Solar radiation estimation using a numerical model

In: Book Title Editor: Editor Name, pp. 1-26 ISBN 0000000000 c 2007 Nova Science Publishers, Inc. Chapter 1 SOLAR RADIATION ESTIMATION USING A NUMERICAL MODEL F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero∗ University Institute for Intelligent Systems and Numerical Applications in Engineering University of Las Palmas de Gran Canaria Abstract A solar radiation numerical model is presented. It is intented to be useful for different purposes like the evaluation of the suitability of possible locations for solar power stations. This model allows the user to evaluate the radiation values in any location easily, and estimate the solar power generation taking into account not only the radiation level, but also the terrain surface conditions considering the cast shadows. The solar radiation model is implemented taking into account the terrain surface using 2-D adaptive meshes of triangles, which are constructed using a refinement/derefinement procedure in accordance with the variations of terrain surface and albedo. The selected methodology defines the terrain characteristics with a minimum number of points so that computational cost is reduced for a given accuracy. The model can be used to find the optimal location for obtaining the maximum power generation. For this purpose, the effect of shadows is considered in each time step. Solar radiation is first computed for clear sky conditions, considering the different components of radiation: beam, diffuse and reflected radiation. The real sky radiation is computed daily starting from ∗E-mail address: gusta[email protected]c.es 2 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero the results of clear sky radiation, in terms of the clear sky index. Maps for clear sky index are obtained from a spatial interpolation of observational data which are available for each day at several points of the zone under consideration. Finally, the solar radiation maps of a month are calculated from the daily results. The model can be also applied in solar radiation forecasting. To do so, a forecasting meteorological model is required. The estimation of daily solar radiation provided by such model is used to adjust the clear sky results and obtain the real sky radiation. Some numerical experiments related to the generation of solar radiation maps in Gran Canaria Island (Canary Islands - Spain) are presented. Keywords: solar radiation, shadows, adaptive meshes, solar power. 1 Introduction Solar radiation affects all physical, chemical and biological processes on earth. Therefore, the knowledge about it is relevant in meteorology, forestry, agronomy, geography, medicine and, of course, in power generation. In this topic, the importance of the renewable energies in our world is greater each day. Factors like the oil shortage or the climate change related to the global warming, have caused a high increase in the development of the different renewable energy technologies. Between them, solar power has become very important due to the support of the authorities and the research in this field. Part of this research is the development of some solar radiation numerical models. These models need to take into account the interaction of the radiation with the terrestrial atmosphere and surface, this is [1, 2]: 1. The geometry of the Earth (declination, latitude, solar hour angle) 2. The terrain characteristics (elevation, albedo, surface inclination and orientation, shadows) 3. The atmospheric attenuation (scattering, absorption) caused by gases, particles and clouds Considering the three factors of atmospheric attenuation in a model, it will produce real sky radiation values. If we omit the cloud attenuation, clear sky (cloudless) radiation values will be obtained. Two main groups of spatial models for solar radiation can be found. On one hand there are those models based on the study of data obtained from satellite observations [3], and, on the other, those based on astrophysical, atmosphere physical and geometrical considerations [1, 2]. Starting from the works of Šúri and Hofierka [1, 2] regarding a GIS-based solar radiation model, the calculation of solar radiation on the terrain [4, 5] is studied. Determining the influence of the orography and the terrain characteristics in the radiation is very important to reach accurate results. In fact, the factors related to the terrain such as the elevation, the albedo, the surface inclination or the cast shadows, are essential to have a precise idea of how the radiation comes into contact with the surface. The shadowing effect over a surface has been studied by Niewienda et al. [6] who