Integrating variable renewables, particularly wind and solar, into the grid has been widely studied, with energy complementarity seen as a key solution for mitigating variability. This concept emerged in the late 1970s when [13] showed that a spatial distribution of wind generators could reduce production variability in California. Using Pearson’s correlation coefficient, Kahn established a foundational metric for complementarity, which remains widely used despite some critiques regarding its adequacy [14] .

The concept of complementarity was later consolidated by other studies, such as [15] - [17] , which created specific indices to assess the spatial, temporal or spatiotemporal complementarities between wind speed and solar irradiation.

Research on the optimal mix of renewable energies often addresses diverse objectives. For example, [18] explored the ideal ratio of solar, wind, and hydropower to minimize supply loss; [19] focused on reducing fossil fuel use; [20] aimed to cut greenhouse gas emissions; and [21] applied economic metrics such as energy cost, ROI, and grid present value.

Achieving the maximum penetration of renewable energies has also become a study criterion, along with the stability of the energy supplied or minimizing the uncertainties of renewable energies, as evaluated by [22] , or [23] studied the possibility of 100% of the energy of a small system being supplied by a mix of solar, wind and hydroelectric energy. In agreement [24] concluded that it is possible to supply 65% of the electricity in the Northeast region of Brazil with wind energy alone. If more energy sources are considered, up to 100% of the demand can be supplied by renewable energies; in the same thematic axis [25] studied the behavior of wind and solar power plants with hydroelectric and chemical battery storage when fully supplying demands of several different sizes, they concluded that the hydraulic source could supply the market with the same reliability and lower cost. With similar objectives, [26] also carried out a multi-criteria optimization in a hybrid system (wind, solar and hydroelectric); the criteria were minimizing fluctuation in energy production, maximizing wind and solar energy and economic criteria. Similarly [27] simulated solar and wind power plants in hydroelectric plants on the Wujiang River in China, and using various plant management strategies such as storage, they obtained economic gains and gains in the stability of the energy produced. [28] , for example, it evaluated the possibility of converting 100% of Chile’s energy matrix into renewable energy using the complementarity between solar and wind energy, which have a complementarity of up to 80% according to Spearman’s coefficient.

Because of this potential, there is a lot of research in the area of complementarity that focuses mainly on calculating the correlation between two or more energy modalities, which can increase the system’s reliability. The most common techniques are graphical analysis [29] [30] , calculation of correlation coefficients [31] - [33] and preparation of correlation maps [15] [30] [34] . There is also research on complementarity with other objectives, such as minimizing energy variability [35] or needing storage [36] .

Once complementarity is estimated, it is possible to use this information to design an optimal system to operate under these conditions, such as, [37] who used neural networks to predict the operation of a hybrid plant or [35] that used complementarity data to minimize the variation in electricity produced. [38] used local complementarity in hydroelectric plants to design a hybrid system to minimize the variation in electricity produced. [25] [39] concluded that combining different sources of electricity production can make integration with the electricity grid easier since combined production leads to a relatively controllable and constant total power output. [40] [41] simulated hydroelectric, wind and solar power plants with batteries for extremely dry climates and achieved satisfactory results in terms of energy stability.

This latest study shows that another advantage of investing in energy sources other than hydroelectric is that in Brazil, for example, it is estimated that climate change could reduce hydroelectric potential by 312 - 430 GWh in the dry season [42] . So solar and wind energy can make up for this drop without ceasing to be clean energy. The same is observed for other regions, such as North Africa [43] .

Due to the size of Brazil, there is a solar complementarity between its regions, as [44] showed in their study that it is possible to generate total solar energy corresponding to irradiation of 5.2 kWh/m²/day, with a minimum variance (risk of deficit) of 0.158, using the Markowitz mean-variance model method, where variance is the measure of risk. There is also complementarity between the hydroelectric energy produced in the different regions, reaching a Pearson coefficient of −0.97 and −0.86 for the wind resources available in the different regions [45] . In addition, the Northeast region is even more important in this context, when [46] concluded that the large reservoirs of its plants would be very important in transforming the energy matrix entirely renewable.

