Renewable Energy 220 (2024) 119620 Available online 11 November 2023 0960-1481/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by- nc/4.0/). A viability index for comparing the binominal return-risk of solar radiation E. Trincado a,*, J.M. Vindel b a Department of Applied Economics, Structure and History, Complutense University of Madrid, Campus de Somosaguas, Pozuelo de Alarcón s/n, 28223, Madrid, Spain b Ministry of Labour and Social Economy, Paseo de la Castellana 63, 28046, Madrid, Spain A R T I C L E I N F O Keywords: Viability index based on Gini DNI Availability of solar resource Variability of solar resource Risk improvement index A B S T R A C T An insufficient amount of solar resource collected and the variability of this resource can compromise the viability of solar plant projects. For this reason, feasibility studies are usually accompanied by seasonal radiation data, which provides evidence of changes throughout the year. However, there is not a single index to quantify at the same time both elements - the amount available and the variability of the resource - which would provide a measure of the viability in a certain place to set up a solar plant. In this work, a new viability index based on the Gini coefficient (VIG) is proposed and its value is estimated for a wide region of the planet. The VIG calculates the binomial return-risk represented by the average-variance relationship. It clearly improves the results for high latitudes provided by the Sharpe index, which describes the ratio between the difference of the average return and a risk-free return and the risk in general. Finally, the article considers the storage option, drawing the value of the risk improvement index (RI) from the temporal series of the cumulated direct-normal irradiance (DNI) and an ample spectrum of constant demands. RI shows the risk improvement associated to storage of solar energy. 1. Introduction The availability of solar radiation measurements has increased both in spatial data and in historical time series. However, it is yet limited and frequently there is no high-quality terrestrial database available in the long term on a user basis. Today, satellite-derived solar radiation is considered as a reliable product for solar resource analysis and also for providing time series of solar irradiance [1–3]. Over the last 40 years, there has been an evolution in the methodology of analysis of the components of solar radiation from satellite images. Scholarship has improved from the first contributions [4–6] to the present days [7–9]. On the other hand, there is an economic risk associated with the unavailability of the primary solar resource and other economic barriers for solar plants, such as excessive minimum investment or high- transaction costs involved [10]. Plant operation is affected by the tem- poral variability of irradiation [11], and thus, variability must be included in the assessment of plant performance. Normally, variability is only represented through the evolution of time series (for instance, in the amounts of solar resource or photovoltaic potential, [12,13]). The probability density function [14], the cumula- tive distribution [15], or other formulations from the standard deviation [16] are also used to quantify the variability. However, these methods are not time-dependent, so they do not quantify temporal variability. For instance, the standard deviation is no more than a measure of the de- viation from the mean value. The variability could be calculated with autocorrelation, as it has a temporal nature. Indeed, autocorrelation represents the degree of cor- relation of the temporal series with the delayed series at different times, that is to say, the relationship between the variable’s current value and its past values. In this sense, autocorrelation allows us to define patterns that characterise the variability of the series. However, it only represents the degree of correlation between the temporal series and the series moved. It does not measure changes in irradiance at different times and thus, it does not assess the need for storage capacity. Temporal variability is also studied by assigning temporal series into classes of a scheme, which is outlined using a variability index along with a clearness index (for instance, a simple classification to distinguish four types of irradiance days could be: clear, overcast, mixed and highly variable) [17]. As clouds determine this variability, the temporal vari- ability is assessed per cloudy conditions. In Ref. [18], temporal auto- correlation and the distribution into classes are combined, applying the autocorrelation on the time series of the clear-sky index per class. Nevertheless, as variability is confined in a frame (for instance, within some of the categories above mentioned), these methods seem to be restrictive. Standard deviations may also be used for calculation. However, they * Corresponding author. E-mail address: estrinaz@ccee.ucm.es (E. Trincado). Contents lists available at ScienceDirect Renewable Energy journal homepage: www.elsevier.com/locate/renene https://doi.org/10.1016/j.renene.2023.119620 Received 16 August 2023; Received in revised form 15 October 2023; Accepted 8 November 2023 mailto:estrinaz@ccee.ucm.es www.sciencedirect.com/science/journal/09601481 https://www.elsevier.com/locate/renene https://doi.org/10.1016/j.renene.2023.119620 https://doi.org/10.1016/j.renene.2023.119620 https://doi.org/10.1016/j.renene.2023.119620 http://crossmark.crossref.org/dialog/?doi=10.1016/j.renene.2023.119620&domain=pdf http://creativecommons.org/licenses/by-nc/4.0/ http://creativecommons.org/licenses/by-nc/4.0/ Renewable Energy 220 (2024) 119620 2 do not quantify the variability in the worst scenarios. For this reason, to assess the availability of the solar resource at a site, a variability study using the exceedance probabilities of these variations, which show the probability that a certain energy value will be exceeded, could be con- ducted [19]. Thus, the viability of a solar plant project considers the worst conditions of irradiance reaching