propose calculating the GSC (geometrical shading coefficient) this is, the proportion of shaded area of an arbitrarily oriented surface surrounded by shading elements, as a function of time and location. Zakšek et al. [7] propose a solar radiation model based on defining the incidence angle by computing Solar Radiation Estimation using a Numerical Model 3 normal-to-the-surface tangent plane and direction of the Sun. Since they use a regular grid in their computations, the computational cost of this approach is higher than others using an adaptive discretization. Other methods [8, 9] do not consider a solid surface and, thus, need a high density of sample points in order to obtain accurate results. In this chapter, a solution for the estimation of the solar radiation on the terrain with a low computational cost is presented. We will use an adaptive mesh of triangles [4, 5] to represent the terrain and its actual orography as a solid surface that really casts shadows and so it is not as sensitive as the former to the density of sample points. Mesh refinement/derefinement techniques which have been widely used in other scientific problems [10, 11, 12, 13] have been applied. For example, the implementation of the 4-T Rivara’s refinement algorithm [14] and a derefinement algorithm [15], developed by Ferragut et al. [11] are good choices to reach our terrain mesh. For solar collectors, the application of our model is straightforward. It can determinate the terrain-induced shading on collectors. To summarize, this adaptive model allows the refinement of the results of a GIS-based model, which would have a high computational cost, in accurate local area simulations. In addition, this model may be connected to a GIS tool as a local solver. 2 Modelling of geographical features As said above, the terrain characteristics, including albedo, shape or inclination, are very important for the knowledge about the hourly and daily radiation values. Related to this, we will have to take into account two main issues: the construction of an accurate terrain mesh, and the detection of shadows. 2.1 Terrain surface mesh The terrain surface mesh must be constructed taking into account both, orography and albedo. As Montero et al. [4] described, an adaptive procedure of mesh refinement/derefinement has been carried out using two different derefinement parameters. This is not the only applicable method since a Delaunay triangulation or other more complex methods can be applied to get specific features like ridges or valley bottoms. However, nested meshes are used because we need to transfer information along an evolution process (shadow and radiation) from mesh to mesh. The first step to make the mesh generation is the determination of nodes allocated on the terrain surface. Their distribution must be adapted to the orographic and albedo characteristics in order to minimize the number of required nodes for a given accuracy. Starting from a regular triangulation τ1of the rectangular region under study, a sequence of nested meshes Γ={τ1<τ2< ... < τm}will be builded. The triangulation is such that the level τj is obtained by a global refinement of the previous level τj−1with the 4-T Rivara’s algorithm [14], this is, each triangle of level τj−1is divided into four subtriangles inserting a new node in the middle of the edges and linking the node in the longer edge with the opposite vertex and with the remaining two new nodes. The number of levels mof the sequence is determined by the degree of discretization of the terrain, so we can ensure that this regular mesh is able to capture all the orographic and albedo information by an interpolation of the heights and albedo in the nodes of the mesh. 4 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero Once this is done, we have to define a new sequence Γ′={τ1<τ′ 2< ... < τ′ m′},m′≤m, byapplyingthederefinement algorithm [11, 15]. By means of twoderefinementparameters, εhand εa, we can determine the accuracy of the approximation to terrain surface and albedo. The absolute difference between the height obtained in any point of the mesh τ′ m′and the corresponding exact height will be lower than εh. 2.2 Detection of shadows The accurate estimation of the solar radiation on a terrain surface needs to take into account the cast shadows. This problem is, after all, a geometrical one: a triangle will be shadowed when, looking at the mesh from the sun, we can find another triangle that totally or partially covers the former. One way to face this issue is constructing a reference system x′,y′and z′, with z′in the direction of the beam radiation (see Figure 1). After this, the mesh needs to be