Tilt Model

Perez Algorithm (Diffuse Sky)

1.) Find the factors that account for the respective angles of incidence of circumsolar radiation on the tilted and horizontal surfaces. The circumsolar radiation is considered to be from a point source.

Figure 8. Coefficients of Slope Irradiance

4.) Compute the diffuse sky irradiance on the tilted plane:

Hay Davies Algorithm (Diffuse Sky)

Compute the diffuse sky component on the plane of array using the direct beam transmittance from the DISC model:

Common Algorithm (Beam and Diffuse Ground Components)

1.) Compute the beam component on the tilted plane as a function of the incidence angle and the tilt angle. Due to symmetry, incidence angles greater than 90° will have the same effect as those less than 90°. To avoid negative numbers, the absolute value is applied.

Reference

Muneer, T. Solar Radiation and Daylight Models, Elsevier, 2004, pp 102, 155, 156.

R. Perez, P.Ineichen, R. Seals, J. Michalsky, R. Stewart. Modeling Daylight Availability and Irradiance Component from Direct and Global Irradiance. Solar Energy 44, no 5, pp 271-289, 1990.

Hay, J.E., Davies, J.A., Calculations of the solar radiation incident on an inclined surface. Proc. of First Canadian Solar Radiation Data Workshop, 1980, 59. Ministry of Supply and Services, Canada.

GTI DIRINT

This function derives direct normal irradiance (G_B,n) and diffuse horizontal irradiance (G_D) from the global tilted irradiance (G_Ti) using modifications to the DIRINT decomposition model and Perez transposition model. This allows for G_Ti to be input into an energy prediction, largely avoiding decomposition and transposition model errors (with the exception of calculations that require the direct/diffuse components, i.e. shading and IAM).

Inputs

Outputs

Algorithm

This model implementation calculates a solution for incidence angles (θ) over and under 90° separately to allow solutions at times when the G_Ti measurement does not include any component of G_D.

1. GTI DIRINT model for θ < 90°

Note that this model uses a modified (tilted) clearness index as described in the Clearness Index section. Unmodified DIRINT equations (see DIRINT Model) are used to calculate G_B,n. G_D can then be calculated using:

Known errors are introduced when using the G_Ti as an input into the GTI DIRINT model (i.e. differences in proportions of diffuse and direct irradiance between global horizontal irradiance and G_Ti and the fact that G_Ti also includes ground reflected radiation). So, after G_D and G_B,n are calculated from the measured G_Ti using the reverse decomposition of GTI DIRINT, the forward transposition using Perez is used to calculate G_Ti as a check. The measured G_Ti and calculated G_Ti can then be compared and used as the iteration criteria in an optimization loop.

In the optimization loop, the initial calculation uses the measured G_Ti to calculate K_t. Subsequent iterations adjust the G_Ti until the K_t provides a G_B,n and G_D value such that a modeled G_Ti matches the measured G_Ti. The G_Ti value is adjusted using:

Where i is the index corresponding to the iteration number, D is the difference between the modeled and measured G_Ti, and C is a coefficient to ensure an iterative solution (1.0 for i <3, 0.5 for 3 ≤ i < 10, 0.25 for 10 ≤ i < 20, and 0.125 for 20 ≤ i < 30). Often, only two iterations are needed. The following illustrates the flow chart for the model when θ < 90°.

2. GTI DIRINT model for θ ≥ 90°

After G_B,n and G_D are calculated for θ < 90°, average early morning and late afternoon K_t‘ values are determined by averaging K_t‘ values for morning and afternoon instances when θ is between 65° and 85°. Besides K_t‘, the solution for G_B,n requires K_t which is rearranged from the original DIRINT equation to solve for K_t instead of K_t‘:

The G_D can then be determined using:

Reference

Marion, B., A model for deriving the direct normal and diffuse horizontal irradiance from the global tilted irradiance.  Solar Energy 122 (2015), pp. 1037-1046.