The main objective of all these studies is to increase the share of renewable energies in the energy matrix. Table 1 summarizes the main studies specific to Brazil that have already been commented on. Few published articles and many knowledge gaps exist regarding complementarity metrics, time scales, renewable resource databases, etc. And analysis including three renewable sources. Furthermore, no work has evaluated the three energy modalities at the same time in such a large and relevant region in Brazil.

For all the above reasons, in this study, we propose to analyze the solar, wind and hydroelectric energy complementarity available in the São Francisco River basin, where the eight largest hydroelectric plants are located, all in the semi-arid region in the São Francisco River sub-middle. Complementarity was assessed by different metrics on different time scales, showing that the results can vary dramatically depending on this last criterion alone. The combination of all these resources in such a wide geographical area and with such massive energy potential as the São Francisco River basin has not yet been studied in Brazil and very little in most of the world; there are only three recent studies in China [27] [41] [48] and one in Brazil [49] . Both explored a large geographic region, but the first two were restricted to intra-daily time scales and the third to monthly scales.

Thus, it was developed according to the flowchart shown in Figure 1 . It began with the evaluation of the existing renewable energy databases for the region at different time scales. Subsequently, the correlations between these different sources were evaluated at different time scales and with several different metrics: Pearson, Spearman and Kendall coefficients.

Four databases were used to carry out this work: a database for the evaluation of hydroelectric power (banco Operador Nacional do Sistema El’etrico—ONS) and three meteorological databases (Instituto Nacional de Meteorologia-INMET, National Aeronautics and Space Administration—NASA e Copernicus Atmosphere Monitoring-CAMS). These last three were compared to select the most suitable one, taking the criterion of truth as the database measured on the earth’s surface. The data used from the INMET database corresponded to the meteorological stations geographically closest to each hydroelectric power plant ( Table 2 ).

The plants in the Paulo Afonso, Moxotó and Xingó complex are physically close to each other, all within a maximum radius of 40 km, so the same station was considered representative of all of them. The comparison for the validation of the NASA and CAMS banks was made by directly comparing incident solar irradiation and wind speed with data from the INMET terrestrial weather station.

Table 3 shows the main hydroelectric plants on the river, as well as their reservoir sizes and installed capacities.

The hydropower data used here was provided by the National Electricity System Operator (ONS) [10] ) which is the body responsible for coordinating and controlling the operation of electricity generation and transmission facilities in the National Interconnected System (SIN). This database is very robust, with daily operational data for practically all the generation projects in the SIN. From the start of its operation to the present day. The metric used to evaluate the hydroelectric resource is Affluent Natural Energy (ENA), the hydroelectric energy potentially available at a given hydroelectric plant. It is calculated based on various factors, such as precipitation, evaporation, storage conditions and the geographical characteristics of the river basin. In short, ENA is the energy produced by the plant considering its affluent natural flow and its generation capacity with its quota of 65% of the useful volume [10] .

To assess the available solar and wind energy, data on horizontal global solar irradiance and wind speed provided by the National Meteorological Institute (INMET), the National Aeronautics and Space Administration (NASA) and Copernicus Atmosphere Monitoring (CAMS) were used:

1) Database of the National Meteorological Institute (INMET)—Weather stations collect this data on the earth’s surface. Weather stations collect meteorological variables from minute to minute that are integrated and made available in hourly averages. A typical station is shown in Figure 2 and consists of the following sensors: thermometers and psychrometers to record air temperature and pressure; anemometers installed at 10 meters to record wind speed and direction; and a pyranometer to record incident global solar irradiation.

2) NASA-POWER database, whose solar irradiance has been modelled with satellite images. The data is available in various temporal resolutions (hour, day, month or year), and the spatial resolution of the data is 0.5˚ × 0.5˚ latitude and longitude or approximately 111.12 × 112.12 km at the equator. The wind speed is available for 10 or 50 meters.

3) The CAMS—Copernicus Atmosphere Monitoring Service database has the same characteristics as the previous one but differs in origin. The databases modelled with satellite images cover practically the entire earth’s surface and are long-lived, with initial availability before the 2000s and ending at present.

There are different ways of conceptualizing and quantifying complementarity between renewable energy sources, so other methods were used in this work to assess the various facets of complementarity. Complementarity on an hourly scale was evaluated only for solar and wind energy due to the need for hourly data for hydroelectric plants. The various metrics used are presented in the following sections.