projected on the plane x′y′. y (North) x (East) 'y 1 y 'z P ' P (Sun) 1' x x 0 q0 q 0 z0 A 0 h 0 z 0 q 1 z z Figure 1. Reference systems and Euler angles The sun position with respect to a horizontal surface is given by two coordinates, the solar altitude h0and the solar azimuth A0(see Figure 1), which are calculated as, Solar Radiation Estimation using a Numerical Model 5 sinh0=cosϕcosδcosT+sinϕsinδ(1) cosA′ 0=sinϕcosδcosT−cosϕsinδ sinh0(2) if sinT>0,A0=−A′ 0 if sinT≤0,A0=A′ 0 Tis the hour angle (rad) obtained from equation (5), ϕthe latitude in radians and δthe sun declination in radians obtained according to Gruter [16], sin δ=0.3978 sinj′−1.4+0.0355 sin(j′−0.0489)(3) with j′being the day angle represented in radians as follows, j′=2πj 365.25 (4) Here jis the day number which varies from 1 on January 1st to 365 on December 31st. The hour angle T(rad) is calculated from the local solar time texpressed in decimal hours on the 24 hour clock as T=π 12(t−12)(5) As said above, we need to make a coordinate transformation to get z′in the direction of the beam radiation. Thus, we need to know a vector that defines the solar beam direction for any time and position. In the literature we can find many ways to reach this. Blanco-Muriel et al. proposed the so called PSA algorithm [17], developed at the Plataforma Solar de Almería. Niewienda and Heidt stated a simple vector to define this direction [6]. vsol =  cos h0sin A0 cos h0cos A0 sin h0   For each triangle two angles are computed, the azimuth AN(angle between the horizontal normal projection and East), and γN(angle between the normal to the triangle and the horizontal plane). The solar incidence angle δexp is then computed for each triangle [18, 19]. Please note that these new references will only be used for the determination of this angle. sinδexp =cos ϕ′cos δcos(T−λ′) +sin ϕ′sin δ(6) where sin ϕ′=−cos ϕsin γNcos AN+sin ϕcos γN(7) tan λ′=−sin γNsin AN sin ϕsin γNcos AN+cos ϕcos γN(8) Once this is done, the intersection between triangles is checked. This analysis involves a high computational cost. To diminish this cost we have considered four warning points for each triangle as can be seen in Figure 2, left. This is not the only possibility since we can, if 6 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero 1 2 4 3 Figure 2. Warning points for shading analysis needed, add more precision to the shade detection simply increasing the number of warning points as can be seen in Figure 2 right, where now we have sixteen warning points instead of the former four points. We assign a different level of lighting or shade to each triangle of the mesh depending on the number of warning points that lie inside other triangles. Specifically, a triangle ∆ will have an associated lighting factor which, for the case of four warning points, will be Lf=4−i 4(9) with i=0,1,2,3,4 being the number of warning points inside other triangles that are in front of ∆. Clearly, Lfis a quantity between 0 and 1. This factor is used below in the estimation of beam and diffuse irradiances. 3 Solar radiation modelling Once the terrrain is discretized by means of adaptive meshes, it is time to calculate the radiation values on every triangle. This will be done using a solar radiation model based on the work of Šúri and Hofierka [1, 2]. The obtained values are computed for each time step, taking into account the shadows over each triangle of the surface [4], by using the method described above. The results that we would have right now, are the clear sky radiation values, this is, the solar radiation values considering clear sky conditions. However, since the weather characteristics are a tremendous influence on the true radiation values that reach the surface, it is essential to take them into account. This way we will obtain the real sky values. Due to its random nature, the approach to this matter needs the knowledge of some of the actual existing meteorological variables in a place. The simplest and most accurate way to face this issue is using observational radiation data available from different measurement stations. We can now analyze the real sky radiation values for a chosen day or month for the whole spatial desired zone, or we can decide to make a previous statiscal analysis of the observational data to get more accurate results about what real sky radiation we can expect in a place in a certain date and time of the year. The statiscal analysis of the available data will give a Typical Meteorological Year (TMY) [20, 21] which will be the starting point to convert the clear sky into real sky. Finally, the calculations flow would be: 1. Solar radiation calculation for all the mesh, assuming clear sky conditions Solar Radiation Estimation using a Numerical Model 7 2. TMY calculation for all the involved mesurement stations 3. Correction of the solar radiation values using the measured values to reach the real sky conditions Steps one and three are repeated for each time step and finally the total solar radiation value is obtained integrating all the instantaneous values. 