed data measured every day at 10 m AGL at Calvi into wind speeds at 50 m, *υ*^{50}_{i6}, as described in the previous subsection, we have calculated the corresponding daily mean wind speeds at this height to obtain a database homogeneous with data available for the mast at Punta Aja.

The result of such a comparison is shown in Fig. 3, which shows a better agreement between measured and calculated values for intensities greater than 5 m s^{−1}, whereas for weaker values the calculated wind speeds usually are more intense than the measured ones. The reason of this mismatch is better explained in Fig. 4, which shows the differences between measured (Punta Aja) and calculated (Calvi) wind speeds as a function of time. The black line, which represents the polynomial regression of order five of these differences, shows that, apart from random fluctuations, the daily mean wind speeds at Calvi are greater on average from March to mid-October, while the opposite happens during the rest of the year. This behavior, that is, the higher wind speeds at Calvi than at Punta Aja in spring and summer, might be caused by the land and sea breezes that dominate the atmospheric circulation during those periods of the year when thermal flows prevail (see also Burlando et al. 2008), and are particularly strong just along the coasts. This is also clear analyzing the total wind blown at Calvi and Punta Aja, shown in Fig. 5, defined as the cumulative distance (km) traveled by the wind during a given time. Indeed, the two curves are nearly coincident until early May, when they begin diverging since the wind speeds at Calvi are higher than at Punta Aja. From late October, due to the reduced influence of sea breezes and the prevalence of synoptic-driven circulations, the wind speed is on average higher at Punta Aja than at Calvi so that the gap between the two curves decreases.

The percentage error, defined as the ratio of the absolute difference between the two winds blown at Punta Aja and Calvi over the wind blown at Punta Aja, is about 7% at the end of the year. This value is obviously conditioned by the particular year taken into account, that is, 2004, but we do not expect, in principle, very different values if another year were considered.

### c. Calculation of wind energy output

Once wind speed time series at the hub height become available, these can be converted into time series of wind power produced by whatever turbine if the corresponding power curve is known. Wind turbine manufacturers usually issue the power curve of their wind turbines so that the conversion from wind speed to wind power is almost straightforward. It is worth noting that, in fact, measured power curves by field measurements consist of a swarm of points spread around the curve issued by the manufacturer. The reason for that is twofold: on the one hand, the anemometer that measures the wind speed is placed on a mast reasonably close to the wind turbine, but not on the turbine itself; on the other hand, the wind speed always fluctuates, and one cannot measure exactly the column of wind that passes through the rotor of the turbine. This is why power curves are based on measurements in areas with low turbulence intensity, whereas for a wind turbine placed on complex terrain, local effects may mean higher turbulence intensity and wind gusts. It may therefore be difficult to reproduce the power curve exactly in any given location. In the context of the present study, however, it is not so relevant to use the exact power curve of the chosen wind turbine, since, as we have already stated, the purpose of the research concerned the establishment of the methodology to evaluate the optimal spatial distribution of wind power over Corsica, rather than an accurate evaluation of the wind potential of the island.

The conversion from wind speed data to wind power and wind energy has been tested with the wind speed data at 50 m AGL calculated at Calvi (see section 3b). The transformation of wind speeds, *υ*^{50}_{i6}, into wind powers, *P*^{50}_{i6}, is performed with the power curve of the wind turbine Enercon E40/600 of nominal power 600 kW and hub height 50 m. This is the same kind of turbine with which the Punta Aja wind farm is equipped (http://www.suivi-eolien.com/). The power curve of this turbine is shown in the right panel of Fig. 2. Then, *P*^{50}_{ij} data have been converted into energy output, *E*^{50}_{ij}, multiplying the wind power time series by the discrete time interval Δ*t* = 3 h, according to the sampling time of the wind speed database. The corresponding wind energy output distribution is shown in the left panel of Fig. 2.

### d. Comparison between measured and calculated wind energy outputs

We have evaluated the consistency of the transformation of wind speeds at 50 m AGL at Calvi into the corresponding wind energy output *E*^{50}_{i6} of one wind turbine E40/600 through the comparison between calculated and actually produced daily cumulated wind energy.