the design site (the instances when a lower quantity of solar resource is available). To assess the viability of solar projects due to discontinuity in the production rate, there are some economic indices. The Inventory-sales ratio assesses the cost of uncertainty by comparing storage - inventory - with final price of the resource. This ratio alerts managers about scarcity of stock resources which put at risk the stability of operations [20]. There are other economic parameters that determine the feasibility of a solar project with the objective of self-sufficiency, such as the Self-consumption ratio (SCR) or the Self-Sufficiency Rate (SSR). In other cases, the consumer demand is assumed to be satisfied with a ratio of total energy demand met locally, or through the Simple Payback Time of System (SPBT) - how long a project takes to recover its upfront in- vestments. The Levelised Cost of Storage (LCOS), which assesses the total amount of energy discharged from storage with the addition of charging cost, and the Levelised Cost of Energy (LCOE) are also available [21]. This latter parameter includes all costs that take place during the project’s lifespan. However, it does not consider the risk of not storing energy, which is an actual opportunity cost and, so, the real price of energy availability. In any case, the two aspects of interest for assessing the resource availability in a solar installation (amount and variability of radiation) have not been combined in solar radiation studies in a unified manner (unique index). Other well-known statistical indices, as the seasonality index, the interquartile range, or the coefficient of variation, consider only the effect of variability. Assessing both availability of the resource and risk could be of great interest, given that there are some places with large amounts of resource, but where there is high variability; conversely, there are places where resource is scarce despite being quite stable throughout the year. For this reason, the aim of this work is to define a new index able to assess the degree of suitability of a location for placing a solar plant by considering the amount and variability of the solar resource. The present work is devoted to direct-normal irradiance (DNI), that is, it stands for a CSP technology, but this methodology can also be applied to global horizontal irradiance (GHI) for a PV technology. The work is structured as follows: first, in the Material and methods section, the DNI dataset used in the study is introduced, and the different methodologies applied in the work are presented. Next, the theory/ calculation section explains the way how the average amount of resource and associated risk are combined in financial studies by indifference curves and Sharpe index, as well as Gini index, which measures the inequality of a distribution. Then, a modification of the Gini index is proposed in order to obtain a measure of the degree of goodness in a certain place to set up a solar plant without storage, ac- cording to the amount and variability of solar energy. Finally, in the theory/calculation section, another index is considered, the Risk improvement index [22], to take storage into account. Then, the results obtained by applying the proposed methodologies are discussed. After conducting a global scale research, a more detailed study for 6 locations corresponding to regions with different availability of solar resource is included. In the last section, conclusions of the work are drawn. 2. Material and methods Mean daily DNI values for 2022, from 65◦ S to 65◦ N latitudes and from 65◦ W to 65◦ E longitudes, were drawn from Eumetsat’s Satellite Application Facilities on Climate Monitoring (CM-SAF) with a resolution of 0.5 × 0.5◦ (260x260 points) [23]. After comparing 71331 times with the measured data at 17 BSRN stations, this product gives a bias of 0.33 W/m2, a mean absolute bias of 31.1 W/m2, and a standard deviation of 43.3 W/m2 [24]. In order to assess the availability of radiation in the study area, in both terms of quantity and variability, various methodologies are used in this work. In the first place, the relationship between the average values of radiation and its variance are obtained, calculating these values from the daily DNI data estimated by the satellite throughout the year of study at each point of interest. The mean-variance representation provides us with a first approximation of the availability of solar re- sources according to the quantity and variability of the resource. Next, as a second criterion to assess the availability of the resource, the so- called Sharpe index has been calculated for each point. This index combines both effects, quantity and variability, by determining the ratio between the difference in the average return and in a risk-free return (the difference in the average with and without variability), and the variability itself (represented by the standard deviation). Thirdly, the paper presents a new criterion to assess the resource availability, which provides us with a much more precise and concrete estimate than the previous criteria. The new index, called VIG (viability index based on Gini) is obtained in a similar way to the traditional Gini index, used to assess the distribution of income. However, in order to calculate the variability and amount of radiation, the proportion of population is replaced with time and the proportion of accumulated income is replaced with accumulated solar radiation. In order to obtain an index that allows us to compare the viability in different places, an optimal radiation is taken as a reference. This optimal radiation responds to a constant distribution over time whose value coincides with the maximum radiation observed in the study territory. This constant dis- tribution translates into a line of accumulated radiation over time. Finally, the work also includes an analysis of the capacity for improve- ment that each point of the study territory presents, storing the fore- seeable surplus of radiation at that point. To do this, the so-called risk improvement index (RI), defined later in Equation (3), is used. As we will see, this index shows the risk improvement associated with storage as compared to absence of storage. 