3.1 Solar radiation equations for clear sky The global solar irradiance comprises three different components: beam,diffuse and reflected irradiances. The first one, though partially attenuated by the atmosphere, is not reflected or scattered and reaches the surface directly. The second one is the scattered irradiance that reaches the terrain surface and that goes in all directions and produces no shadows of the inserted opaque objects. The third one is the irradiance which is reflected from the surface onto an inclined receiver and depends on the ground albedo. The relative importance of these three kind of irradiances depends upon the sky conditions. In sunny days, the diffuse radiation would be no more than the 15% of the global radiation, while on overcast days its importance will be much greater. The reflected or albedo irradiance is the one with the lowest contribution to the global. 3.1.1 Beam radiation According to J.K. Page [22], we will take the solar constant outside the atmosphere at the mean solar distance, I0, as 1367 (W/m2). Due to the earth’s orbit eccentricity, a correction factor εis needed for the calculation of the extraterrestrial irradiance G0. G0=I0ε(10) where ε=1+0.03344cos(j′−0.048869), with j′being the day angle; see equation (4). The beam irradiance, normal to the solar beam, Gb0(W/m2) is attenuated by the cloudless atmosphere, and calculated as follows: Gb0c=G0exp{−0.8662TLKmδR(m)}(11) The term 0.8662TLK is the dimensionless Linke atmospheric turbidity factor corrected by F. Kasten [23]. Subindex cshows that we are calculating clear sky irradiances. The parameter min (11) is the relative optical air mass calculated using the formula [24], m=p/p0 sinhref 0+0.50572(href 0+6.07995)−1.6364 (12) wherehref 0isthesolaraltitudeindegrees corrected bytheatmosphericrefractioncomponent ∆href 0, and p/p0is a correction for a given elevation z. Taking into account what written above, the beam irradiance on a horizontal surface for clear sky conditions Gbc(0)becomes, Gbc(0) = Gb0cLfsinh0(13) 8 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero whereh0isthe solar altitude angle and Lf, the lightingfactor thatcorrectsthebeamirradiance as the surface is sunlit or shadowed. Then the beam irradiance on an inclined surface for clear sky conditions Gbc(β)is obtained as, Gbc(β) = Gb0cLfsinδexp (14) where βis the angle between the inclined surface and the horizontal, and δexp is the solar incidence angle measured between the sun beam direction and its projection on an inclined surface. Note that, for horizontal surfaces, δexp coincides with h0. 3.1.2 Diffuse radiation The estimation of the diffuse component in horizontal surfaces Gdc(0)(W/m2)is carried out using the equation, Gdc(0) = G0Tn(TLK)Fd(h0)(15) As can be seen, Gdc(0)is a function of the diffuse transmission Tnwhich, at the same time, depends on the Linke turbidity factor TLK. Also, we have the function Fdwhich depends on the solar altitude h0[25]. The transmission function Tn(TLK)will be, Tn(TLK) = −0.015843+0.030543TLK +0.0003797T2 LK (16) And, Fd(h0), Fd(h0) = A1+A2sinh0+A3sin2h0(17) where the values of A1,A2and A3are, A′ 1=0.26463−0.061581TLK +0.0031408T2 LK A1=0.0022 Tn(TLK)if A′ 1Tn(TLK)<0.0022 A1=A′ 1if A′ 1Tn(TLK)≥0.0022 (18) A2=2.04020+0.018945TLK −0.011161T2 LK A3=−1.3025+0.039231TLK +0.0085079T2 LK To obtain the clear sky diffuse irradiance on a inclined surface with an angle γN, Gdc(γN)(W/m2),both, sunlit and shadowed surfaces (see section 2.2), have to be considered following the equations proposed in [26]. For sunlit surfaces (Lf=1) the equations are, If h0≥0.1 radians Gdc(γN) = Gdc(0)F(γN)(1−Kb)+Kbsinδexp sinh0(19) If h0<0.1 radians Gdc(γN) = Gdc(0)[F(γN)(1−Kb) +(KbsinγNcosALN)/(0.1−0.008h0)] (20) Solar Radiation Estimation using a Numerical Model 9 where A∗ LN =A0−AN if −π≤A∗ LN ≤πthen ALN =A∗ LN if A∗ LN >πthen ALN =A∗ LN −2π if A∗ LN <−πthen ALN =A∗ LN +2π For shadowed surfaces (Lf<1). Gdc(γN) = Gdc(0)F(γN)(21) where F(γN)is a function defined for the diffuse sky irradiance that may be computed as, F(γN) = ri(γN)+NsinγN−γNcosγN−πsin2γN 2(22) being ri(γN)a fraction of the sky dome viewed by an inclined surface, ri(γN) = (1+cosγN)/2 (23) The value of Nfor shadowed surfaces is 0.25227 while, for sunlit surfaces under clear sky, it