The Amplitude-related partial complementarity index i_a, assesses the relationship between the values of the differences of the maxima to the minima of two energy resources availability functions. Based on the work of Beluco et al. [10] , Equations (1) and (2) were adapted for calculating i_a. in these equations, d_w [1] is the difference between maximum and minimum value of wind energy availability, d_s [1] is the difference between maximum and minimum value of solar energy availability, E_{w max} [Wh/m²] and E_{w min} [Wh/m²] are respectively the maximum and minimum values of daily available wind energy through the year, E_{s max} [Wh/m²] and E_{s min} [Wh/m²] are respectively the maximum and minimum values of daily available wind energy through the year.

(2)

It can be easily observed that if the differences δ_w and δ_s are equal, the index i_a would be equal to 1. For this work, a scale factor of 10 will be used, in order to make i_a equal to 10 when δ_w and δ_s are equal, and tending to 0 with the increase of their difference.

According to Beluco et al. [1] , the index i_a was created to account for the difference between maximum and minimum energy availability of the sources in the complementarity evaluation. One energy source cannot be considered for complementarity purposes if it has no available energy during the period of interest. If the two sources present the same energy availability range, they are ideally complementary and the index i_a should be equal to 10. For all intermediate cases, with unequal differences δ_w and δ_s, the index i_a values will range between 0 and 10 for complementarity less than ideal.

Two complementarity components are included and combined in the Time- and Energy-related partial complementarity index i_et. Beluco et al. [1] explain that Time-related complementarity is used to evaluate the time interval between the minimum availability of the two energy sources. If this interval is exactly half the period, the index will equal to one. If this interval is null, the index will equal to zero. These authors also explicate that the Energy-related complementarity assesses the relationship between the average values of the energy availability functions. If the average values are identical the index should equal to one. If those values are different the index will be smaller, and will tend to zero with the increase of the difference.

Rio Grande do Sul is a subtropical state of Brazil, and because of that, is subject to four seasons a year. Based on that, the Time-related complementarity for this work will be considered between seasons (six different combinations). The Equations (3) and (4) are used for calculating i_et: In these equations, E_w [1] is the weighted wind eneergy, E_s [1] is weighted solar eenrgy, E_{w avg} [Wh/m²] is the average value of daily available wind energy through the year, E_{s avg} [Wh/m²] is the average value of daily available solar energy through the year, E_{w A} [Wh/m²] is the available wind energy through the year, E_{s A} [Wh/m²] is the available wind energy through the year.

(3)

The 1 and 2 in the subscript indicate the two seasons under consideration (e.g. 1-spring and 2-fall). As it was the case for the partial complementarity index i_a, a scale factor of 10 is applied to the index i_et found. Therefore, if i_et = 0, it means there is no complementarity, in time and energy, of solar and wind energy sources between these two seasons of the year. If i_et > 0, it represents that there is some complementarity, in time and energy, between these two seasons of the year. When i_et < 0, it will be assumed that no complementarity takes place. Finally, if i_et = 10, it implies a maximum complementarity, both in time and energy, of solar and wind energy sources.

As explained by Beluco et al. [1] , the total complementarity index i_T, indicates the degree of complementarity between two energy sources. It combines the two partial complementarity indices (i_a and i_et) previously described, according to the Equation (5).

Based on the range of values used in this work for i_a and i_et, it can be easily observed that the values of i_T will be between 0 and 100, with 100 denoting an area with full energetic complementarity.

Colors are used for representing numerical values of geographical features in maps. The digitization of these maps allows using the pixel information, given by the palette, for extracting data that can be combined with other sources for compiling new maps using GIS software.

A computer program was developed at the UFRGS Solar Energy Lab to perform the pixel color reading of the maps, by using the getpixel (x,y) function of Visual Basic.NET, which returns the numeric value corresponding to that of the pixel color, by comparing it to a reference palette.

For each nine pixels group was calculated the average of the numeric values associated to them, and then, this average substitutes each individual value.

The computer program starts reading the pixel at the second column of the second row, and continues reading from left to right, every other three pixels, until it finds a pixel with a color defined in the reference palette, and based on this, the average is calculated for this and the other eight surrounding pixels, continuing this process until the end of the row. After this, the reading restarts at the second pixel three rows below, always from left to right, following the same previous method until getting to the last row, according to what is shown in Figure 1 and Figure 2.

When the computer program finds one or more pixels with a color not defined in the reference palette, these are not considered for calculating the corresponding nine pixel group average.

The images used as data sources for this paper, which analyzes complementarity in Rio Grande do Sul, are maps with the average wind speed at 50 meters height, surface roughness and solar radiation on a horizontal surface. It is required to transform wind speed from 50 m to 10 m high, in order to use small wind turbines, with lower towers. For a better estimation of available solar energy, a correction is made for calculating the solar radiation on a 45˚ sloped surface.

The information related to solar radiation on a horizontal surface for Rio Grande do Sul is taken from the work of Martinazzo [11] , and data concerning wind speed at 50 m and surface roughness from Camargo et al. [12] . The maps from the latter author were digitally modified, so as they match the spatial resolution of the maps from Martinazzo [11] .

The computer program explained on the previous subsection was used to extract the information, from these three types of maps, for the average day of the central month of each season.

The resulting images are maps with 254 rows (each encompassing 1'35" of latitude) and 251 columns (each encompassing 1'54" of longitude). The information stored on the cells of these maps can be used in the equations cited in this work.