3. Theory/calculation 3.1. Average and variance In financial analysis, a risk-free return of assets is theoretically an inflation-free return, which is required by investors to compensate them for the postponement of consumption of their resources while they are tied up in the investment. However, when they are not absolutely risk averse, people will accept some risk with the hope of obtaining higher returns. The final choice will ultimately depend on the investor’s atti- tude towards risk, which can be described by his utility function. The utility function describes the degree of satisfaction for an individual associated with the possession of a good or the return of an investment. Depending on the risk aversion of every individual, the utility provided by a project will be greater or lesser. The construction of this function is very difficult for each decision-maker; however, hypothetically the ex- pected utility of a project is a function of the average and the variance [26–28]. The drawing of this utility functions makes us understand the basis for the possible selection of best places to install a solar project. According to the [29] model, a map of independent indifference curves, corresponding to different investors, between the expectation of return and the risk can be considered, as the one represented in Fig. 1. In financial analysis, the map of indifference curves include all combina- tions of return-risk that give the investor the same expected utility and, so, are indifferent for him. A higher risk must be compensated with more than proportional returns to keep a risk averse investor indifferent. However, I4 is preferred to I3 and this to I2 and this to I1, as the first gives more returns per equal levels of risk. For example, the return-risk choice A is preferable than the return-risk choice C (with an equal risk but lesser returns) or than the return-risk choice B (with equal returns but a higher risk). E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 3 Assuming that the investor: (1) is risk averse, (2) knows the expected return and risk of the project, and (3) is not able to obtain an expression for the utility functions between two projects with the same return and different σ2, the one with the lowest σ2 will be chosen. Then, the relative, not the absolute, risk of the project should be considered. Between two projects with the same σ2 and different expectations, the one with the highest expectation (propensity to return) will be chosen. If the project with the highest profitability presents a higher risk, σ2, a decision cannot be made without knowing more precisely the degree of risk aversion of the decision-maker [30]. Therefore, to make sound investment decisions, both variability and expected returns need to be taken into account: if we do not reach a certain threshold, the resource could be made unavailable, if this threshold is exceeded, we may store the surplus, making the future less risky. In this sense, for the selection of an investment, the expected utility of the project must consider both the expected value and the variance of the returns. Since an individual normally obtains greater utility the greater his return is, the slope of the utility function will be monotonically increasing with the return. In the case of radiation, this implies trying to ascertain the expec- tation of return and the risk of unavailability of the resource. Actually, this is the mean-variance binomial; its calculation has the advantage of not requiring knowledge of the entire probability distribution, but only of two moments, mean and variance. With these two physical phe- nomena, we may know how a decision maker can act depending on his attitude towards the risk. 3.2. The Sharpe index As said earlier, the investor will prefer an investment with a lower expected return if the associated risk is also lower. Then, the mean- variance criterion establishes comparisons through profitability-risk binomials, which allows less efficient projects to be discarded. A rational decision maker will maximise the expectation of profit adjusted to risk. Financial studies offer an index, the Sharpe index, including both average expectation and risk. In effect, to assess the risk-adjusted per- formance, William F. Sharpe proposed in 1966 a ratio, originally called reward to variability ratio [31,32]. This ratio compares the excess return of an investment over a period non attributable to risk with its risk. In its simplest form, the formula is RSh = ( Rp − Rf ) / σp (1) where RSh is the Sharpe ratio; the numerator is the difference over time between the returns, realised or expected, Rp, and a reference variable such as the risk-free return (Rf ). The denominator is a measure of volatility or risk (the standard deviation of the returns over the same period - σp). So this ratio normalises the return, comparing the historical or projected returns of an asset relative to a benchmark with the his- torical or expected variability of the returns. The Sharpe ratio provides a measure of the risk-adjusted perfor- mance not attributable to the market. In finances, it represents the premium risk of an investment versus a safe asset and can help explain whether excess returns are attributable to good investment decisions or simply luck and volatility. The ratio determines the degree that the excess returns are accompanied by excess volatility, which cancel out the advantages of returns. A higher Sharpe ratio is assumed to be better than a lower one, ceteris paribus. However, in the assessment of goodness, the Sharpe ratio has some drawbacks. A negative Sharpe ratio means that the risk- free rate is greater than the asset return, but then a greater risk in the denominator makes the ratio less negative, and the investment more eligible. Except of the ratio’s denominator, the standard deviation cal- culates volatility based on a normal distribution and understates tail risk, not capturing rare events. It also understates serial correlation, which tends to lower volatility. In negative ratios, very high risks make the ratio tend to zero. So, the Sharpe ratio does not show the excess return per unit risk when there are negative returns [33]. Finally, the Sharpe index can be manipulated by lengthening the return measure- ment intervals, which results in a lower estimate of volatility – daily returns are more volatile than monthly or annual returns. 