The comparison between the calculated (Calvi) and actually produced (Punta Aja) daily cumulated energies is shown in Fig. 6. Analogous to the behavior already presented in Fig. 3 and discussed in the previous section 3b, there exists a pretty good agreement on average for high energies, while for low values, likely associated with land and sea breeze regimes, the calculated energy is overestimated with respect to the measured one. In general, however, the dispersion of data around the bisector is larger than the dispersion of wind speeds in Fig. 3. This is also evident from Fig. 7, which shows the comparison between the time series of the daily energy produced by a turbine at Punta Aja and the corresponding daily energy output calculated from wind speed measurements at the Calvi station, “transported” to the height of 50 m AGL. Once again, the calculated values are higher from mid-March to mid-October, especially in case of low wind speed, coherently with the hypothesis of stronger breezes at Calvi than at Punta Aja. On the contrary, the peaks associated with synoptic-driven conditions are reproduced quite well.

Finally, the cumulated energy outputs for the calculated daily energy output at Calvi and the actually produced values at Punta Aja during year 2004 are plotted in Fig. 8. In this case, the produced cumulated energy is higher than the calculated one until late June, when the two curves cross each other because of the growing contribution of the sea breeze at Calvi. This pattern is somewhat different from that in Fig. 5, however, since the wind power output exhibits a cubic dependence on the wind speed and it is conditioned by the cut-in and cut-out values (3 and 26 m s^{−1}, respectively) of the chosen wind turbine. This is particularly important during summer when below cut-in winds blow in Punta Aja and do not produce any wind energy output, whereas at the same time above cut-in land or sea breezes contribute to the cumulated wind energy at Calvi. The cut-out value also has influence in the present analysis since, even if the considered daily mean wind speeds are always lower than this threshold (see Fig. 3), maximum intensities up to 35 m s^{−1} have been recorded, clearly affecting power production. From early November the influence of thermally driven winds decreases and the gap between calculated and produced energy starts reducing because wind speeds are slightly higher, on average, at Punta Aja than at Calvi during the winter season (see Fig. 4).

Following these considerations, the production of wind energy, higher at Punta Aja in the first half of the year, higher at Calvi in the second one, turns out to be the combined and antithetical effect of higher wind speeds during the summer at Calvi and higher wind speeds during the winter at Punta Aja. As a result, the percentage error between calculated and produced cumulated energy at the end of the year is only 1%. The exact value of the error is not essential, however, in the context of this research because the explicit consideration of uncertainties is not required to solve the present optimization problem and, in particular, to evaluate the most convenient repartition of wind power. On the contrary, if the present procedure were applied to obtain the most accurate evaluation of the wind potential of a territory, a detailed study of the effects of the topography to distinguish different turbine positions would be recommended. Indeed, in the present paper, all sites surrounding an anemometric station are characterized by the same anemology, but this drawback could be overtaken if more sophisticated numerical tools were adopted to estimate directly the spatial correlation between stations and specific sites. Moreover, in the case of a wind farm, if the same power curve were used for all turbines, wake effects should be explicitly considered. However, provided that the distance between the turbines is more than 8–10 times the diameter of the rotor along the prevailing wind direction and more than 6 times the diameter along the direction perpendicular to the prevailing one, wake effects should be less than 10% of the total (Lissaman et al. 1982). Therefore, the energy output *N*_{j}*E*^{50}_{ij} can be used as a rough estimation of the energy produced by *N _{j}* turbines distributed around the corresponding

*j*th anemometric station.

## 4. Minimization methodology and results

The procedure we propose here aims at calculating the optimal allocation of fractions of wind power over a territory through the minimization of either the wind energy variability or the ratio between energy variability and energy input into the power system. In such a way, undesired power fluctuations in the power system, due to the intermittent nature of the wind energy generation, can be reduced.

To check if the minimization procedure is turbine independent, as expected, and to evaluate how much the performances change by increasing the turbine nominal power, two different kinds of wind turbine have been considered in the analysis: the Enercon E40/600, which is the same model already considered in section 3, and the Enercon E48/800 of nominal power 800 kW. For both turbines the hub height is 50 m AGL.

The methodology can be applied to the 10 areas around single anemometric stations as well as to the three clusters of stations. The possibility of grouping stations based on an objective classification criterion is particularly important to reduce the overall computational time when a large number of stations is available, especially if very close to each other.

In the present work, we have applied the minimization procedure in both cases:

to the three distinct anemological regions of Corsica, identified by clusters of anemometric stations (section 4a);

and to the single 10 anemometric stations, as a generalization to the case of clusters with just one element (section 4b).

Finally, it is worth noting that all the following analyses have been performed with the 3-yr-long datasets described in section 2.

