Computer-Simulation of Terrestrial Solar Radiation
The simulation allows it to prognosticate the efficiency of any terrestrial solar-power system which is determined by the value of the horizontal positioning angle (Azimuth) and the vertical positioning angle (Inclination) as well as to play through different and alternative models of sunray-collection - to avoid later reconstruction of your solar-power system.

It will be possible to minimize the effect of a particular sky-line's shielding of the sun just by fine-tuning the azimuth. On the other hand, a roof-direction other than south can be compensated for in large part by adjusting the system's corresponding inclination. Furthermore, the individual direction can be optimized either for the summer or winter periods or a full year collection. And - taking into account unfavorable environmental conditions, the simulation can be helpful in deciding whether or not to invest in solar technology!
 

The mathematics model is based on the following parameters:

• the latitude of installation
• the declination determined by the calendar date
• the solar-power system's inclination in relation to the horizon
• the solar-power system's azimuth in relation to south (in the southern hemisphere to north)
• the influence of weather conditions and atmosphere

the model takes into consideration the influence of the topographical horizon in order to minimize the sun's hidden-time.
 

Simulation's Benefits
Some results will amaze you! No doubt, one of the most amazing results will be that even in the northern hemisphere an optimal direction can face north (in the southern hemisphere facing south), depending on the latitude and weather conditions. Here is why:

The contents of the daily energy collection is not only determined by the intensity of the sunrays and but also by their duration. The classic direction faces to zenith (sun-position at 12 noon) which only gives a maximum value at 12 noon, leading to a loss of solar hours in the early morning as well as in the evening. Why is that? Let's have a look at the diagram Sun's Path.

On the 21^st of June (summer solstice) for example, everywhere in the world at northern latitudes of 50° degree (for example, Frankfurt, Prague, Kiev, Winnipeg, Vancouver etc.) the sun rises at about 4 am in the north-east (with an azimuth degree of about  -110°) sunset will take place at 8 pm in the north-west (azimuth +110°). Due to this fact the sunrays will be effective full-time (for 16 hours) on a horizontal collector. A collector facing south however, will have sunrise and sunset to its "back" and therefore is active only about 13.5 hours a day.
 

Potential Energy
The potential energy reaching the earth is determined by the geographical latitude, the annual-season, time of day and weather conditions - which can be changed only "if you could create your own weather or move the world on its axis".

The potential energy which can be collected by an terrestrial horizontal system follows nearly a sinus-curve valid in its positive area (day-time). Due to summer- and winter-cycles an offset will shift the neutral axis up (summer) or down (winter), measured in degree of declination. Twice a year only, at the spring / autumn equinox, this declination will be "0".

The diagram Solar Radiation shows the energy trajectory for
     -   21 June                  (summer solstice / longest day)       declination +23.5°
     -   21  December          (winter solstice / shortest day)        declination  -23.5°
     -   21 March / 22  Sept.       (spring / autumn equinox)        declination     0.0°
not taking the atmospheric influence into consideration, therefore these curves are only of a theoretical nature.

• The area bordered by theses curves is a measurement for the potential day's energy. For a sinus half wave this is "2/pi" = ca. 64% of the corresponding rectangle.
   
• Outside of the atmosphere there is a the potential solar radiation of about 1.353 kW / m² (Solar-constant by Thekaekara). Without taking the atmosphere's influence into consideration, the energy contents of a 12 hour sunny period would be approximately 2/pi * 12h * 1.353 kW /m²  = 10.336 kWh/m². So this value is the unattainable maximum of "sub-solar" radiation.
   
  Sub-solar: 90° sun elevation which gives a perpendicular solar radiation on horizontal ground. Each tropic latitude will obtain an approximate sub-solar radiation for a few weeks at 12 noon, however a true sub-solar radiation can be reached only twice a year!

Due to this, the mathematical model has to take into consideration the influence of atmosphere and weather - which is considerable  (for details please see: Atmosphere's and weather's influence). These atmospheric parameters (used by the simulation) are essential for designing the right size of the collector / generator.
 