3.3. The Gini index (G) The Italian statistician Corrado Gini devised in 1936 the Gini coef- ficient as a measure of inequality [34–36]. The result of the Gini index is a number between 0 and 1, where 0 corresponds to perfect equality (everyone has the same income) and 1 corresponds to perfect inequality (one person has all the income and the others none). The Gini coefficient can be estimated from the Lorenz curve. This curve is obtained by plotting the proportion of a population in the horizontal axis and the proportion of the cumulative income in the vertical axis. Lorenz curves join the points (0,0) and (1,1). If A is the area between the line of perfect equality and the Lorenz curve, and B is the area below the Lorenz curve (Fig. 2), then the Gini coefficient, G, is: G=A/(A+B) (2) The Gini coefficient describes the inequality of the distribution of income of a certain country. Besides, it may also be used to describe variability, as the equi-distribution line of the accumulated values of the figure (red line in Fig. 2) relates to a constant distribution in not accu- mulated terms. Then, the distance between the Lorenz curve and this line can be used as a measure of variability. However, international comparisons of Gini coefficients can be misleading, as an equal distri- bution of poverty is not the same as an equal distribution of wealth. As the Gini coefficient refers to a proportion of population related to the proportion of income this population earns (the inequality of income for each country), it does not consider the total income of the world – although we may assume that a greater income of the country makes greater variability of income possible, as more income is to be distrib- uted between the population. In order to be able to make comparisons in different sites, a reference income value for all those places must be established. The percentage of income for each site is calculated in relation to this reference value (as explained in the next section for ra- diation analysis). When there is no variability, that is to say, when in- come is the same for all population, cumulative distribution turns out to Fig. 1. Map of indifference curves of a risk averse investor. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 4 be a straight line (of a more or less steeper slope depending on the total income of the country). The greater the variability, the greater the convexity of the curve. 3.4. The viability index based on Gini (VIG) The modification of the Gini index, previously explained, can be used to measure the variability and the amount of solar radiation. To that end, the population is substituted for the time variable, and the accu- mulated income is substituted for the accumulated solar radiation. Thereby, the Lorenz curve (blue curve in Fig. 2) is replaced by the cu- mulative solar radiation. Radiation is ordered from days with lesser to days with greater radiation, the same as happens with the Gini index for the income population. On the other hand, the accumulated income in case of equi-distribution for the Gini index (red line in Fig. 2) is replaced by the accumulated radiation in case of constant radiation (the slope of the line depicted will depend on the value of this radiation). This con- stant value corresponds to a reference value, or optimum, for comparing the different sites of the area. In this sense, the maximum daily radiation observed in the area will be considered the optimum. After those sub- stitutions are done, the final points of the curves will not coincide (as against what happens in the Gini index –Fig. 2) except for an hypo- thetical point whose total radiation coincides with the optimum accu- mulated for all the period. On the one hand, the distance between both curves due to convexity provides us with a measure of the inequality of radiation received throughout the period analysed, which is a measure of temporal variability. On the other hand, the distance due to the dif- ference between both final points provides us with a measure of the amount of the incident radiation. Value A of Eq. (2) corresponds to the area between both curves. To obtain the index, that value is divided between (A + B), which represents the area of the reference curve. In brief, in our study, to estimate this new index, instead of the Lorenz curve, the cumulative DNI will be represented (blue curve in Fig. 3), along with the line corresponding to the optimal radiation (red line in Fig. 3). The population is replaced with time (measured in days), and the cumulative income is replaced with the cumulative radiation (DNI). Then, the proposed new Gini index would still follow the expression (2), but with a Lorenz curve that does not reach the value 1 as the proportion of accumulated DNI is estimated with regard to the total reference value, not with regard to the total radiation of the site, following the structure of Fig. 3. 