### a. The case of three regions

As discussed in section 2, we have identified three different wind climate regions along the coasts of Corsica: the eastern region including stations 1, 2, and 3, the northwestern region including stations 4, 5, and 6, and the southwestern region including stations 8, 9, and 10. In this analysis, station 7 (Ajaccio) has not been considered, since the clustering technique revealed that it is a transitional station between NWR and SWR.

The minimization procedure requires a single time series of wind energy for each anemological region. Therefore, the wind energy time series, *E*_{ij}, obtained for the single anemometric stations (see section 3c) have to be arranged according to a weighted averaging to obtain the corresponding time series for each region. For instance, the time series of wind energy of ER *E*_{i,ER} can be calculated as Σ_{j}*w*_{j}*E*_{ij}, where *j* = 1, 2, 3 and Σ* _{j}w_{j}* = 1. The value of weights

*w*corresponds to the repartition of wind turbines among the areas surrounding stations 1, 2, and 3. In the present study, in the absence of other information, we have assumed a uniform distribution of the wind turbines within every region, namely

_{j}*w*= 1/3.

_{j}Suppose we now place *αN* turbines into ER, *βN* into NWR, and *γN* into SWR, where *N* is the total number of turbines all over the territory of Corsica and the coefficients *α, **β*, and *γ*, which add up to unity (*α* + *β* + *γ* = 1), represent the distribution of *N* among the anemological regions. The time series of the overall energy output can be written as

where *E*_{i,ER}, *E*_{i,NWR}, and *E*_{i,SWR} are the wind energy time series of the three anemological regions, calculated from the time series of the corresponding anemometric stations.

It is now straightforward to calculate the corresponding total energy output:

its variability, defined as follows:

and their ratio, 〈Δ*E*〉/〈*E*〉, as a function of the three parameters (*α*, *β*, *γ*). The calculation is performed assuming different triplets in the space of the parameters *S* = {*α*, *β*, *γ* ∈ *R*:0 ≤ *α* ≤ 1; 0 ≤ *β* ≤ 1; 0 ≤ *γ* ≤ 1}.

The equilateral triangle shown in Fig. 9 represents the space of the *α*, *β*, and *γ* parameters, which range from 1 in the corresponding vertex to 0 on the opposite side along the medians: the condition *α* + *β* + *γ* = 1 is then satisfied, as the sum of the distances from any point inside an equilateral triangle to the sides is constant. The shaded contours represent the values of the three calculated variables 〈*E*〉, 〈Δ*E*〉, and 〈Δ*E*〉/〈*E*〉, normalized with respect to the corresponding maximum value.

The distribution that maximizes wind energy output (〈*E*〉 = max) is trivial, as it shows a maximum for the triplet (*α* = 0, *β* = 1, *γ* = 0), which corresponds to locate all the turbines into the windiest region, that is, the NWR (see the left panel in Fig. 9). The minimum shown in the center panel corresponds to the distribution (*α* = 0.46, *β* = 0.26, *γ* = 0.28) that minimizes the wind energy variability (〈Δ*E*〉 = min). It can be seen that in this case the highest percentage of turbines is found in the least windy region, that is, the ER. Finally, the minimum shown in the right panel in Fig. 9 corresponds to the distribution (*α* = 0.27, *β* = 0.48, *γ* = 0.25) that minimizes the ratio between energy variability and energy output (〈Δ*E*〉/〈*E*〉 = min). We have verified that all these triplets are turbine independent, as they do not change if the two considered different power curves are used.

The mean annual energy production per turbine (*N* = 1) corresponding to these three distributions is 1568, 1075, and 1235 MWh, respectively, if Enercon E40/600 wind turbines are considered. Applying the same methodology, but considering 800-kW nominal power turbines (Enercon E48/800), the mean annual energy production for the three distributions rises to 2270, 1570, and 1795 MWh, respectively. Note that by adopting the higher nominal power turbine, the values of annual energy production increase by a factor about 1.45.

The 〈Δ*E*〉/〈*E*〉 = min distribution permits a larger energy production together with relatively low power fluctuations, whereas the 〈Δ*E*〉 = min distribution reaches this goal at the expense of the energy output. Indeed, for the 〈Δ*E*〉 = min distribution, the variability reduction with respect to its maximum value (corresponding to have all the turbines in NWR) is about 37% and the energy production loss is 31%; for the 〈Δ*E*〉/〈*E*〉 = min distribution the variability reduction is quite similar, 32%, but the energy loss turns out to be only 21%.^{1} These values are almost the same for both E40/600 and E48/800.