Pearson’s coefficient assesses the correspondence between any two data series. It varies between +1 and −1, where +1 means that the quantities are completely similar. When it is equal to −1, the quantities are completely complementary, and when the coefficient is equal to zero, it implies the absence of a linear relationship between the series. In practice, this index considers the first derivative of the curves to which it is applied. It is one of the most widely used techniques for exploring the association between variables in numerous research fields, from the social sciences to the natural sciences. It is the literature’s most commonly used metric for assessing energy complementarity. Some of the advantages of this Correlation are:

The disadvantages of using this metric include:

Pearson’s Correlation is defined by Equation (1):

(1)

where:

Although Pearson’s Correlation is a powerful tool for analyzing linear relationships between variables, it is essential to consider its limitations and suitability for the specific analysis context. In situations where the data does not meet Pearson’s Correlation assumptions, other correlation measures, such as Spearman’s or Kendall’s Correlation, may be more appropriate for capturing the underlying relationships between variables.

Spearman’s Correlation, also known as rank index, is a non-parametric measure that assesses the monotonic relationship between variables, i.e. how much the variables involved tend to vary simultaneously but not at a constant rate. Instead of taking into account the exact values of the variables, it is based on the order or ranking of the data. This method is especially useful when the data does not follow a normal distribution or when there are outliers, making it more robust when the assumptions of Pearson’s coefficient are not met. The advantages of Spearman’s Correlation include:

The disadvantages of using this technique include:

Spearman’s correlation formula can be seen in Equation (2):

$r_{\delta }=\frac{\mathrm{cov}\left(r{g}_X,r{g}_Y\right)}{{\sigma }_{r{g}_X}{\sigma }_{r{g}_Y}}$ (2)

where:

In summary, Spearman’s Correlation offers a robust and versatile approach to assessing the relationship between variables, especially when the data does not meet the assumptions of Pearson’s coefficient. However, it is important to consider its limitations and suitability for the specific analysis context before opting for this method over Pearson’s coefficient.

Kendall’s Correlation, also known as Kendall’s tau, is a non-parametric measure that assesses the agreement of classifications between variables. Like Pearson’s Correlation, Kendall’s Correlation is better suited to capturing monotonic associations that may not necessarily be linear, so it has similar applications. Among the advantages of the Kendall correlation are:

The disadvantages of using this technique include:

Kendall’s coefficient is defined by considering $\left(x_1,y_1\right),\left(x_2,y_2\right),\cdots ,\left(x_n,y_n\right)$ a set of observations of variables X and Y, respectively, such that all the values of $\left(x_i\right)$ and $\left(y_i\right)$ are unique. Any pair of observations $\left(x_i,y_i\right)$ and $\left(x_j,y_j\right)$ is concordant if the classifications of both elements agree with each other, i.e. if $x_i>x_j$ and $y_i>y_j$ or $x_i<x_j$ and $y_i<y_j$, otherwise the pair is discordant. The equation for Kendall’s coefficient is given by Equation (3):

$\tau =\frac{\left(\text{quantidade}\text{de}\text{pares}\text{concordantes}\right)-\left(\text{quantidade}\text{de}\text{pares}\text{discordantes}\right)}{n\left(n-1\right)/2}$(3)

where n is the number of pairs.

In summary, Kendall’s Correlation also offers a valuable alternative to Pearson’s coefficient, especially when a rank order better captures the relationship between variables than a direct linear relationship.

Due to the similarity between the Spearman and Kendall correlations, there is a need to explain the difference between them: Unlike Spearman’s Correlation, Kendall’s Correlation takes into account the magnitude of the variables, which is unimportant for the analysis of gross resources because the variables are in different units. In addition, Kendall’s Correlation is more suitable for small samples. Therefore, all three relationships were used in this study.

The similarity between the databases was checked using the Pearson correlation coefficient. The NASA and CAMS solar irradiance and wind speed databases are practically identical, as the Pearson correlation coefficient between them is always greater than +0.98% in all 4 locations presented, as seen in Table 4 and Table 5 .

