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Solar Day


The "Solar Day" and "Solar Year" windows calculate energy characteristics of the same collector system of the latest active 3D View window. They display warnings if there are no 3D scenes (if the last 3D View window was closed), or if the latest active scene does not contain solar collectors.


The "Solar Day" window shows how the energy characteristics of your solar collectors (including the self- and full-shading) vary during a given day, and calculates their daily sums or daily average values.


All the calculations are implemented according to the climate model of the scene. To specify the climate model of the scene, read the next section and use the "Climate & Irradiance" window. Note also that the "Solar Day" window is an additional way to verify your climate model if you have the detailed hourly statistic for the average days of the months.


Create the climate model for the scene and transfer the climate data


If you did not specify the climate model of the scene, the default climate model is applied. Note that the default climate model uses the plain 50% probability of clear sky for all 12 months of the year. So to have a more realistic results, you need to adjust at least this setting according to the actual solar radiation statistic of your particular site.


The climatic information is a part of the scene record like any other scene parameters. The structure of this information is same as the structure of the input data of the "Climate" window. When the scene is firstly created, this structure is filled by the same default climate settings that appear in the firstly opened "Climate" window. You can change these default settings in two ways: either creating and adjusting the climatic model of your site in the "Climate" window and then transferring data from the "Climate" window into your scene, or retrieving the climatic model from another scene into the "Climate" window and then also transferring data from the "Climate" window into your scene. See details of your work with the climate model in the topic "Climate & Irradiance" of this chapter.


You can transfer the climatic parameters of the "Climate" window into the latest active 3D scene in two ways. If the "Climate" window is active, use the key "T" (export To the scene). If a 3D view is active, use the key "C" (import the Climate). Note that in both cases the parameter "Latitude" is not copied from the "Climate" window to the 3D scene. The climatic parameters are updated in the dynamic memory of the scene immediately after the transfer. Later, when you save the scene, the climatic information will be saved in the scene file too.


To retrieve the climatic parameters from the latest active 3D scene into the "Climate" window, use the key "F" (retrieve From the scene) when the "Climate" window is active. Note that in this case the parameter "Latitude" is copied from the 3D scene into the "Climate" window.


So you can use the "Climate" window as an intermediate data storage for the exchange of the climatic data between scenes. Use the key "L" in this case to lock the "Climate" window, and to prevent it from the occasional changing of the transferring data.


The "Solar Day" window


We assume that reading this text you see the "Solar Day" window on your screen, so we explain and comment just what you see now.


The name of the associated latest active scene is written in the title bar of the "Solar Day" window.  




1. Radiation, kW/m2

2. Shading, dimensionless units and %

3. Power, kW


To switch pages, use the keyboard keys "1", "2", "3", or the right mouse click, when the window is active.


The argument of functions of all pages is the same: time in hours from true solar noon (HFN -- Hours From Noon).

See "Set Geography..." for details about true solar time and local time.


To see the numerical values of the calculated function curves, use the "Solar Table" window.


There are no specific control elements or input data for the "Solar Day" window. It reacts on the changes provided in the latest active scene displayed in the latest active 3D View window. So to change the day in the "Solar Day" window, activate the associated 3D view, and use the control elements of the "3D View" toolbar or the keyboard keys as it is described in the sections "A typical session", "Use the toolbar 3D View", and "Use the keyboard modes" of the topic "Visualize Shadows".


Use keyboard


Most of the switches between different show and calculation modes of the "Solar Day" pages are represented by the keyboard keys. Only some of them are duplicated by the menu items. The symbolic description of the keyboard keys is shown in the "Solar Day" window after the list of named curves -- you can show / hide it with the key "K".


Show the description of the keyboard keys



The symbolic description is a very short hint. It contains the key upper letter and a word that denotes the sense of the switch, or the chosen mode.


Switch between the PV-system and individual PV-strings



These switches are common for all pages. The hint for all three switches occupies only one line. It shows symbols of three keys and the word "string".


Switch the type of the Relative Fresnel Transmittance



This switch is shown on Page 1. It determines which one of the curves "T_Tot", "T_Dir", or "T_Dif" to show.


Note, the shading and power calculations are provided always for the case "Total" and do not depend on this switch.


Switch the type of shading calculations



This switch is shown on Pages 1 and 2. It determines which one of two shading types "Full" or "Self" to calculate.


Switch the spline and sky modes



These switches are common for all pages.


Both the switches have same meaning as ones of the "Climate & Irradiance" window. However the "Climate" window does not have the "random clouds" mode. The switches of the "Solar Day" window work independently from the corresponding switches of the "Climate" window.