is defined as, N=0.00263−0.712Kb−0.6883K2 b(24) Kbis a proportion between beam irradiance and extraterrestrial solar irradiance on a horizontal surface, Kb=Gbc(0)/G0(0)(25) where G0(0) = G0sinh0(26) 3.1.3 Reflected radiation The last component to take into account is the ground reflected irradiance under clear sky (Gr(γN)). According to Muneer [27], this will be proportional to the global horizontal irradiance Gc(0), to the mean ground albedo ρgand a fraction of the ground viewed by an inclined surface rg(γN). Gr(γN) = ρgGc(0)rg(γN)(27) where rg(γN) = (1−cosγN)/2 (28) Gc(0) = Gbc(0)+Gdc(0)(29) 16 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero Figure 7. Coarse (left) and fine (right) triangular meshes adapted to topography and albedo HourlyClearSkyradiation calculationforalldays NumericalIntegration ClearSky Indexand Interpolation Irradiationfor RealSkyconditions Figure 8. Actions flow to obtain irradiation values these numerical data into a picture. For example, we can see the irradiation map of Gran Canaria Island for December (TMY), for clear sky conditions. In Figure 9, beam (left) and diffuse (right) radiation maps (J/m2) are presented for December TMY. In Figure 10 we can see the reflected and global radiation for the same month. If we take a look at the obtained results, we can see that, under clear sky conditions, beam, diffuse and reflected radiation values are about 82−87%, 13−18% and 0−0.5% of the mean global radiation respectively. Figure 11 shows the global irradiation map for the island under real sky conditions for December considering a typical meteorological year. The monthly daily average real sky global radiation, for the whole studied region of the example (Gran Canaria Island), varies from 10.6, MJ/m2per day in December, to 25.6, MJ/m2per day in June (see Figure 12). In Solar Radiation Estimation using a Numerical Model 17 Figure 9. Beam (left) and Diffuse (right) radiation maps (J/m2) for December TMY Figure 10. Reflected (left) and Global (right) radiation maps (J/m2) for December TMY this Figure we can see the annual evolution of the computed monthly average per day for both, clear sky and real sky global radiation. Taking a look at the differences between both curves we obtain Figure 13, where the percentage decrease from the computed radiation is presented. In this Figure we observe the radiation behaviour for our example, where the most clear days over the whole island are those from Spring, especially during the months of May and June. We can see the typical behaviour of the cloudiness produced by the Trade Winds over the island during Summer. The months of July and August show a separation from the trend that would be expected when we are talking about this season. That decrease in the radiation level over the whole island is caused by the above mentioned cloudiness which affects in Summer to the northern part of the island. Below (Figure 14) we can see the global real sky irradiation maps for the months of 18 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero Figure 11. Global Real Sky radiation map (J/m2) for December TMY 10 15 20 25 30 35 (MJ/m2) Monthly dialy average radiation Real sky Clear sky Trend (Real sky) Trend (Clear sky) 0 5 10 15 20 25 30 35 1 3 5 7 9 11 (MJ/m2) Months Monthly dialy average radiation Real sky Clear sky Trend (Real sky) Trend (Clear sky) Figure 12. Monthly average radiation per day June and September TMY and a southeast three-dimensional view of the same radiation map for the TMY month of March (Figure 15). Solar Radiation Estimation using a Numerical Model 19 5,00% 10,00% 15,00% 20,00% 25,00% 30,00% Real Sky to Clear Sky Decrement RS to CS Trend 0,00% 5,00% 10,00% 15,00% 20,00% 25,00% 30,00% 1 3 5 7 9 11 Real Sky to Clear Sky Decrement RS to CS Trend Figure 13. Percentage decrease in computed radiation Figure 14. Global Real Sky radiation map (J/m2) for June (left) and September (right) 5 Irradiance graphs The previous section showed the distribution maps for irradiation as described by ISO 9488, this is, the radiation energy density that comes into contact with the terrain. Irradiation maps are very useful to find the best places in a zone where to install a solar power station from the perspective of the energy maximization, or, for example, simply for climate analysis or for farming production objectives. However, solar radiation incidence on the terrain is not constant along the day. As everybody knows, radiation increases from zero at dawn, up to a maximum at noon and then decreases until the sunset, when it becomes zero again. Of course, what we have computed in previous sections was the integration of these radiation curves along a day. This irradiation per time unit is the irradiance, usually given in W/m2.The analysis of irradiance values and graphs is of utmost importance when it comes to studying the electric power generation through solar radiation conversion. An irradiance graph for a TMY particular day, allows us to know the electric power production per hour (or per another time step) what can be of big 20 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero Figure 15. 