Regarding the stations closest to the hydroelectric power stations, such as Três Marias and Sobradinho, the satellite banks have greater similarities than the more distant stations. The similarity between the satellite databases for all the quantities evaluated was always very close to +1, which means that these data are strongly similar. When compared with INMET’s terrestrial data, NASA’s data has always a slight advantage, achieving faintly higher rates than the CAMS data. Table 4 shows that similarity in solar irradiation is very strong in all locations, even where the weather stations are further away from the ground station. Regarding wind speed, Table 5 shows that the similarity is slightly lower than solar irradiation. Still, it remains very strong in all locations, with the exception of the Paulo Afonso Complex, where the Correlation reached the lowest value of +0.61, which can still be considered a good similarity. The great geographical distance between the hydroelectric plant and the INMET station explains this.

As seen above, the satellite databases were validated with the terrestrial database. So, for the rest of the work, the data used was from the NASA database, as it has a slight advantage over CAMS. The basic idea behind the validation was to use NASA’s solar irradiance and wind speed data, which would make it possible to use it to generalize throughout the semi-arid region of Northeast Brazil.

The complementarities of renewable energy resources were analyzed for co-located plants and between spatially distributed plants using the Pearson, Spearman, and Kendall index, calculated between the years 2000 and 2023 and called global for clarity and conciseness. Figures 3-5 show the global Pearson, Spearman, and Kendall correlations, all on a daily scale.

Figures 3-5 show that all the correlations behave similarly, and there is no drastic difference between them. The correlation values are rounded to one decimal place for better visualization in the graphs, so the biggest difference is 0.2. Therefore, in this section, the most detailed analysis was based only on Spearman’s Correlation, Figure 4 , which, as you know, is the most robust of the three: it tolerates “outliers”, does not require the assumption of linearity and does not require the distribution to be normal. It should be remembered here that the distribution of wind speed is of the Weibull type.

In principle, by analyzing the energy resources for each location separately (co-located analysis), it was possible to see the following relationships:

[30] calculated a Pearson index of up to −0.98 (clearly overestimated) between wind and hydroelectric energy in regions of the São Francisco River, which contrasts sharply with the results obtained here, which are a maximum of −0.7. The discrepancy can be explained by the much older database used by Cantão, which dates from 1961 to 2013, and was created in old INMET stations that could only take three measurements a day. Another notable difference was the use of monthly hydroelectric flow measurements.

Other authors, such as [24] or [5] , have not calculated these indexes and also captured a relationship of energy complementarity involving solar, wind, and hydroelectric energy in some locations along the São Francisco River. The overall Correlation corroborates this, showing some degree of complementarity between solar and wind power, as in Paulo Afonso, for example. It is also worth pointing out that some complementarity relationships will be more easily exposed when analyzed using other time metrics.

About the analysis of spatially distributed renewable energy resources (intra-plants), the following conclusions were reached:

There is also complementarity between hydroelectric power in Sobradinho and wind power in Luiz Gonzaga, reaching −0.4. Wind power in Sobradinho is also complementary to hydroelectric power and solar power in Luiz Gonzaga, with correlation indexes ranging from −0.3 to −0.5. Hydroelectricity in the Paulo Afonso complex also has complementarity ranging from −0.4 to −0.5. From the above, it can be concluded that there is a significant complementarity of −0.5 between different renewable resources for spatially distributed hybrid systems.

Kendall’s Correlation was chosen for this analysis because of the advantages above over Pearson’s Correlation. The series analyzed for each month only had 24 pairs of points, which could cause problems when calculating the Spearman correlation due to the small sample size. It should also be noted that the analyses carried out here refer only to co-located plants. All Figures 6-9 were made in the same way. In them, the monthly averages of each month of the 24 years were calculated for the energy resources in all modalities and grouped in a list with the averages of the 24 January, 24 February and so on. Then, the Kendall correlation index was calculated for each month of the list for all possible combinations of the resources taken two by two.

The monthly Kendall correlation for Três Marias is shown in Figure 11 . Where hydroelectric and solar energies, Figure 6(a) , have strong complementarity throughout the spring and summer (Sept-Dec and Dec-Mar), reaching a maximum of approximately −0.6. Between hydroelectric and wind energies, Figure 6(b) , for half the year, these energy modalities are complementary, although the index is low or average; only October is complementarity strong. Between solar and wind energy, Figure 6(c) , there is complementarity for six months, but it is weak, reaching −0.2 in the best of the cases.