In the "constant month" mode (when "M" shows "flat month"), all the monthly input parameters of the climate model stay the same within the corresponding months. The "constant month" mode is useful when you need to interpret experimental data of a particular day.


In the "month spline" mode (when "M" shows "climate spline"), all the monthly input parameters of the climate model are parabolically splined. To see the parameter value of a particular day, transfer the climate model from the scene to the "Climate" window, open the corresponding input page of the "Climate" window (where the parameter is displayed graphically), and then open the "Solar Table" window.


The switch "V" between "100% clear sky", "mean cloudiness", and "random clouds" modes affects both the daily dynamics and the daily sums of solar radiation. The "random clouds" mode have an additional numerical switch with the "Y" key, which simulates 30 different random sequences of "sunny" and "cloudy" 10-minutes time intervals during a particular day for 30 different years. See the next section for details.


Use the "view" switches to change the appearance of text and curves


These switches are common for all pages of both  "Solar Day" and  "Solar Year" windows.



These switches can be useful, when you prepare illustrations for a report with the "Copy To Clipboard" options.

See also the "Increase Line's width" command of the "Edit" / "Copy To Clipboard" menu.


Basic formulas for calculations of solar radiation


To specify which a surface we are speaking about, we use the following suffixes:


"_N" -- for the direct solar radiation Dir_N measured on the "normal" surface,

"_H" -- for the components of solar radiation measured on the horizontal surface,

"_S" -- for the components of solar radiation measured on the arbitrary oriented surface S. 


The main assumption that we use in the calculations of the "Solar Day" and "Solar Year" windows is that the diffuse radiation on an arbitrary oriented surface S is approximately equal to the diffuse radiation on the horizontal surface. So the value of Dif_S is substituted by the value of Dif_H. See "Comments" in the topic "Bird Clear Sky Model".


So we express the total solar radiation on an arbitrary oriented surface S as follows


Dir_S = Dir_N * cos(i);

Tot_S = Dir_N * cos(i) + Dif_H;      (instead of Tot_S = Dir_N * cos(i) + Dif_S),


where "i" means the incidence angle (for a horizontal surface i = Z, and suffix "_S" becomes "_H").


In the "Climate & Irradiance" window, we introduced two variables:


Prob -- the probability of clear sky in % (or in dimensionless units from 0 to 1),

Dcs -- the correction coefficient for the diffuse solar radiation.


For the "100% clear sky" conditions, we apply the same formula as shown above:


Tot_S = Dir_N * cos(i) + Dif_H.


For the "mean cloudiness" conditions, we modify the formula as follows:


Tot_S = Prob * Dir_N * cos(i) + Dcs * Dif_H.


For the "random clouds" conditions, we modify the formula in two ways:


Tot_S = Dcs * Dif_H;      (within "cloudy" time intervals),

Tot_S = Dir_N * cos(i) + Dcs * Dif_H;      (within "sunny" time intervals),


and the parameter Prob determines the fraction of "sunny" time intervals within the entire light day. 




The case of "random clouds" explains the sense of the correction coefficient Dcs for the diffuse solar radiation in the case of "mean cloudiness" conditions.


The "random clouds" mode simulates the variable cloudiness within one particular day by a random sequence of "cloudy" and "sunny" 10-minutes time intervals. The total duration of all "sunny" intervals divided by the duration of the light day is approximately equal to the parameter Prob (the probability of clear sky) of the "mean cloudiness" mode. However, "sunny" does not mean that the sky is totally free of clouds, as well as "cloudy" does not mean that the sky is totally covered by clouds. Clouds are moving over the sky, and "sunny" means that we see the solar disc clearly between clouds, while "cloudy" means that the solar disc is temporarily obscured by a cloud.


Under such conditions, the diffuse radiation can decline from the Bird model calculations (increasing due to the additional reflection from rare clouds, or decreasing due to the relatively dark sky, when it is almost totally covered by clouds). But what is more important, we should apply the same diffuse radiation to both cases of "sunny" and "cloudy" intervals, because the average brightness or "darkness" of the sky stays approximately the same independently of the fact is the relatively small solid angle around the solar disc temporarily obscured by an occasional cloud or not.


Summarizing both the formulas for "sunny" and "cloudy" time intervals of the "random clouds" mode, and taking into account their statistical weights, we can write the following: 


Average_Tot_S = Prob * Tot_S_sunny + ( 1 - Prob ) * Tot_S_cloudy =

= Prob * ( Dir_N * cos(i) + Dcs * Dif_H ) + ( 1 - Prob ) * ( Dcs * Dif_H ) =

= Prob * Dir_N * cos(i) + Dcs * Dif_H;


that is just the same formula, which we use in the case of "mean cloudiness" conditions.