3D global Real Sky radiation map (J/m2) for March help when trying to make an optimal electric power dispatch in case of having a high solar energy penetration in the electric power system. This fact is increasingly more likely due to the growth of solar power installed, whether it be photovoltaic or solar thermal power. At this point, the process to obtain the irradiance is very simple. Actually, we already do have irradiance values, since we integrated them as showed in Figure 8. In Figure 16, the actions flow for the extraction of the irradiance values is showed. The calculation of the subsequent conversion to electric power is easy introducing the models of photovoltaic and/or solar thermal power conversion. With regard to the real sky irradiance, it will be calculated (Figure 16) using the clear sky index values obtained by means of equation 32 and considering a linear distribution along the day. This is done to avoid an extremely high computational cost. If more accuracy is needed, kccan be calculated for every time step as long as there are available data. In Figure 17, the calculated irradiance in Maspalomas2for clear and real sky are showed for January, 15th (left) and July, 15th (right). All values are calculated assuming a horizontal surface. The abscissa are the Coordinated Universal Time (UTC) for Gran Canaria Island, which is around one hour forward respect to the local solar time. As said above, irradiance graphs allow us to know the evolution of the electric power generated through a photovoltaic or solar thermal conversion. In these cases, the surface that receives the radiation is generally not horizontal. As the objective is to maximize the energy production, photovoltaic panels or solar thermal collectors will have a certain inclination respect the terrain in order 2South of Gran Canaria Island, next to C4 Station -see Fig. 4- Solar Radiation Estimation using a Numerical Model 21 HourlyClearSkyradiation calculationforalldays NumericalIntegration ClearSky Indexand Interpolation IrradiationandEnergyfor RealSkyconditions RealSky Irradiance SolarPV Model Solar Thermal Model ElectricPower Generation ClearSky Irradiance Figure 16. Actions flow to obtain values for irradiance and electric power generation Figure 17. Daily irradiance (W/m2) for Maspalomas. January (left). July (right) to get the maximum solar radiation. The optimal inclination depends on the latitude of the zone under study and, of course, on the month, day and time to be considered. Therefore, this angle is desirable to be changed along the year though it is very common to find solar facilities, especially in photovoltaics, with fixed inclination collectors to reduce costs. Due to the great importance of the solar energy conversion into electric power, it is suitable to use the model with surfaces inclined respect the horizontal. The way to achieve this goal is considering a photovoltaic panel or a solar thermal collector, inserted in the terrain mesh by means of two or more triangles with the desired inclination and then, calculate the irradiance over it. That is how figures 18 and 19 have been done. In them, we can see the daily irradiance for a south oriented collector in Maspalomas, July the 15th with different inclinations and with a time step of 30 minutes. Maximum irradiance (see Table 2 is obtained with a collector inclination of 10oat 13 UTC (12 solar local time). Please note 22 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero that maximum irradiance does not implies a maximum daily irradiation because the panel orientation (south in this case) affects the incidence of the sun rays at the first and last hours of the day what causes small differences between the irradiance integrals. Figure 18. Daily irradiance for Maspalomas, July 15th. Panel inclination: 0o(L), 5o(R) Figure 19. Irradiance for Maspalomas, July 15th. Panel inclination: 10o(L), 15o(R) Table 2. Maximum irradiance. Maspalomas, July the 15th, 13 UTC Collector inclination 0o5o10o15o Irradiance (W/m2) 790 796 798 795 This numerical model allows to evaluate the irradiance for different orientations and Solar Radiation Estimation using a Numerical Model 23 inclinations. As an example, the daily