### b. The case of 10 regions

A straightforward extension of the procedure shown in the previous subsection is the calculation of the 〈Δ*E*〉 = min distribution and the 〈Δ*E*〉/〈*E*〉 = min distribution for the ten areas around the single anemometric stations, each considered an independent anemological region.

Suppose we place *α*_{1}*N* turbines in the Solenzara region, *α*_{2}*N* turbines in the Alistro region, and so forth, until *α*_{10}*N* turbines are in the Figari region, with *N* being the overall number of wind turbines to install and *α*_{j} the fraction of *N* assigned to region *j*, so that *α*_{1} + *α*_{2} + · · · + *α*_{10} = 1.

The time series of the overall energy output can be written as follows:

where *E*_{ij}(*j* = 1, . . . , 10) are the wind energy time series of each anemometric station. The total energy output, 〈*E*〉, and its variability, 〈Δ*E*〉, as a function of the parameters (*α*_{1}, . . . , *α*_{10}) follow analogously to Eqs. (2) and (3), respectively.

The results are summarized in Table 2, where the values of the parameters for the three considered distributions as well as the corresponding values of 〈*E*〉, 〈Δ*E*〉, and 〈Δ*E*〉/〈*E*〉 are reported. The results for the case of three anemological regions, discussed in section 4a, are also reported to highlight the consistency between the higher- and the lower-resolution analysis.

It is worth noting that the energy output is maximized if one places all the turbines in the fourth region (i.e., Cap Corse), characterized by the highest average wind speed (see column 10 of Table 1). For this distribution, if Enercon E40/600 turbines are considered, the annual energy production per turbine would be 1830 MW h, whereas this value would increase up to 2600 MW h if E48/800 turbines are assumed.

For the other two distributions, Fig. 10 shows the values of *α*_{j} for all 10 stations. Furthermore, it is evident that the 〈Δ*E*〉 = min distribution is relatively more uniform than the 〈Δ*E*〉/〈*E*〉 = min distribution. Consistently with the results obtained for three anemological regions introduced in section 2, the highest percentage of turbines is found in the least windy regions (*α*_{1}, *α*_{2}, *α*_{8}, and *α*_{9}) for the 〈Δ*E*〉 = min distribution, and in the most windy regions (*α*_{4}, and *α*_{10}) for the 〈Δ*E*〉/〈*E*〉 = min distribution. Moreover, the sum of the parameters *α*_{j} corresponding to the areas of the three anemological regions gives a value close to the one obtained by the minimization procedure when applied to three regions only. For instance, as far as the 〈Δ*E*〉 = min distribution and the eastern region (consisting of stations 1, 2, and 3) are concerned, *α*_{1} + *α*_{2} + *α*_{3} = 0.45; that is nearly equal to the value of the corresponding parameter *α* = 0.46 found for such a region (see Table 2 and section 4a).

Considering the E40/600 (E48/800) wind turbine, the mean annual energy production per turbine corresponding to the 〈Δ*E*〉 = min distribution is 1010 (1480) MW h, which is not very different from the value found in the case of three regions. On the contrary, for the 〈Δ*E*〉/〈*E*〉 = min distribution, these values become significantly higher than for the three-region case [i.e., 1410 (2030) MW h against 1235 (1795) MW h]. This is not surprising, because we are now taking into account all stations without any kind of averaging. In such a way, local features filtered out by the clustering procedure can be retained, permitting a more efficient repartition of wind power throughout the territory. The variability reduction with respect to its maximum value is pretty high (about 64% for the 〈Δ*E*〉 = min distribution and 58% for the 〈Δ*E*〉/〈*E*〉 = min distribution) and not very different for the two distributions. As far as the energy production loss is concerned, a value of 45% (43%) is found for the 〈Δ*E*〉 = min distribution, whereas for the 〈Δ*E*〉/〈*E*〉 = min distribution it is only 23% (22%). Thus, the 〈Δ*E*〉/〈*E*〉 = min distribution combines a significant damping of power fluctuations with large energy production values.

### c. Considerations about wind power production and variability

In this subsection the spatial distributions for the case of 10 regions, mentioned in section 4b, are analyzed from the point of view of wind integration in the power system and the ability of interconnected wind farms to provide base load power. All results here presented and discussed are per turbine.