This approach is motivated also by the fact that the parameter Prob (the probability of clear sky) actually means the probability of the clear solar disc, because it is usually calculated as the ratio of the total monthly duration of available direct solar radiation to the total monthly duration of the light days. 


Note the curves' overdrawing


There is a common point of possible misunderstanding of the graphical diagrams of "solar" windows. Some users might be confused being unable to find a particular function curve on a diagram. It might occur when one curve is overdrawn partly or completely by another curve. For example, if the collector system consists of 2-axes sun-tracking collectors, for which permanently cos(i) = 1, then the curves of Dir_N and Dir_S are the same, and the curve of Dir_S (as one painted after Dir_N) completely covers the curve of Dir_N.


The order of painting and colors of curves correspond to the order and colors of their names in the list of curves that is shown in the second line of the legend for each page of the "Solar Day" and "Solar Year" windows. 


Fresnel reflection and the Relative Fresnel Transmittance (RFT)


To cover the theme of Fresnel reflection in PV-modules and build a practical calculation scheme for our software we introduce the notion of the Relative Fresnel Transmittance, shortly RFT, or the RFT coefficient.


See the general description of Fresnel reflection at 


The Fresnel transmittance (shortly FT) on the boundary between two media for the not-polarized solar light is the function of the incidence angle a_i and refractive indexes n_i, n_t of first and second media:


   FT = FT( a_i, n_i, n_t ).


The direction of the transmitted (refracted) ray is determined by the Snell's law:


   sin( a_t ) = ( n_i / n_t ) * sin( a_i ), where a_t is the angle of the transmitted ray measured from the normal inside second media.


We define the RFT coefficient as the ratio of the Fresnel transmittance at an arbitrary incidence angle a_i to its value at a_i = 0, when rays are perpendicular to the boundary:


   RFT = RFT( a_i, n_i, n_t ) = FT( a_i, n_i, n_t ) / FT( 0, n_i, n_t ).


One can say that the RFT coefficient is the normalized Fresnel transmittance. Let us consider how RFT works in a PV-module. 


See the general description of the construction and materials of PV-modules at 


Most of silicon PV-modules consist of three-four layers. The first layer is the transparent top surface -- usually low iron Glass. The second layer is the encapsulant -- usually EVA (ethyl vinyl acetate). The last layer is Silicon of the PV-cell itself that can be coated by an additional thin anti-reflection SiN film.


So we have up to five media with the following typical values of the refractive indexes:


   n_Air = 1; n_Glass = 1.5; n_EVA = 1.5; n_SiN = 2.3; n_Si = 3.5;


To illustrate the idea of the RFT coefficient, let us consider the three-media system "Air-Glass-Silicon" that is represented in the following table. First three columns of the table show angles i1, i2, i3 of the ray directions inside each media with respect to the normal vector of the boundary plane. First angle i1 is merely the incident angle i of the direct solar radiation on the upper surface of a PV-module.


Air Glass Si A-G G-S A-G-S A-G A-G-S

i1 i2 i3 FT12 FT23 FT123 RFT12 RFT123

0 0 0 0.960 0.840 0.806 1.000 1.000

15 9.9 4.24 0.960 0.840 0.806 1.000 1.000

30 19.5 8.21 0.958 0.840 0.805 0.998 0.998

45 28.1 11.7 0.950 0.839 0.797 0.989 0.988

60 35.3 14.3 0.911 0.837 0.762 0.949 0.945

75 40.1 16.0 0.747 0.834 0.623 0.778 0.773

90 41.8 16.6 0.000 0.833 0.000 0.000 0.000


We see that the ray direction become nearer to the normal in each next media, because the refractive index is increasing. Fourth and fifth columns show absolute values of the Fresnel transmittance on the Air-Glass and Glass-Silicon boundaries. The sixth column shows the Fresnel transmittance for the entire "Air-Glass-Silicon" system.


Two last columns show that the RFT coefficient of the relative (normalized) transmittance for the entire A-G-S system RFT123 practically repeats the behavior of the RFT curve RFT12 for the very first boundary between Air and Glass with the accuracy not worse than 1%. It occurs because the Fresnel transmittance FT23 on the Glass-Silicon boundary is practically constant for all allowed variations between 0 and 42 of the angle i2 inside Glass. One can say that the ray inside Glass is already so close to the normal that further refractions on next boundaries occur as if the ray is perpendicular to the boundary plane. All other effects like possible multiple reflections inside a media between its boundaries or a weak polarization of the transmitted light play minor roles.