Figure 7 shows the monthly Kendall correlation for Sobradinho. The hydroelectric and solar energies, Figure 7(a) , complement each other for the entire year, except February. Between hydroelectric and wind energy, Figure 7(b) , there is also complementarity for most of the year, although less intense. Between solar and wind energy, Figure 7(c) , complementarity only occurs between June and September and varies between −0.3 and −0.4.

The monthly Kendall correlation for Luiz Gonzaga is shown in Figure 8 ; similarly to Sobradinho, hydroelectric and solar energy Figure 8(a) complement eachydroelectric and wind energy were complementary only in 5 months, in spring and summer. Between solar and wind energy, Figure 8(c) , there was complementarity for most of the year, especially in winter and early spring; the result is approximately −0.4.

The monthly Kendall correlation for the Paulo Afonso Complex is shown in Figure 9 . Due to their proximity, there is a great similarity between the behavior of the monthly Correlation of the Paulo Afonso Complex and Luiz Gonzaga.

It can be seen that on a monthly scale, there is complementarity for practically the whole year in the São Francisco River basin. The highlight is the Três Marias hydroelectric plant, where the Correlation reaches −0.6 with an average of −0.5 in the hottest months of the year. The other locations also have good complementarity indexes, with an average of around −0.3.

It should be noted that analyses on this time scale are unprecedented in Brazil and rare worldwide.

The average monthly energy potential normalized by the average for each renewable energy source is shown in Figure 10 . This graph was made from the monthly average of all resource modalities. For better visualization, each curve relative to each energy modality was divided by its average; thus, the physical unit of the resource was suppressed, making it possible to evaluate the shape of the curve.

The similarity between the annual profiles for all 24 years is remarkable. The peak in hydroelectric energy potential occurs between December and February, and the valley occurs between June and October. The greatest variability also occurs with the end, so the greater constancy of the other modalities has a beneficial effect on this time scale. In addition, the high magnitude of solar and wind energy in the low-energy months also produces a complementary effect.

The graphical analysis of this figure shows that complementary hydroelectricity is most important between April and October, when the ANS is low, which is the case in all locations, although less so in Três Marias.

The daily seasonal profiles for each location (plant co-located) are shown in Figures 11-14 . The behavior of the inlet hydraulic flow is not illustrated in the figures because the temporal resolution of the ONS database does not include hourly data. Solar irradiation is in Wh/m², and wind speed is in m/s at a height of 50 meters, both from the POWER-NASA database. To improve visualization, the units have been normalized with their maximum and minimum values occupying the same location on the ordinate axis of the graphs.

Therefore, the standard day was calculated for each quarter of the year, in order to verify the hourly behavior of solar and wind resources varying with the seasonality of the year.

The quarterly daily profile of the behavior of solar irradiation and wind speed at the Três Marias plant can be seen in Figure 11 . For the sake of complementarity, the further apart the peaks of the wind speed (blue) and solar irradiation (red) curves are, the better the complementarity. Throughout the year, the highest wind speeds occur at night, and the lowest occur close to solar noon, which benefits daily complementarity. In addition, the increase in wind speed at dusk reduces the effect of the “duck curve”.

The quarterly daily profile of solar irradiation and wind speed at the Sobradinho plant can be seen in Figure 12 .

In this case, throughout the year, the resources are very similar; the peak wind speed always occurs just before sunrise and in the first and last quarters, the wind remains strong, which further benefits complementarity daily. When it gets dark, the problem of the “duck curve” is also mitigated by the rising wind speed.

Figure 13 shows the quarterly daily profile of solar irradiation and wind speed at the Luiz Gonzaga plant.

In Luiz Gonzaga, the peak wind speed also occurs with a lag about solar noon, although from April to September, wind speeds are also high close to solar noon. Even so, throughout the year, the rapid increase in wind speed as the sun goes down greatly mitigates the effect of the “duck curve”, and among the locations studied, this is where this effect is most evident, with the maximum magnitude of wind speed occurring in the early hours of the night.

Figure 14 shows the quarterly daily profile of solar irradiation and wind speed behavior for the Paulo Afonso Complex.

Again, the Luiz Gonzaga site is strongly similar due to its geographical proximity, so all the considerations there also apply to the Paulo Afonso Complex.