Diagrams
There are two different kinds of diagrams which show simulations for any location all over the world. Both diagrams refer to the same chosen calendar date.
 

Diagram "Sun's Path"
shows the sun's elevation and its corresponding azimuth (both measured in degrees) constantly changing with the time of day (please see "green lines" in the diagram). It also shows the topographical horizon and the border-line of summer / winter solstices.

This diagram gives useful hints for architects concerning the particular shade-situation and is to be understood as a theoretical background for the diagram Solar Radiation. It also helps to explain a particular sunray intensity which does not always seem logical at first glance.
 

Diagram "Solar Radiation"
The day-time illustrated is measured in "true local time" which means 12 noon with a maximum sun elevation (zenith), it does not take summer/winter-time into consideration. Here is why:
 

The official time (like GMT, and so on) determines people's social life, but not the potential sunrays and its possible collection. Therefore sunrise and sunset will take place about 1 hour earlier at the eastern border of the each time zone compared with its western border. So that, the official time in any case is "wrong". However, each location will get the energy determined by its latitude, which may be half an hour earlier or later.

The red curve shows - in accordance to the chosen parameters - a good approximation of the daytime changing of the direct sunray intensity. As already explained, the square dimension bordered by the curve is a measurement for the energy of a particular day. Without the atmosphere's influence the daytime changing of solar radiation would follow a sinus wave (see above mentioned items) and would be approximately 2/pi * 12h * 1.353 kW /m² = 10.336 kWh/m² at spring / autumn equinox.

The influence of the atmosphere however depends on the sun's elevation angle. With a plane angle (in summer just during sunrise and sunset periods - in winter-time, the full day) the influence increases exponentially, so the curve becomes bell-shaped. Therefore the intensity of the solar radiation never reaches the above mentioned value bordered by the sinus wave. A realistic value for a sub-solar sunray intensity on a horizontal receiver is about 7.5 to 8 kWh m² a day.

During the "high-summer" season this value can be passed over even in middle latitudes by an optimized collector-direction! In equatorial latitudes it will be reached nearly daily.
 

Collector's Power Output
Due to the optical properties of the atmosphere, such as atmospheric transmission, reflection and environmental pollution, the potential solar-power radiation (measured on terrestrial ground) doesn't reach the above mentioned 1.353 kW /m², the true maximum value is even on a clear day approximately 1 kW /m² at 12 noon. Technical data of a solar-power generator or collector relate to this value - if no other definition is pointed out. This means, that a solar-power system's output reaches its "manufacturer's stated value" only if there is an sunray input of 1 kW /m²- which occurs at noon on a cloudless day.
 

The result of the simulation's calculation (please see red note in the diagram) multiplied by the "manufacturer's stated value" shows the system's true daily output. Therefore a "10 kW system" will collect 10 kW /l kW * 7.5 to 8 kWh = 75 to 80 kWh on a clear summer day.
 

Optimized Directing towards Sun
To get maximum energy output it will be necessary to direct the system's active surface in such a way towards the sun - that it is perpendicular (90° angle) to the sunrays at all times. To do so, the system must follow the "sun's apparent movement". The coordinates to track the system are to be found in the diagram Sun's-path.

There is an enormous amount of technical equipment - and also investment - needed for a tracking device which compensates for the various sun-paths. Theoretically, the system must follow the sun's azimuth moving, as well as its corresponding elevation. The adjustment-benefit of such a tracking device however depends on weather conditions and season. In European latitudes, for example these benefits can reach up to 50% on a nice summer day with a cloudless sky (direct rays) - in the winter period this benefit reaches only about 8%. Cloudy weather (just non direct rays) will reduce this benefit to about 25% or less in summer and 3.5% in winter. The theoretical adjustment-benefit calculated by the simulation is shown in the diagram Solar Radiation by the yellow containment curve. For an opacity less than 0.3 (please see below) the adjustment-benefit is additionally pointed out in the yellow note. This value, however, is only an approximate one to show its tendency!
 