The next important circumstance is that the ISO-test of the certification procedure of PV-modules simulates mainly the direct component of solar radiation (of 1 kW/m2) at the incidence angle i near to zero. Although the solid angle of the artificial light source from the PV-module viewpoint is much greater than the solid angle of the visible solar disc, it is also much less than the semi-spherical solid angle of the sky. So that we can assume that cos(i) within the power test is almost equal to 1.  


Therefore, we can express the power generated by one PV-module from the direct solar radiation Dir_N without shading by the formula:


   Power_1_Dir(i) = Mod_kW * ( Dir_N / ( 1 kW/m2) ) * cos(i) * RFT(i);


where RFT(i) = FT12(i) is the relative Fresnel transmittance calculated on the boundary Air-Glass for a given value of the refractive index n_Glass, and Mod_kW is the parameter "Mod_kW" of "solar" array objects discussed in the topic "Define Collector System".


Note that the formula does not contain any data of materials inside the PV-module below the glass. All the details of reflection / refraction are summarized by the Mod_kW parameter measured in the power test. We do not need to know these details to predict the angular behavior of the Power_1_Dir(i) function. Moreover, these details usually are not included in the documentation for the manufactured PV-modules and are not available for users. It was just the main motivation for us, when we were researching and developing this "RFT-method".


The power generated by the PV-module from the diffuse solar radiation Dif_S = Dif_H is given by the formula:


   Power_1_Dif(i) = Mod_kW * ( Dif_H / ( 1 kW/m2) ) * <RFT>;


where <RFT> is the RFT(i) function averaged over the semi-sphere. For example, <RFT> = 0.945 for n_Glass = 1.55.


Use the "T" key to see the relative Fresnel transmittance for "Total" / "Direct" / "Diffuse" solar radiation on the first page of the "Solar Day" window. The case "Direct" shows the behavior of the T_Dir = RFT(i) function during the day. The case "Diffuse" shows the T_Dif = <RFT> constant value (it is constant, because we accept Dif_S = Dif_H). The case "Total" shows the resulting T_Tot function for the Tot_S radiation.


You can see these characteristics for individual PV-strings or for the entire PV-system. If the PV-System consist of more than one "solar" object, then the RFT-curves show the corresponding functions averaged over the system according to the "energy weights" of its parts. The characteristics of the relative Fresnel transmittance are calculated before (and independently of) the shading calculations, so the "energy weights" of the PV-system parts in average values of RFT-functions are calculated by the incoming flux of solar radiation but not by the resulting power production that include "Full" or "Self" shading.


So all you need to model the impact of Fresnel reflection on your solar project is to set the "Refr_n" parameters of your "solar" array objects (that is just the n_Glass values of PV-modules). Moreover, the default value Refr_n = 1.55 is enough for the most of practical cases -- so you can do nothing.


If you want to do your own mini-research on the behavior of RFT-functions, then do the following. Open a blank scene, create the single object of the "A: PVM" type. Set the scene latitude = 0, keep the default date March 21, and open the "Solar Day" window. Note, the incident angle is proportional to the time measured from the true solar noon for a horizontal solar collector at this latitude and date, i = 15 * HFN. So you can consider the grid of the HFN-axis as a grid for i values with the step of 15. Then play with the "T" key of the "Solar Day" window and the "Refr_n" parameter of the object. Open the "Solar Table" window to see the numerical values of functions. You can transfer this table into other applications like MS Excel using the "Edit" / "Copy To Clipboard" menu (see details in the topic Data Transfer of this chapter). 


Page 1: Radiation


The page shows the direct, diffuse, and total irradiance on the collector surface (remember that we assume Dif_S = Dif_H).


If there are no active collectors in the associated scene, then the irradiance is calculated for a horizontal surface.



The first line of the legend is common for all pages. It shows general data for the entire PV-system or a chosen PV-string. Now you see the system data of the scene "s623": four PV-strings, fifty one PV-modules, installed power = 8.415 kW (peak). On the right, you see the date 03/21.


The second line of the legend shows the list of names of the displayed curves. Colors of names correspond to the colors of curves.





Below the curves you see hints for the keyboard keys. You can hide them with the key "K".


Then you see a small table titled "Daily sums ...". It shows sums or average values for the displayed curves. The order and colors are same as in the list of curves. Average values are shown in brackets.