3.5. The risk improvement index (RI) In the radiation data, increases above the average makes the storage of solar energy possible – solar energy has a direct relationship with radiation. Then, an increase in the expected return from the average is clearly different from a decrease in the expected return from the average. So, it is of great interest to consider the possibility of using this storing capacity when assessing the viability of a certain location for the installation of a solar plant. In particular, the Risk Improvement index (RI) is available for this purpose [37]. This index, as previously com- mented, shows the risk improvement associated to storage as compared to the absence of storage. The RI assesses the degree of improvement in viability with storage based on a “hypothetical” demand that ensures the perpetual energy availability. A probability of satisfying different demands all the days of the year –different thresholds-, is estimated. From these probabilities, the RI is defined as the difference between the probabilities reached by storage and those reached in absence of storage. Surely, the excess output of a day with respect to this threshold is stored and it may be used when there is a deficit. Then, a new temporal series can be obtained and the exceedance probabilities of the outputs, and of their variations, can be estimated using the available storage. The index is represented by the area between the demand curves, being the different probabilities a price to be paid for surpassing each threshold of the output. These areas being normalised provide an index that indicates the percentage of improvement that can be reached by storage, that is: RI = ∫d=dmax d=0 ( Pd S − Pd NS ) ∫d=dmax d=0 Pd S (3) where RI is the Risk Factor Improvement Index; Pd S is the probability of satisfying the demand every day of the year considering storage and a demand d (the demand varies between 0 and dmax) and Pd SN is this same probability without the storage condition. A structure like the one shown in Fig. 3 can be drawn to estimate the RI, although in this case not only the optimal radiation distribution must be considered, but all the constant radiation distributions from 0 to the optimal one. Indeed, for each of the energy demands contained in the interval [0, dmax], a graph like Fig. 4 can be drawn. As there is a direct Fig. 2. The Lorenz curve. Fig. 3. Proportion of accumulated DNI and of optimal accumulated DNI. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 5 relationship between energy and radiation, we will assess it in terms of radiation. Given the wide spectrum of constant demands [0, dmax], there are three possibilities for the cumulative DNI that, as proposed, replaces the Lorenz curve: a) all values are below the line that represents the accumulation of a certain amount of constant radiation or, in this case, of demand of radiation; b) all values are above the demand line; c) some values are above and the other below (case represented in Fig. 4). In the first case, there is no possibility of storage, since the demand is always greater than the radiation received. In the second case, storage is not necessary, since radiation is always higher than the required demand. In the third case, the surplus obtained for the days to the right of P (zone E in Fig. 4) can be stored. This storage turns into an increase in radiation on days with lower radiation (included in zone D of Fig. 4), and a decrease on days with higher radiation (included in zone E), so that the new curve of cumulative proportion of radiation will rise in zone D and fall in zone E, although always maintaining the same total area under it and reaching the same end point. Accordingly, Pd NS can be estimated considering the proportion of the total days in zone E, i.e., the area where the demand is reached without the need for storage. For its part, Pd S can be estimated considering, besides the total days of area E (days in which demand is already covered), days within zone D for which de- mand is reached by contributions from zone E. The proportion of these total days will be taken as an estimation of Pd S. The higher RI is, the more benefit taken from storage. RI is the premium risk of not storing. It is important to note that the risk of un- availability of the resource makes investors not fully rational and they will tend to undervalue renewables [38], underreacting or overreacting [39]. Variability and uncertainty make planning more difficult [40] with greater needs for storage [41]. Fig. 4. Proportion of accumulated DNI and of demand. Fig. 5. Panoramic view of total DNI. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 6 4. Results and discussion Fig. 5 shows the total amount of DNI received at the surface of the Earth within 2022. These results obtained are aligned with the expected results drawn from the analysis of the general atmospheric circulation [42–46]. Indeed, the general atmospheric circulation in each hemi- sphere is characterised by three cells in both hemispheres, as shown in Fig. 5. From the Equator to around latitude 30◦, the Hadley cell is located, covering the so-called Intertropical Convergence Zone (ITCZ). There the air near the ground converges towards the Equator. Then, warm air rises, producing dense clouds and precipitations and thus, there is an important solar attenuation. Between latitude 30◦ and 60◦, there is the Ferrel cell: the air in high altitudes converges at latitude 30◦, increasing pressure at the surface and producing a corresponding sub- sidence, which inhibits the generation of clouds. At this latitude, at the surface, the air diverges and when it reaches around the latitude 60◦, it rises producing abundant clouds and precipitation. Finally, above 60◦ latitude, there is a polar cell, in which, at the surface, the air coming from the Pole rises at 60◦ and then returns to the Pole. Due to the con- servation of the angular momentum, there is an increase in speed to- wards the Poles and then, low pressures at high latitudes are much more vigorous than at lower ones. Consequently, the polar front, which ap- pears as a result of cold polar air meeting warm tropical air at about latitude 60◦, goes with clouds and abundant precipitations, producing a higher attenuation of solar radiation than expected in the ITCZ. According to Fig. 5, radiation within the ITCZ (equatorial area of Africa, especially its western section, as well as part of the Near East) is quite low. This radiation increases as we move away from the equatorial zone, especially in the south of the African continent, as well as in the strip that joins Brazil and South Africa. High radiation is also observed in North Africa and southern Europe, although at lower intensity. For latitudes above 45◦, the DNI is drastically reduced, especially in the