Figure 11 shows the frequency of given intervals of wind power input into the power system for five different spatial distributions. More precisely, these frequencies represent the time during which a given wind power per turbine is inserted into the grid with respect to the total time. In abscissa, the considered power intervals, which refer to the Enercon E48/800 wind turbine, are divided into classes of 100 kW from 0 < *P* ≤ 100 kW to 700 < *P* ≤ 800 kW, plus the first class corresponding to the case of *P* = 0 kW. The first class corresponds to all the wind speeds below the cut-in or above the cut-out thresholds, while the last class 700 < *P* ≤ 800 kW comprehends the wind speed interval between the saturation of the power curve at the nominal power of the wind turbine and the cut-out threshold. Two distributions correspond to place all the turbines in one region: the 〈*E*〉 = min distribution, in the least windy region of Pila Canale; and the 〈*E*〉 = max distribution, in the windiest region of Cap Corse. The other three distributions are “scattered” ones: the “uniform” distribution, where the total number of turbines is equally subdivided among the 10 regions, the 〈Δ*E*〉 = min, and the 〈Δ*E*〉/〈*E*〉 = min distributions, whose repartitions of the total number of turbines among regions are reported in Table 2.

It is worth noting that only the two distributions corresponding to the total number of turbines placed into a single region show non-null frequencies in the first class *P* = 0 kW. They also show relatively high frequencies in the last interval, corresponding to 700 < *P* ≤ 800 kW. On the contrary, the repartition of wind turbines around the perimeter of the island reduces to zero the probabilities that the wind speed is null or above the cut-out everywhere. Analogously, also the probability of high, but below the cut-out, wind speeds contemporarily all around the island is low, so that the frequency of class 700 < *P* ≤ 800 kW is almost zero for distributions different from the 〈*E*〉 = min and 〈*E*〉 = max ones. As far as the intermediate intervals are concerned, however, the frequencies are in general much higher when the wind turbines are not concentrated within a single region. Finally, note that the 〈Δ*E*〉 = min distribution shows higher frequencies for the interval 0 < *P* ≤ 300 kW when compared with the 〈Δ*E*〉/〈*E*〉 = min distribution, which shows higher frequencies for *P* > 300 kW instead, accounting for the higher values of annual energy production (see section 4b).

Figure 12 shows the frequencies of the wind power variability over 3 h, that is, Δ*P*_{i} = (*E*_{i} − *E*_{i−1})/Δ*t* (where Δ*t* = 3 h) as a function of the same wind power classes as those reported in Fig. 11. The frequencies of class Δ*P* = 0 kW are greater than zero only for the 〈*E*〉 = min and 〈*E*〉 = max distributions, which correspond to placing all turbines in the same region. Although in a single region the probability that the wind speed does not change over a period longer than 3 h is rare, these frequencies describe the situations when wind speed variation over 3 h remains below the cut-in or above the cut-out so that the power output persists being zero. The same situation does not occur for the scattered distributions because their power output is always *P* > 0 kW, as shown in Fig. 11, and the probability that the wind speed does not change over 3 h contemporarily all around the island is practically zero. As far as all the other classes are concerned, the frequencies of power variability for 0 < Δ*P* ≤ 200 kW are almost doubled for the scattered distributions relative to the 〈*E*〉 = min and 〈*E*〉 = max distributions, comparable for 200 < Δ*P* ≤ 300 kW, and much lower for Δ*P* > 300 kW. In particular, about 70% of the cases belong to the class corresponding to 0 < Δ*P* ≤ 100 kW for scattered distributions, so that a more regular and slowly varying power output is expected when interconnected wind farms are considered.

Finally, Fig. 13 shows generation duration curves as defined by Holttinen and Hirvonen (2005). This kind of graph represents a reversed cumulative probability distribution, in which each point of the abscissa represents the probability (in terms of number of hours in a year) of wind power production greater than or equal to the corresponding wind power value on the curve. The area below the generation duration curve represents the total energy (kW h) produced in a year by the corresponding spatial distribution. As already seen in Fig. 11, the power generation is 0 kW for about 20% (39%) of the hours of the year for the 〈*E*〉 = max (〈*E*〉 = min) distribution, while it is never equal to 0 kW in the other cases. For about 10% (40%) of the hours the 〈*E*〉 = min (〈*E*〉 = max) distribution produces more energy than scattered distributions. For the 〈*E*〉 = min distribution, this higher production over 10% of the time is largely made up for by the higher production of scattered distributions over the remaining 90% of the hours. On the contrary, the larger production for 40% of time by the 〈*E*〉 = max distribution is not completely compensated, because such distribution still produces a larger amount of energy than scattered ones as reported in Table 2.