Solar radiation data can be available at several different forms, and each one of these can be used for a multiplicity of purposes in the design and development of solar energy systems. Jakhrani et al. [13] explain that monthly total solar radiation on a horizontal surface can be used for designing such systems. However, the process performance is generally not linear with solar radiation, and because of that, the solar radiation measurements on tilted surfaces are essential for more accurate inputs to solar photovoltaic (PV) systems or collectors.

The method used in this work for estimating the solar energy availability (monthly incident solar radiation on a sloped surface) was the isotropic sky model described in Duffie and Beckman [14] . The following values were used:for the southern hemisphere, the slope β = −45˚, for Rio Grande do Sul, the latitude ϕ = −30˚ and the diffuse reflectance ρ was considered equal to 0.2.

The value chosen for representing the solar energy availability was considered to be that of the average day of the central month of each season. Table 1 shows the declination δ and recommended average day for each month.

The declination δ was found from the approximate equation of Cooper [15] , according to Duffie and Beckman [14] , shown in Equation (6) where n is the n-th day of the year.

The sunset hour angle ω_S is determined by means of Equation (7).

The monthly average daily extraterrestrial radiation H_o, is found using the average day for a particular month, because on that day, the H_o value is the closest to the average extraterrestrial radiation value of the month. With solar constant GSC equal to 1367 W/m² (adopted by the World Radiation Center), the equations given by Duffie and Beckman [14] for calculating H_o is shown in Equation (8). In this equation, G_on [W/m²] is the extraterrestrial radiation incident on plane normalto radiation.

Irradiation distribution along the year presents a seasonal trend with daily overlapped fluctuations. For this reason, it is necessary to make this distribution independent of the season, by means of dividing the monthly average daily total radiation on a horizontal surface H by the monthly average daily extraterrestrial radiation H_o. This results in the monthly average clearness index K_t, which can be calculated with Equation (10).

On a cloudy day, the global radiation on horizontal surfaces received would indicate the diffuse radiation by means of an suitable index, K_d, correlated to K_t. The Equation (11) allows estimating the diffuse radiation fraction K_d:

For ω_S ≤ 81.4˚ and 0.3 ≤ K_t ≤ 0.8, Duffie and Beckman [14] use the Equation (12) to calculate K_d. and Equation (13) when ω_S > 81.4˚ and 0.3 ≤K_t ≤ 0.8.

The sunset hour angle on a sloped surface, ω_St, is initially found by means of computing the intermediate value ω_S2, using the Equation (14).

Once the ω_S2 value is available, this is compared with ω_S. If ω_S < ω_S2, then ω_St = ω_S2; else ω_St = ω_S2.

The geometric factor R_b, which expresses ratio of the average daily beam radiation on the tilted surface to that on a horizontal surface for the corresponding month, can be used to estimate the daily radiation on a sloped surface. For the southern hemisphere, Duffie and Beckman [14] use the Equation (15).

With the features previously found, it is possible to estimate H_T [MJ/m²], the monthly average daily radiation on the sloped surface. The expression for calculating H_T, at the average day of the central month of each season, is available at Duffie and Beckman [14] and shown in Equation (16).

As mentioned before in a previous section, for this work, the information related to the monthly average daily radiation on a horizontal surface at Rio Grande do Sul can be obtained from Martinazzo [11] . Figure 3 shows the resulting maps with the average daily radiation on the 45˚ sloped surface for each season, and Figure 4 displays this average for the year, based on the arithmetic mean of the seasons.

Like almost every type of energy on the planet, energy coming from the wind is an indirect form of solar energy. The heating of the atmosphere, due to the absorption of solar radiation, combined with the earth’s rotation generates global and local wind patterns.

According to Leishman [16] , the spatial variation of wind speed with height can be modeled by the logarithmic law, which includes a surface roughness

length parameter, z₀, representing the effects of the terrain on the upstream boundary layer development. The logarithmic law equation for the time-aver- aged wind profile is shown in Equation (17), where V(h) [m/s] is the wind speed as a funciont of height above ground level h [m], V_ref [m/s] is the reference wind speed, h_ref [m] is the reference height and z_o [m] is the roughness length.

Table 2 presents some typical surface roughness lengths, as presented by Burton et al. [17] . Something important to highlight is that the logarithmic law

has the limitation of not considering temperature or atmospheric pressure, factors that also have influence on the wind speed.

For this work, wind speed at 10 m above ground level is estimated based on available wind speed data at 50 m and surface roughness information, derived from Camargo et al. [12] . Figure 5 shows the resulting maps with the time- averaged wind profile for each season of the year, and Figure 6 displays this average for the year, obtained from the arithmetic mean of the seasons.

The average power per disk areas wept out by the blades of the wind turbine, P_W/A [W/m²], can be written as shown I Equation (17), where C_P [1] is the power coefficient for a wind turbine, η_C [1] is the mechanical conversion efficiency, ρ_air [kg/m³] is the air density at standard atmospheric conditions, k_e [1] is the Weibull shape parameter and V [m/s] is the average wind speed at 10 meters height.

In this work, in order to estimate the average daily wind energy availability derived from the average wind speed at 10meters height, the following values were adopted for these parameters: C_P equal to 100%, η_C equal to 100%, k_e equal to 2.5 and ρ_air equal to 1.225 kg/m³.

The value of 100% assumed for C_P and η_C are obviously not realistic (in fact, the maximum value of Cp is usually less than 50%), but this simplification was made for making the result independent of the wind turbine quality or type. The k_e factor of 2.5, uniform for the whole State, was deemed the best fit for the analyzed region.