Simulation's Parameter
For the input of these values please select the worksheet Parameter,
in addition the 'Simulation Time & Space' will offer a fascinating  'Time machine's solar flight'  from the Equator to the  pole; everything moves 'on click'.

Calendar Date
You can choose any date you like. To get comparable values we suggest using
     -   21st  of March           (spring equinox)                           declination     0.0°
     -   21st  of June              (summer solstice / longest day)     declination  +23.5°
     -   22nd of September     (autumn equinox)                         declination     0.0°
     -   21st  of December     (winter solstice /shortest day)         declination  -23.5°
of any year.
 

Geographical Latitude
Pleas consult a map to find out the latitude (measured in degrees °) of the location to be simulated. For the southern hemisphere don't forget the negative sign of the latitude's degree.
 

Atmospheric and weather's influence
Due to the fact that the optical properties of the atmosphere change with the weather, this is the big unknown with an extreme influence. Long-term weather statistics can be supplied by the regional weather-bureau, however these statistics are mostly very complex,  the level of science is not specifically focused on practical solar requirements.

To avoid this inconvenience the simulation takes into consideration the atmospheric influence by using a special opacity-factor (including atmospheric transmission, reflection and environmental pollution) which can be found in a special table contained in the worksheet, Parameter. This opacity-factor can take values starting with 0.2 (for a clear sky in mountain areas) up to more than 1 ("an overcast sky"). This will show the limits of the practical value of each solar-power system.

To establish the full range theoretical investigations are possible by using an opacity-factor of "0" which means "without atmosphere". This can be helpful for a better understanding of seemingly "non-logical" results. This investigation will explain for example why even in a northern subtropical hemisphere and African latitudes an optimal direction can face north.

Remember: As already explained, this influence depends not only on the atmosphere itself but also on sun's elevation. With a plane elevation angle (sunset, sunrise and winter time), this negative influence increases exponentially, - and the radiation curve becomes bell-shaped.
 

Receiver's Surface, Azimut
Directing the receiver's surface towards the south (Azimut = 0) is normally the best solution (except for above mentioned situations) and should be done whenever possible - however this is not always possible! The angle-deviation between the receiver's azimuth-direction and that of true south is measured in degrees (°), (please enter western values with a positive sign and use a negative sign for an eastern deviation).
 

Receiver's Surface, Vertical Inclination
The vertical inclination is the principal parameter for optimization, as it is normally the only one which can be freely chosen. A collector, lying horizontal to the ground has a vertical inclination of 0°, if its position is perpendicular, the value is 90°.

The classic collector-direction towards zenith (12 noon) with an azimuth of 0° is simply calculated by subtracting the declination angle from the latitude. Even this gives a maximum at 12 noon its not the optimum for a full day.
 

Simulation Mode

• Analysis for a particular date
  The analyzes for a single particular date can be found instantly by entering the above described parameters while taking the topographical horizon into consideration.
  For the results please see the diagrams:
  -  Solar Radiation  (1)            (hourly changing output / adjustment benefit)
  -  Solar Radiation  (2)            (theoretical background)
  -  Sun's path                         (hidden sun situation and theoretical background)
   
• Seasonal Analysis and Optimization
  In the SOLAR-menu you have the choice to run a
  -  Day's optimization
  -  Year's optimization            (Estimated Quality)
  -  Season's optimization (1)    (Date from / to          First Approximation)
  -  Season's optimization (2)    (Date from / to          Precise Approximation)
  -  Season's analysis               (Date from / to)
  
  The results of the season's analysis and optimization can be found in the table of worksheet Statistics which will appear automatically.

The two bar-indicators in worksheet Parameter show the progress of the optimization process. This process can be interrupted. Interruption takes place at the end of every scan-period indicated by the bars. To interrupt the simulation please press the ESC-key for a few seconds in the end of the scan-period. An restart is available.
 