northern hemisphere. On the other hand, as mentioned, the seasonal behaviour of radiation is also of great interest. For this reason, in Fig. 6 the DNI received in the study area is represented for four particular dates of the year corre- sponding to the four seasons (the 21st of March, June, September and December). The figure shows that during the winter corresponding to each hemisphere (south hemisphere in Fig. 6b. and north hemisphere in Fig. 6d.), the solar attenuation increases with latitude, while in the opposite hemisphere the radiation received increases with latitude. Then, on June 21st the DNI increases with latitude in the northern hemisphere, and decreases with latitude in the southern hemisphere (Fig. 6b). The same observation can be done, but interchanging the hemispheres, on December 21st (Fig. 6d). For these two months the mid- latitude radiation can move from values around 200 W/m2 to values close to 500 W/m2; at higher latitudes, the solar intensity can go from near 0 W/m2 to values near 600 W/m2. On the contrary, in spring and in autumn the radiation remains fairly constant at all latitudes (Fig. 6a and 6c). Then, Fig. 7 displays the average DNI (W/m2) versus the DNI vari- ance all over the study territory in the year 2022. Besides, in the figure some hypothetical indifferent curves have been included. Actually, there are as much curves as there are investors, as each curve reflects the behaviour of a particular investor. The curves depicted show behaviours which seem to be frequent. Fig. 6. DNI; a) March 21; b) June 21; c) September 21; d) December 21. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 7 Fig. 7. Average-variance relationship for risk-aversion pattern of indifference curves. Fig. 8. DNI variances. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 8 As we see, high averages of radiation are linked to a lesser variance (lesser uncertainty) and high variances (bigger uncertainty) are linked to more stable averages. Indeed, Fig. 7 shows a large cloud of dots in the area of low variances (approx. less than 1500 W2/m4),with averages that go from approx. 270 W/m2 to more than 390 W/m2. But in case of high variances (approx. bigger than 30000 W2/m4) averages are quite stable, with values near 330 W/m2. Seasonal variability associated to latitude, shown in Fig. 6, affects variance, what is evident for high latitudes, as shown in Fig. 8 (which represents the variances of the DNI series in each site of the territory). There, values of DNI are quite similar along the year in low latitudes. Places with high averages and low variance are the best locations for placing solar plants; however, as low variances (approx. lower than 1500 W2/m4 as above indicated) are more eligible for the investor, for low variances we have all the average spectrum. Then, the combination of averages and variances does not allow to specify the best points for installing solar plants as there are places with high average and low variance and with low average and high variance that are not comparable. In fact, in the figure several indifference curves for a PV investor have been overlapped, each of which draw points where binominals return-risks of solar radiation would be similar. Then, several options are available and equally preferable, and the final de- cision depends on the arbitrage between consumers. Besides, when there is distance from different uses of energy and transport is costly, the opportunity costs and the different possible uses in the installation areas could be assessed [47,48] and a von Thünen framework could be used. In particular, von Thünen assumes that the possible uses of land may be described by concentric rings of activity depending on the transport cost to the market and on the yield of the land that allows the tenant to pay a rent [49]. A selection from the graph average-variance based on the indiffer- ence curves entails high levels of subjectivity, so we can use the Sharpe index previously mentioned. With our daily radiation data (365 data for each point), the Sharpe index is presented in Fig. 9. In financial studies, one drawback from the Sharpe index is that, in negative ratios, very high risks make the ratio tend to zero. This is the case in Fig. 9 with high latitudes, where the variance is very high and the index seems to tend to zero. Then, the Sharpe index does not discretise the values of different zones from middle latitudes upwards. The stan- dard deviation in the formula of the Sharpe ratio considers abnormally low returns in the same way as abnormally high returns. Nevertheless, it is alleged, these two events must not be considered equally desirable. Thus, in financial studies to avoid this drawback a variation of the Sharpe ratio has been proposed, the Sortino ratio [50]. This index ig- nores the above-average returns to focus solely to downside deviation, including only the variance of negative returns in the denominator (Downside Risk). So, it is formalised as RS = ( Rp − Rf ) / σd (4) where RS is the Sortino ratio; the numerator is the same as the numerator of the Sharpe index, but σd is the variance of negative returns. However, after checking the result of the radiation map in the case of the Sortino index, we see that it is very similar to the Sharpe one, not obtaining any further improvement. Indeed, as in the Sharpe index, in very high lati- tudes, the Sortino index does not capture the high seasonality presented in Fig. 6 (and shown in Fig. 8 with variance data). Therefore, given the lack of concretion of the average-variance cri- terion and the evident lack of accuracy shown by the Sharpe index and the Sortino ratio for high latitudes, we define the VIG, which can combine both effects, quantity and variability of radiation, allowing us to measure the degree of suitability of a place to instal solar plants. The value of the index in each of the grid points of the study area is esti- mated, obtaining the result shown in Fig. 10. Fig. 9. The Sharpe index. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 9 As previously explained, VIG can be used to measure the variability and the amount of solar radiation. Indeed, the difference between the final points of the reference curve and the DNI curve provides us with a measure of the amount of the incident radiation, and the variability is shown by the convexity of the accumulated DNI curve. In this sense, this index displays better qualities than the Sharpe index, which does not reflect variability adequately. On the one hand, the total amount of ra- diation received (Fig. 5) is evinced both in the new index (Fig. 10) and in Sharpe index (Fig. 9), as more radiation goes along lower VIG index values in much of the territory. However, Fig. 10 reflects the same seasonal variability as Fig. 6, as against Fig. 9 of the Sharpe index that only presents similitudes with the radiation represented in Fig. 5 and does not illustrate the high variability existing in high latitudes. In Fig. 6, low latitudes present a low seasonal variability, as against what happens in high latitudes. This behaviour is also shown in Fig. 10, in which low values of the index (associated to a high radiation or to a low variability, or to both effects) prevail in low latitudes, while high values (associated to places that seem not to be much suitable for the installation of solar Fig. 10. The Viability index based on Gini. Fig. 11. Temporal evolution of DNI in different locations. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 10 plants) are in high and mid-latitudes. A low VIG index is good, as it goes along with high DNI and low variability. When DNI and variability play in opposite direction for the index (high DNI and high variability or low DNI and low variability), it depends on the corresponding levels of the values if the index is high or low. For instance, in southern Europe, the value of the index is low (0.25 in the case of Spain) but variability is middle and DNI is high. In other regions, such as northern Europe, the index presents very high values, as in those places a very high variability and a low total DNI affect the index in the same direction. The region corresponding to the ITCZ, in which the total DNI and the seasonal variability are low, clearly presents low values of the index (close to 0.25 in a good part of the region). This proves that the index is also influenced by variability as low DNI tends to increase the index, but here the index value is low. After having performed an analysis in the ample study region considered, a more detailed study for 6 locations with different radiation features is presented. The locations selected are shown in Fig. 10 as black dots, and temporal evolution of DNI along 2022 is presented in Fig. 11. According to Fig. 11, the four points with greatest seasonal vari- ability are the ones with greatest latitude (60.25◦ in the Northern and Southern hemispheres, where this variability is especially pronounced, and then 40.25◦ N and 30.25◦ S, although these two points present a very different variability from that of the first two). Finally, the two points closest to the Equator (15.25◦ N and 2.75◦ S) are those with more stable behaviour throughout the year, especially the first, given that in the second, located in the equatorial zone, the radiation is attenuated from May to September. As for the total radiation received, whose value is shown in Table 1 with the indices corresponding to each point, they do not depend so much on latitude as in the case of variability. In fact, the three largest quantities, which range between 134550 W/m2 and 122800 W/m2, are received at three points with very different latitudes (30.25◦S, 40.25◦N and 60.25◦S), and the two smallest quantities, which are very similar, correspond to 60.25◦N and 2.75◦S. Similar to Fig. 3, Fig. 12 represents the proportion of accumulated DNI for these six points. From this figure, VIGs are obtained comparing those curves to that of the optimal accumulated DNI, and the convexity of the curves evinces the variability observed in Fig. 11. Indeed, from looking Fig. 12, curves corresponding to the points (15.25◦N, 10.25◦W) and (2.75◦S, 16.25◦E) present the least convexity (the least variability). Next are those corresponding to the points (30.25◦S, 25.25◦E) and (40.25◦N, 4.25◦W) and, finally, the most convex and, therefore, most variable are those corresponding to the points (60.25 ◦N, 15.25◦W) and (60.25◦S, 15.25◦E). On the other hand, the final point reached by the curves is indicative of the total amount of radiation, whose order cor- responds to that shown in Table 1: (30.25◦S, 25.25◦E), (40.25◦N, 4.25◦W), (60.25◦S, 15.25◦E), (15.25◦N, 10.25◦W), and, finally, practi- cally equal (as also seen in Fig. 12) the points (2.75◦S, 16.25◦E) and (60.25◦N, 15.25◦W). According to the VIG estimated for each point, the best location from the point of view of resource availability is the one corresponding to the coordinates (30.25◦S, 25.25◦E) located in southern Africa. This place receives a large amount of radiation throughout the year and its vari- ability is not high. The next best location has a VIG not far from the one corresponding to the previous location − 0.22 versus 0.15. It is in the Iberian Peninsula (40.25◦N, 4.25◦W) and its characteristics in terms of quantity and variability of radiation are not far from those of the first point. Next with a very similar VIG to the previous one (0.26 versus 0.22) is the point (15.25◦N, 10.25◦W) located in tropical Africa. This location is the one with the least variability, although the total amount of radiation is somewhat less than that received in the Iberian Peninsula. The following location presents a VIG (0.37) somewhat further the previous one. This point is in the equatorial zone of Africa (2.75◦S, 16.25◦E), and is characterised by a reduced annual variability, although not as reduced as in the case of the previous tropical location, and due to a very low amount of annual radiation received from a significant cloud attenuation in the area. On the contrary, the following location, (60.25◦S, 15.25◦E), presents a high annual radiation (like that of the Iberian Peninsula), but enormous variability. According to