irradiance on a collector inclined 60oand oriented east or west, is presented in Figure 20. The left figure shows the east oriented collector, and the right one the west oriented. The studied point is in Las Palmas de Gran Canaria (north of the island) and the date is January, 15th. It is obvious that both curves are now asymmetrical. The east oriented panel receives higher irradiance values during the first hours in the morning, with a maximum at 10:30 UTC (time steps of 1/2 hour) of 566.3 W/m2. On the contrary, the west oriented one receives more irradiance during the second half of the day, with a maximum value of 567.8 W/m2at 16 UTC. Figure 20. Irradiance for Las Palmas de Gran Canaria, January 15th. Panel inclination: 60o. Orientation: East (L), West (R) In case of photovoltaic or solar thermal power exploitation, especially with high solar energy penetration in the electric power system, an optimal power dispatch is needed. The problem is that we really do not know the exact electric power that can be produced tomorrow or the day after tomorrow. Moreover, to adjust the electric generation to the particular Load Curve of a system, we need to know the solar power along the day, the actual daily solar power hour by hour, but that is simply impossible. Most that can be done is a prediction of the power generated hour by hour. This prediction would have to be based on a weather forecast that estimates some local weather variables which allow us to know if it is probable to have a cloudy or sunny day at any hour. Therefore, what we need is to adjust the clear sky index kchour by hour, to draw an irradiance graph that takes into account the presence of clouds in the hours that they are expected, this is, to calculate different kc depending on the hour. What is new in this approach respect to what was done above? The answer is easy: Our first approach was applying the clear sky index in a linear way along the day. Now, that index can have different values for each hour. The rest of the procedure would be the same as explained above. 24 F. Díaz, J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero 6 Conclusions A numerical model for estimating the solar radiation on a surface is presented. Needed requirements are the location, topography, albedo and observational data. Solar radiation on a surface is estimated taking into account the shadow distribution in each time step. For this purpose, the adaptivity of the triangulation related to the topography and albedo is essential. Adaptive meshes lead to a minimum computational cost, since the number of triangles to be used is optimum [4]. ATypical Meteorological Year (TMY) [21] is needed to convert the clear sky into real sky radiation values. To calculate these, we propose an interpolation method which is suitable when a considerable number of stations is available and they are well distributed in the zone under study. Another procedures, such as spline functions for interpolating the clear sky index on the surface (see e.g. [32]) can be applied. Solar power generation, photovoltaic or solar thermal, can be estimated from real sky values, taking into account the models of the different power station parts. Moreover, rectangular collectors can be included in the model as composed by two triangles in the same plane, with a given inclination and orientation. The model allows estimating the irradiance for any day (TMY) with the desired time step. This could be a big help in forestry, agronomy or electrical engineering. It can be an interesting tool regarding to the optimal power dispatch in case of having solar power generation. For this purpose we need to add meteorological information (for example from MM5, WRF, HIRLAM) to obtain a predictive solar radiation model. 7 Acknowledgments This work has been supported by the Dirección General de Investigación,Ministerio de Ciencia e Innovación, Spanish Government, grant contract CGL2008-06003-C03-01/CLI. References [1] M. Šúri, J. Hofierka, “A New GIS-based Solar Radiation Model and its application to photovoltaic assessments”, Transactions in GIS 8 (2), 175–170. 2004. [2] M. Šúri, J. Hofierka, “The solar radiation model for Open source GIS: implementation and applications”. In: Benciolini, B., Ciolli, M., Zatelli, P. (Eds.), Proceedings of the Open source GIS-GRASS users conference, Trento, Italy, 1–19, 2002. [3] E. Cogliani, P. Ricchiazzi, “Generation of operational maps of global solar irradiation on horizontal plan and of direct normal irradiation from Meteosat imagery by using SOLARMET”, Solar Energy 82 (6), 556–562, 2008. [4] G. Montero, J.M. Escobar, E. 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