SOLAR-Menu
A click on the sun-symbol starts the SOLAR-menu consisting of

• Analysis and Optimization
• Simulation Time & Space
• Topographical Horizon

In the case, that due to the topographical horizon the sun will be hidden for a time the simulation proposes to run a combined optimization of inclination-angle and azimuth-angle. If you don't agree to accept this, the inclination-angle will be optimized based on the chosen azimuth. The optimization's result depends on the chosen period. You can chose any period starting with a particular day, a particular season or a full year.

The effort involved in calculating a true optimization "is quite a job", therefore an estimated optimization is offered additionally (which only takes 10 - 30 sec.) to get a rough approximation for a full year period. For a true optimization you can select either a first approximation or a precise approximation which takes a few minutes. The simulation scans the daily solar radiation so the time of calculation depends on the chosen season and the scan-interval. Reducing the interval will increase the quality of the simulation but also the running time. A combination of inclination-angle and azimuth-angle optimization will double that calculation-time.
 

Sky-Line's Topography and the resulting Shade
In mountainous areas as well as in urban surroundings we have to keep in mind the influence of trees, houses, mountains etc. To do that the topographical horizon has to be recorded. The directing equipment needed for that is very simple:

• To direct the topographical horizon you need either a camera with a powerful telephoto lens or a binocular lens - however if a rifle scope is available, that's the best solution. One of theses optics is to be fixed on a semi-professional (Photo/Video) tripod constructed to swivel independently, both horizontally (azimuth) and vertically (elevation) and must be equipped to measure both angles in degrees.
   
• Fix the directing equipment to the lowest point of the (eventual) position of the solar-power system, adjust it with a spirit level to a horizontal position and direct it also to true south with a compass. The above gives the zero-position, next adjust both the horizontal and vertical readings to zero.

Now you can aim it at the borderline between the sky and horizon. The azimuth's and elevation's values (measured in degrees) should be taken in pairs and recorded in the Horizon Database.
 

Horizon Database
The Database Manager of the menu Topographical Horizon is quite similar to the windows^® file management.

There are 124 data-pairs available to record a particular horizon, however you are not forced to use them all. You are also free to use a sequence of your choice for the pairs input, and you are not required to enter the sequences in numerical order as the database manager will sort it by Azimut from east to west.

The active data-set can be increased if necessary. If one and the same azimuth value is combined with more than one elevation angle, the last one entered is valid.

The border line's elevation between two neighboring azimuths' values is defined by the left elevation's value. So you only have to enter the coordinates where a change takes place. To construct a sloping border line just use some more pairs with smaller azimuth distances.

The simulation draws the silhouette of the chosen horizon in addition to the curves of the diagram sun's path. If the sun's path moves below the topographical horizon, the sun is hidden and therefore the generator does not receive any direct sunray - it is shaded. The diagram Solar Radiation shows a downward movement of the direct sunrays intensity to zero during the time, the receiver is shaded. During this period, only the non direct rays will have some small solar-effects. Normally the shade-period is not symmetric to 12 noon, so the simulation proposes a combined optimization of inclination-angle and azimuth-angle - which you can either accept or not!

Useful Hints and Tips

• You may keep track of the database manager's work with the worksheet Top-Hor (Topographical Horizon). In addition, it is possible to change and add some values of the active data-set "by hand", however this should be done by Excel-experts only. The database manager is designed to do this job, furthermore the shape of diagrams are optimized by the manager while performing this task.
   
• In connection with this, you should try to avoid azimuth-values between -120° and -180° respectively +120° and +180° whenever possible: If these values are used, the manager compresses the acuity of diagram sun's path.
   
• It is preferable to use the option Max-Screen offered in the SOLAR-Menu, if you use the similar task from the original Excel some graphic's elements will appear incorrectly shaped. To correct this if it ever occurs, please click the button Max-Screen from SOLAR-Menu up and down a few times.
   

Do your own experiments

• Last not least 
  A  'Time machine's solar flight'  from the Equator to the  pole;
  everything moves 'on click'. The 'Simulation Time & Space' will be started from the SOLAR-Menu.
   

Enjoy yourselves,
enjoy the Solar-Power-Simulation!