these char- acteristics, VIG is similar to that of the equatorial zone (0.40), which makes it an unsuitable place for the installation of the solar plant. Finally, the location in which this installation is least viable (VIG of 0.53) is in the coordinates (60.25◦N, 15.25◦W), characterised by very low annual radiation and high variability. So far, we have tried to compare the feasibility of different sites in terms of solar resource availability when installing CSP. However, the possibility of making use of radiation surpluses that can be obtained at times of higher radiation to alleviate deficit situations has not yet been considered. For this reason, an analysis of the capacity of each point in the study territory is finally carried out to improve its viability through the storage of surplus energy (given the direct relationship between energy and radiation, it is assessed in terms of radiation). To do this, the so-called risk improvement index (RI), defined in Equation (3), is used (Fig. 13). It is to be reminded that the higher the RI, the greater the capacity for improvement attributable to storage. As we see, at higher latitudes this capacity of improvement is bigger than at lower ones, although as happens with variance (Fig. 8), high and mid-latitudes make evident the latitudinal differences. This result should be expected, since at high latitudes the demand can be widely covered in certain periods (in Fig. 6b. corresponding to June, it can be covered in the North hemi- sphere and in Fig. 6d. corresponding to December, it can be covered in the South hemisphere) as surpluses can be used in periods where the availability of radiation is much lower. Therefore, Fig. 13 presents important similarities with Fig. 10, although the viability index obtained from Gini gives greater detail of specification possible within the study territory, which indicates that the capacity for improvement provided by Table 1 Total DNI, VIG and RI in different locations. Latitude Longitude Total DNI VIG RI 60.25◦N 15.25◦W 98353 0,53 0,33 40.25◦N 4.25◦W 125503 0,22 0,10 15.25◦N 10.25◦W 111322 0,26 0,04 2.75◦S 16.25◦E 99337 0,37 0,09 30.25◦S 25.25◦E 134550 0,15 0,08 60.25◦S 15.25◦E 122800 0,40 0,32 Fig. 12. Proportion of cumulative DNI in different locations. E. Trincado and J.M. Vindel Renewable Energy 220 (2024) 119620 11 storage is quite similar in most of the territory. Finally, Table 1 provides with the value of RI corresponding to the 6 locations of the complementary analysis. In Fig. 13 the RI is shown to be very dependent on latitude. Indeed, the four locations with less improvement capacity due to storage are the ones with lower latitude (15.25◦N, 30.25◦S, 2.75◦S and 40.25◦N), all of them with very similar values of RI. However, in the two higher latitudes considered (60.25◦N and 60.25◦S), the improvement capacity increases substantially due to the high capacity to store during part of the year. 5. Conclusions In this article, a new index based on the well-known Gini index was defined. The study was centred in assessing the availability of DNI in a study area and thus, for appraising the suitability of the territory for installing a solar facility with CSP technology. The index considers both the amount of solar radiation received and its variability. Other well- known statistical indices, as the seasonality index, the interquartile range, or the coefficient of variation, consider only the effect of vari- ability. On the contrary, the exceedance probabilities provide estima- tions of the total amount of available power generated, but they do not deal with the variability of the resource. The viability index based on Gini, VIG is obtained from the com- parison of curve of cumulative DNI and a reference curve considered as an optimum. The distance between both curves due to convexity pro- vides us with a measure of temporal variability. The distance due to the difference between both final points provides us with a measure of the amount of the incident radiation. Then, the viability index based on Gini is a more complete tool than the simple Gini index, since it allowed us to characterise with a single value the suitability of a place for the establishment of a solar plant considering the solar resource. In addition, this index has the advantage of being perfectly limited between 0 and 1, making the comparison of suitability between different locations easier. A low VIG index is good, as it goes along with high DNI and low variability. When DNI and vari- ability play in opposite direction for the index, it depends on the cor- responding levels of the values if the index is high or low. On the other hand, the case of storage during times of higher radi- ation has been considered in the work. This implies a possibility of improving the viability of a solar installation in terms of the availability of solar resources. To assess this capacity of improvement, the Risk Improvement Index (RI) has been used, whose value has been estimated by comparing the corresponding temporal series of accumulated DNI with the entire possible spectrum of demands expressed in terms of ra- diation. Although both indices, VIG and RI, present important similar- ities in terms of their spatial distribution, the VIG can be specified in greater detail in the study territory, which implies that the capacity for improvement provided by storage is quite similar in most of this terri- tory. This new index, VIG, is applicable to other disciplines, notably eco- nomics, from which the Gini index comes. We leave this possible extended application to future research. CRediT authorship contribution statement E. Trincado: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Validation, Roles, Writing – original draft. J. M. Vindel: Conceptualization, Data curation, Formal analysis, Investi- gation, Methodology, Supervision, Validation, Roles, Writing – original draft. Declaration of competing interest The authors declare that they have no known competing financial Fig. 13. The Risk improvement index. E. Trincado and J.M. 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