For a scene without collectors, the corresponding values are calculated for the case of a horizontal surface, so the values of cos(i), Dir_S, Tot_S are substituted by cos(Z), Dir_H, and Tot_H.


Page 2: Shading


The page shows the shading of the collector system. The shading is expressed in the dimensionless units in range from 0 to 1. It means the part of the energy that is lost. We distinguish self-shading (only by elements of the system) and full shading (a result of self-shading and the shading by other objects) -- the case is controlled by the "S" key.


If there are no active collectors in the associated scene, the page shows the warning.



The first line of the legend says us whose curves are displayed -- of the entire PV-system or a chosen PV-string.





The table titled "Shading losses ..." shows losses (in percents) of daily productivity for the cases of "blocked" PV-strings, "blocked" PV-modules, and MPP-tracking. Colors help to distinguish the sense of values.


Curves of solar radiation Dir_S and Tot_S are added to help you understand what namely is shaded (the Dir_S component is shaded) and how high is the "relative energy weight" of the shaded Dir_S component in the total solar irradiance Tot_S on the collector surfaces of an individual PV-string or the entire PV-system.


Page 3: Power


The page shows the output of your collector system (the generated power in kW) for four cases: without shading (Free), with "blocked" PV-strings (BStr), with "blocked" PV-modules (BMod), and with the MPP-tracking.


If there are no active collectors in the associated scene, then the page shows the warning.



The first line of the legend says whose power curves are displayed -- of the entire PV-system or a chosen PV-string.





The table titled "Daily productivity ..." shows the daily productivity in kWh and percents (in compare with the "Free" case). Colors help to identify curves and their daily values.


PV-strings, bypass diodes, and MPP-tracking


Pages 2 and 3 of the "Solar Day" and "Solar Year" windows display curves with symbolic names "BStr", "BMod", and "MPPT". The curves of page 3 show the generated power. The curves of page 2 show the shading. The shading-curves of page 2 are calculated on the base of the power-curves of page 3 that are compared with the curve "Free" that means the power profile calculated without any shading (neither "Full" nor "Self").


The curve "MPPT" shows the result of the "Maximum power point tracking" (MPPT). The MPPT mechanism is embedded in the most of contemporary PV-inverters. Curves "BStr" and "BMod" play an auxiliary illustrative role -- they help you understand how the PV-system reacts on the shading and how the MPPT works in your solar project. 


Read about MPPT at 


Read more about shading, bypass diodes, inverters, and MPPT at 


Here is a very brief and extremely simplified summary of the facts concerning the PV-strings, bypass diodes, and the MPP-tracking. It is an "image" (but not a rigorous description) addressed to users that are new to the theme.


A PV-string is a set of serially connected PV-cells. For the sake of convenient mounting, PV-cells are constructively united into PV-modules. So a PV-string is a set of serially connected PV-modules that in turn is connected to an inverter.


A PV-cell generates electrical current that is proportional to the coming solar radiation, however as a semi-conductor device it can conduct only that amount of current, which it generates by itself. When all PV-cells are equally irradiated, the current flows through the circuit without problems. However, if one PV-cell is shaded from the direct solar radiation and receives only the diffuse component, it "blocks" the extra-current that could be generated by other fully irradiated PV-cells, and the extra-power generated by irradiated cells is transforming into heat energy on the shaded cell.  


The bypass diodes are used to protect PV-cells against the hot-spot heating. Usually, each PV-module is equipped at least by one bypass diode. The bypass diode switches the shaded module off -- it completely excludes the shaded module from the circuit.


The bypass diodes do not "regulate" a partly shaded PV-string -- they merely switch off all "weak" PV-modules knowing nothing about the reason why these modules are "weak" -- they operate in this way also with damaged or mismatched modules.


In a contrast, the MMP-tracker is designed just to actively regulate the behavior of a PV-string and to find the "Maximum power point" on the I-V-curve of the string. It "decides" what is better: either to adjust the current to the shaded modules (so that all modules are working as if they receive only the diffuse radiation), or to adjust the current to the fully irradiated modules and allows bypass diodes to completely exclude shaded modules from the PV-string.


For the simplicity of our symbolic notation, we name the first case as the "blocked" string "BStr" and the second case as the "blocked" modules "BMod". If your PV-system consist of the single PV-string, the "MPPT" curve completely overdraw the corresponding span of the better one of two curves "BStr" or "BMod". If your PV-system consist of two or more differently shaded PV-strings, then you can see all three curves. If there is no shading at all, you see the single "MPPT" curve that overdraws other ones.