The yearly averaged wind power of the considered distributions ranges from 118 kW for the 〈*E*〉 = min distribution to 297 kW for the 〈*E*〉 = max distribution, corresponding to capacity factors, defined as the fraction of the rated power actually produced in a year, of 0.15 and 0.37, respectively. The uniform and 〈Δ*E*〉 = min distributions have values between 185 and 169 kW, respectively, whereas the yearly power of the 〈Δ*E*〉/ 〈*E*〉 = min distribution is 232 kW and its capacity factor is 0.29.

The firm capacity of the distributions, which is the fraction of installed wind capacity (here 800 kW) that is online at the same probability as that of a coal-fired power plant, is then considered so as to evaluate the base load wind power. We shall assume the threshold of 87.5% as the probability that coal plants are free from scheduled maintenance (see Archer and Jacobson 2007). At this threshold, the distributions that correspond to place all the turbines in one region do not guarantee any power generation. Instead, the guaranteed power generation for uniform and 〈Δ*E*〉 = min distributions is 33 and 32 kW, respectively, and 38 kW for the 〈Δ*E*〉/〈*E*〉 = min distribution. These values correspond to firm capacities of 0.04, 0.04, and 0.05, or analogously, to 18%, 19%, and 16% of the yearly power produced, respectively. The latter values, in principle, represent the percentages of yearly averaged wind power that can be used as reliable base load electric power.

## 5. Conclusions

In the present paper, we have proposed a procedure to calculate the optimal allocation of wind power plants over a territory to minimize the variability of energy input into a power supply system. The optimization can be performed under the constraint of obtaining the absolute minimum temporal variability or the minimum ratio of the temporal variability over the overall wind energy input.

The adopted methodology makes use of wind measurements at ground level, and the conversion from wind data at 10 m AGL to wind aloft at the hub height is based on the use of three-dimensional numerical simulations.

We have shown that some kind of statistical technique can be applied to group different anemometric stations that belong to the same anemological region, instead of analyzing each station independently. This is particularly relevant when a large number of stations is available whose spatial representativeness partially overlaps, and a reduction of the total number of degrees of freedom of the problem could be recommended to reduce the computational time required for the calculation of the aforementioned optimal distributions.

We tested the procedure over Corsica, the fourth largest island in the Mediterranean, located in the northwestern part of the basin. In the beginning we subdivided the territory of Corsica into three anemological regions by means of a cluster analysis of the wind data, so that the spatial distributions of wind turbines have been calculated considering three parameters only. The distribution that minimizes the ratio between energy variability and energy output, 〈Δ*E*〉/〈*E*〉 = min, permits a pretty high annual energy production (1795 MWh for an 800-kW turbine) together with rather low power fluctuations (911 MW h). On the contrary, the distribution that minimizes the variability, 〈Δ*E*〉 = min, reduces the power fluctuations (850 MW h) but at the expense of the energy output (1570 MW h).

Then, it was a straightforward extension to apply the calculation of such distributions to ten independent anemological regions, each corresponding to a single anemometric station. The results have shown that the increase of the spatial resolution of the analysis has provided a refined repartition of the overall power, although in agreement with the lower-resolution one. As far as the 〈Δ*E*〉/〈*E*〉 = min distribution is concerned, in particular, a significantly higher mean annual energy production per turbine has been found (2030 versus 1795 MW h). Moreover, this distribution, which splits the total number of turbines among 10 regions, was found to have 16% of yearly averaged wind power to be used as a reliable base load into the power system. It is worth noting that this value is in between the corresponding values obtained by Archer and Jacobson (2007) for the cases of 7 (14%) and 11 (23%) interconnected wind farms.

Finally, the procedure can be easily applied assuming the contemporary use of different wind turbines, using different wind power curves and/or hub heights when wind measurements are converted into wind energy outputs. This further improvement has been already implemented and tested in software that we have prepared on behalf of ADEME.

## Acknowledgments

This research has been funded by Agence De l’Environment et de la Maîtrise de l’Energie (ADEME) and Collectivité Territorial de Corse. In particular, we warmly thank Dr. Philippe Istria from ADEME (Délégation Corse), with whom we had valuable and constructive exchanges of ideas and results throughout this research activity. We thank Prof. Roberto Festa for his numerous and helpful comments and suggestions. We are grateful to Mrs. Marina Pizzo for her assistance in the drafting of this paper.

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## Footnotes

*Corresponding author address:* Federico Cassola, Department of Physics, University of Genoa, Via Dodecaneso 33, 16146 Genoa, Italy. Email: cassola@fisica.unige.it