# Effective solar angles

The equations above show that at a given location and a given day, the solar angles depend entirely on the true solar time. Since the radiation measurements are made over a certain period of time during which the true solar time and therefore the solar angles change, what are the solar angles to be assigned to each measurement? Measurements or estimates of solar radiation are integrated over a certain time, called the integration period or summarization. Therefore, the above equations cannot be applied as such.

Why should you care? It is very often necessary to assign solar angles to each measurement. An example of such a need is the computation of the corresponding extraterrestrial radiation which is necessary to control the quality of measurements and should be known for the integration period. Another example is the computation of the angle *9* under which the sun is seen from an inclined surface which is the major factor in the calculation of the received solar radiation. In the two examples, a solution consists in computing the radiation every 30 s or 1 min within the integration period, then summing the results. Is there a simpler and faster way of doing it?

In cases where the integration period is 1 min or less, then it can be considered with good accuracy that the solar angles are constant over this duration. During such short periods, the angles may be considered as varying linearly and they can be calculated for the time in the middle of the period. For example, if the measuring interval is between 09:25 and 09:26, angles can be calculated at 09:25:30.

This is no longer the case when the integration period is greater than 1 min. The angles vary greatly during such a period, in particular at the time of sunrise or sunset. In this context, how to calculate the corresponding solar zenithal angle *в$* and azimuth ¥$? The concept of effective angles is defined to answer this practical problem. The effective solar angles are fictitious and correspond as best as possible to the solar radiation measured during the period. The effective solar angles are noted *в ^{,e}f *and

*'Vf.*

## Methods for computing effective angles

The problem comes down to the assessment of the effective solar zenithal angle *ef. *Once *6 ^{e}£* is estimated, the effective solar azimuth 4^ can be deduced by the equations already presented. I do not know of a general solution to this daily problem. Philippe Blanc and I did a fairly simple study in 2016 by looking at some common methods and suggesting others. The resulting publication

^{2}does not provide a definitive answer to this question, but provides some concrete, easy-to-implement solutions, which are described below.

The concept of the 2016 study is based on the assessment of the direct component of the solar radiation that will be detailed in the following chapters. For now, I just have to say that this component, noted *B,* is the part of the solar radiation that appears to come from the sun. It therefore excludes the other part of the solar radiation coming from the other directions of the sky vault and called diffuse component. Because it appears to come from the direction of the sun, the angle of incidence of *В* onto a horizontal plane is equal to the solar zenithal angle *ef.* The direct component *В* is related to the direct component *В*д? at normal incidence by the following equation:

In this study, Philippe and I used very accurate 1-min measurements of the direct component *B^* at normal incidence available at two measuring stations, from which we computed 1-min *B.* Then, for the sake of simplicity, we assumed hourly measurements, i.e., with an integration period *At* of 1 h. We thus computed hourly averages of *B^* and *B.* Each hourly average of *В* was assumed to be a measurement for which *6 ^{L}f* must be computed by one of the tested methods. Finally, the hourly

*B*was estimated by the equation

where cos^{_1}() is the arc cosine function and was compared to the actual value.

Three common methods were tested first. Let *t* be the time at the end of the integration period, expressed in h. In Method #1*,6^* is equal to *0$* at half-hour, i.e., at (Z—0.5):

Blanc P., Wald L., 2016. On the effective solar zenith and azimuth angles to use with measurements of hourly irradiation. *Advances in Science and Research,* 13, 1-6, doi:10.5194/asr-l-l-2016.

In Method #2, *Bf* is equal to the average of *9s* over the hour, i.e., between (f-1) and *t*:

In Method #3, *6 ^{e}f* is equal to the average of

*9s*over the hour, with limitations to the times of sunrise and sunset:

Comparisons with actual measurements showed that the relative error on the solar radiation (i.e., *B^)* depends on *9$-* The closer the sun to the horizon, the greater the error. In any case are the errors negligible. They are the greatest for Methods #1 and #2. The root mean square error is about 30 %-40 % relative to the average radiation. The bias is the average of the errors and denotes a systematic error. It is of a few percent in relative value; it may reach 16 % when *9$>* 75°. The best results are given by Method #3. The relative root mean square error is about 4 %-5 %, though it may reach 24 % when *9$>* 75°. The relative bias is small, except when *9$>* 75°, where it may reach 23 %.

