EME 810
Solar Resource Assessment and Economics

4.6 Using Components for a Tilted Aperture


Reading Review

Please make sure you read all of Ch 8 in SECS for this lesson, still related to "Empirical Correlation for Components," but paying attention to the isotropic and anisotropic sky models and this page content. This is the third page for which we have included review of the Gueymard paper, so you should be familiar with the findings by now. The Perez et al. paper will be useful to you in Learning Activity 4.2 and for your Lesson Quiz.

Earlier, we discussed the different components of light, beam and diffuse on a horizontal surface. Now, we will discuss how these components can be estimated for tilted surfaces through isotropic or anisotropic diffuse sky/ground models of light source components.

Estimating the Plane of Array (tilted surfaces)

We shall see that we do not need to measure every component of light (scattered and unscattered) to make estimations on the contributions of each component to the total irradiation incident on the aperture of interest. We can rely somewhat on decades of historical observation and empirical correlation by solar scientists and engineers for hourly, daily, and monthly average day data.

The main tools we need are the equations for hourly and daily extraterrestrial irradiance (Air Mass Zero, or AM0) and the integrated energy density (irradiation: $MJ/m^2$) gathered from a horizontally mounted pyranometer, which you learned of in the last section. We shall also find that we can infer more than just the components of light from the ratios of measured irradiation to AM0 calculated irradiation--we can describe the fractions of days in a given month where lighting conditions will be clear or overcast/cloudy.

Following our step to break apart the beam horizontal component from the diffuse horizontal components, we then estimate the components on a tilted surface.

Procedure for Components

For a tilted plane of array,

Total Radiation = beam + diffuse, sky + diffuse, ground

G t = G b,t + G b,d + G g,t

A simple calculation of the beam component can be achieved using

G b,t = G b,n .cosθ

Radiation on a sloped surface can be calculated for the beam component of irradiation by the geometric scaling factor of

R b = cos(θ) cos( θ z )

In order to estimate the diffuse component, we use alternate models that become increasingly better fits with the empirical data. We can integrate any of these equations over an hour or a day (irradiation). I prefer to offer the irradiance version as a bit easier to read. Note: all of these estimation models use irradiation values that were measured from a pyranometer mounted along the horizontal plane, and then estimated for beam and diffuse components from data correlation or directly measured using a shadow band measurement and energy balance equations.

Isotropic Diffuse Model: Liu & Jordan (1960)

The isotropic sky model was developed in the 1960s to estimate the diffuse sky on a tilted surface, complemented by an estimate for diffuse light from the ground. This model assumes that the sky is uniform in composition across the sky dome.

The following expression gives the total solar irradiance incident on a tilted surface as

G=  beam+diffuse sky +diffuse ground G total = G b R b + G d ( F surfacesky )+Gρ( F surfaceground )


F surfacesky =  1+cosβ 2 F surfaceground =  1-cosβ 2 G d,t  = G d .  1+cosβ 2 G g,t = ρ g ( G b + G d ).  1-cosβ 2

The fraction proportional to the collector tilt is called the diffuse sky irradiance tilt factor for an isotropic sky model, and the reflectance of the ground is called the albedo (a fraction between 0 and 1), and is multiplied by the GHI and the diffuse ground irradiance tilt factor for an isotropic sky model.

Note: "Surface": the aperture. ρ : is the collective reflectivity of the ground (the albedo). ρ reduces the irradiance G by a value between 0--1. On an inclined surface, Gd,ground increases, relative to a horizontal collector.

HDKR Model:

This model incorporates isotropic diffuse, circumsolar radiation and horizontal brightening. It also employs an anisotropic index A defined mathematically as

A= G N,I G

The total irradiance on a tilted surface is then calculated by using

G T =( G b + G d A) R B + G d (1A) 1+cosβ 2 ×[ 1+ G b G sin 3 β 2 ]+Gρ 1cosβ 2

Go to Kalogirou (Solar Energy Engineering) Ch 2 (pdf from Library) this will be labeled the "Reindl model"

Perez Diffuse Model: Richard Perez (1990)

This is an anisotropic diffuse sky model that takes into consideration the real observations of subcomponents of diffuse light. The Perez model adds the circumsolar diffuse component and the horizon diffuse component to the diffuse$_{sky}$ component of the isotropic model. Notice how the beam component is not mentioned here--it doesn't change.

Sidenote: Richard Perez is a Senior Research Associate in the Atmospheric Sciences Research Center in SUNY Albany. He has a great website.
G d = G iso + G cir + G hor  +diffus e ground G d =[ G d (1 F 1 ) 1+cosβ 2 ]+[ G d ( F 1 ) cosθ cos θ z ]+[ Gd( F 2 )sinβ ]+[ G d ρ 1cosβ 2 ] The shape factors (F) in this model can be reviewed in the original article by Perez et. al (1990). However, we can inspect the equations and observe in the equation that $F_{surface-sky}$ is reduced by a proportion of $F_1$ (circumsolar radiance), and $F_2$ can either increase or decrease the contribution of horizon radiance.

Try This! Tilted Surface Radiation Model

Software: SAM from NREL
  1. You can open up SAM at this point and click the button "Create a New File."
  2. Click on "Photovoltaics (Detailed)" on the left of the pop up window ("Choose a performance model").
  3. Click on "Residential (Distributed)" on the right of the pop up window ("then choose from the available financial models").
  4. Hit "OK."
  5. You will now have a default residential PV project, based in Phoenix, AZ. There will likely be a SunPower (California company) PV module, and an SMA America dc-ac power inverter listed on the left column of shortcut tabs. They're not really "tabs" like the tab at the top of the window, but I'm going to use that term for convenience in the class.
  6. Click on the shortcut tab called "Location and Resource" (will list "Location," latitude ($\phi$), longitude ($\lambda$), and elevation above sea level in the shortcut.
  7. At the bottom of the screen on the right is a large "+" sign labeled "PV Albedo and Radiation". Please expand the section. I want to draw your attention to the little section called "Diffuse Sky Model (Advanced)." Yep, it's so advanced that you don't even realize this is where a powerful data transformation sits!

You can see that one may select "Irradiance Components used for Calculation": this is specifying the type of horizontal irradiation components that you will use in your tilted model. In a data set called the Typical Meteorological Year, the data for the beam is often not actually a measured value.

You can also see that one may select three diffuse sky/ground transposition models to transform the "Irradiance Components" (horizontal) to tilted values. The default is the Perez model that we describe below (and in your supplemental reading). The isotropic model is not used in practice, but it contains the basis for the other anisotropic models of Hay-Davies-Klutcher-Reindl (HDKR) and Perez et al. 1990.

Diffuse Sky Model (Advanced)

  • Tiny box hidden within a plus sign, whole lotta power. Keep that in mind.
  • Also, when in doubt: Read the Fine Manual. SAM has a detailed Help manual with links to the literature for every aspect of the program.


Liu, B.Y.H., Jordan, R.C., 1960. The interrelationship and characteristic distribution of direct, diffuse, and total solar radiation. Solar Energy 4(3),1-19

Perez, R., Ineichen, P., Seals, R., Michalsky, J., Stewart, R., 1990. Modeling daylight and Irradiance components from direct and global irradiance. Solar Energy 44(5), 271-289