Two novels methods were proposed by Philippe and I, where *9$* is not computed from averages of *9s-* Taking advantage of equation (2.28), *9 ^{е}/* is given by models computing the direct components

*В*and over the hour for the extraterrestrial radiation (Method #4) and cloudless atmosphere (Method #5). In Method #4, the hourly

extraterrestrial radiation *B _{0}* and

*B*are computed, and their ratio yields

_{0N}*Q*

^{e}:

In Method #5, the hourly radiation *B _{c/ear}* and

*B*for cloudless conditions are computed by a so-called clear-sky model, and their ratio yields

_{Nclear}*df*:

Method #4 provides similar errors to those of Method #3. The smallest errors are obtained by Method #5, for which the bias is almost negligible, whatever *9 _{S}.* The relative root mean square error is 2 %, except when

*9*75° where it may reach 9 %.

_{S}>## How to practically use one of these methods

Which lessons can be learnt from this study? Philippe and I recommend not using Methods #1 and #2. The errors are too large and include an important bias that may reveal troublesome in subsequent calculations. Methods #3 and #4 offer smaller but still noticeable errors. This should be accounted for in subsequent calculations. Errors tend to dramatically increase when the sun is low on the horizon, at sunrise and sunset. Method #5 gives the best results by far, with small bias and root mean square error. It shows a limited degradation of performance when *9s* increases.

How to practically use one of these methods? Methods #1-3 need the calculation of *9s* every minute. To do so, one may use the equations given above, or already mentioned software libraries. One may also use Web services such as the SoDa Service specialized in solar radiation (www.soda-pro.com). This site proposes services to which you can provide inputs such as beginning and end dates or geographic coordinates, and that return time series of solar angles for various integration periods in tabulated form. They can be copied and pasted in a spreadsheet for further use.

Method #4 requests estimates of the solar radiation at the top of the atmosphere on a horizontal plane and at normal incidence. They can be calculated by means of the equations given in the following chapter, or by means of Web services, including those available in the SoDa Service.

Of course, it is best to use Method #5 because it is the most accurate. It requires that a clear-sky model is available that provides estimates of *B _{c}i_{ear}* and

*В^ыеаг*i

^{n}cloud- free conditions. You may wish to implement one yourself. The book by Daryl Myers documents several such models

^{[1]}, and several codes are available on the Web.

^{[2]}Then, you have to find sources of data about the clear atmosphere such as the water vapor content, the Linke turbidity factor or the aerosol optical depth as it will be seen in the following chapters. These data are input to the clear-sky model. The direct component of the radiation is computed every minute within the integration period for the horizontal plane and at normal incidence. Then, the results are summed yielding

*B*and Вnclean and finally

_{c}/_{ear}*6*

^{e}

*Another practical means for exploiting Method #5 without writing a code dedicated to a clear-sky model and obtaining the necessary inputs to the model is to use online services. One example is the McClear application in the SoDa Service, also part of the Radiation Service of the European Copernicus Atmosphere Monitoring Service (CAMS). It exploits the McClear clear-sky model and returns time series of irradiances, including £B^ _{c}i_{ear}* Users must enter the place of interest or geographic coordinates, desired period of time, time system, and integration period from 1 min to 1 month. The application itself reads the database of estimates of constituents of the cloudless atmosphere carried out daily by the CAMS. Connoisseurs may be interested to know that the McClear application is a Web processing service, a standard from the Open Geospatial Consortium (OGC). It can be invoked automatically from a computer code written, for example, in Python or other language. It thus offers an opportunity to fully automate the process of computing effective angles.

Methods #4 and #5 call upon models that are often available for the total radiation, i.e., integrated over the whole solar spectrum. What should you do if you deal with other spectral ranges such as the UV or visible? Nothing more. These models for total radiation are suitable in these cases as the solar angles do not depend on the wavelength. The effective solar zenithal angle *6 ^{e} is computed from the ratio B_{c}/_{ear}lBp/_{c}i_{ear }which is independent of the wavelength as shown by equation (2.28).*

*[1] Myers D.R., 2017. Solar Radiation: Practical Modeling for Renewable Energy Applications, Boca Raton:CRC Press.**[2] https://github.com/EDMANSolar/pcsol, written by Oscar Perpinan Lamigueiro. [The author gives Rcodes for several tens of clear-sky models.]*