Clear Sky Radiation

The clear_sky_radiation module contains all functions related to the calculation of (instantaneous) clear sky radation. Most of these functions are based upon Šúri and Hofierka1.

extraterrestrial_irradiance_normal(I0, ied)

Computes the extraterrestrial irradiance normal to the solar beam.

\[G_{0} = I_{0} \cdot \varepsilon\]
Parameters

  • I0 (float) – solar constant, \(I_{0}\) [W m-2]

  • ied (float) – inverse earth sun distance, \(\varepsilon\) [AU-1]

Returns

G0 – ext rad normal to solar bea, \(G_{0}\) [W m-2]

Return type

float

inverse_earth_sun_distance(day_angle)

Computes the inverse earth sun distance

\[\varepsilon = 1 + 0.03344 \cdot \cos(j^{\prime} - 0.048869)\]
Parameters

day_angle (float) – day angle, \(j^{\prime}\) [-]

Returns

ied – inverse earth sun distance, \(\varepsilon\) [AU]

Return type

float

day_angle(doy)

Computes the day angle. 0 is january 1st, 2\(\pi\) is december 31st.

\[j^{\prime} = \frac{2 \cdot \pi \cdot j} {365.25}\]
Parameters

doy (float) – day of year, \(j\) [-]

Returns

day_angle – day angle, \(j^{\prime}\) [rad]

Return type

float

solar_constant()

Returns the solar constant. The solar constant is defined as the flux density of solar radiation at the mean distance from Sun to Earth. The solar constant is estimated to be 1367 W m-2.

Returns

I0 – solar constant, \(I_{0}\) [W m-2]

Return type

float

declination(day_angle)

Computes the solar declination. The solar declination is computed according to Gruter2.

\[\delta = \arcsin\left(0.3978 \cdot \sin\left(j^{\prime} - 1.4 + 0.0355 \cdot \sin \left(j^{\prime} - 0.0489\right)\right)\right)\]
Parameters

day_angle (float) – day angle, \(j^{\prime}\) [rad]

Returns

decl – declination, \(\delta\) [rad]

Return type

float

relative_optical_airmass(p_air_i, p_air_0_i, h0ref)

Computes the relative optical air mass. It is calculated according to Kasten and Young3.

\[m = \frac{\frac{p}{p_{0}}}{\sin h_{0}^{ref}+0.50572 \left(h_{0}^{ref}+6.07995\right)^{-1.6364}}\]
Parameters

  • p_air_i (float) – actual instantaneous air pressure, \(p\) [hPa]

  • p_air_0_i (float) – air pressure at sea level, \(p_{0}\) [-]

  • h0ref (float) – solar elevation angle corrected for refraction, \(h_{0}^{ref}\) [degrees]

Returns

m – relative optical airmass, \(m\) [-]

Return type

float

solar_elevation_angle_refracted(h0)

Computes the solar elevation angle corrected for refraction.

\[h_{0}^{ref} = h_{0} + \Delta h_{0}^{ref}\]

where:

\[\Delta h_{0}^{ref}=0.61359 \cdot \frac{0.1594 + 1.123 \cdot h_{0} + 0.065656 \cdot h_{0}^{2}}{ 1+28.9344 \cdot h_{0} + 277.3971 \cdot h_{0}^{2}}\]
Parameters

h0 (float) – solar elevation angle, \(h_{0}\) [degrees]

Returns

h0ref – solar elevation angle corrected for refrection, \(h_{0}^{ref}\) [degrees]

Return type

float

hour_angle(solar_time)

Computes the solar hour angle.

\[T = \frac{\pi}{12}\left(t-12\right)\]
Parameters

solar_time (float) – solar_time, \(t\) [hours]

Returns

ha – solar hour angle, \(T\) [rad]

Return type

float

solar_elevation_angle(lat, decl, ha)

Computes the solar elevation angle.

\[h_{0}=\arcsin\left(C_{31} \cdot \cos(T) + C_{33}\right)\]

where:

\[\begin{split}C_{31} &= \cos(\varphi) \cdot \cos(\delta) \\ C_{33} &= \sin(\varphi) \cdot \sin(\delta)\end{split}\]
Parameters

  • lat (float) – latitude, \(\varphi\) [rad]

  • decl (float) – declination, \(\delta\) [rad]

  • ha (float) – solar hour angle, \(T\) [rad]

Returns

h0 – solar elevation angle, \(h_0\) [degrees]

Return type

float

rayleigh_optical_thickness(m)

Computes the Rayleigh optical thickness at airmass \(m\). It is calculated according to the improved formula by Kasten4.

if \(m\) > 20:

\[\delta_{R}(m) = \left(6.6296 + 1.7513 \cdot m - 0.1202 \cdot m^{2} + 0.0065 \cdot m^{3} - 0.00013 \cdot m^{4}\right)^{-1}\]

if \(m\) < 20:

\[\delta_{R}(m) = \left(10.4+0.718 \cdot m\right)^{-1}\]
Parameters

m (float) – relative optical airmass, \(m\) [-]

Returns

rotm – Rayleigh optical thickness at airmass m, \(\delta_{R}\) [-]

Return type

float

beam_irradiance_normal_clear(G0, Tl2, m, rotm, h0)

Computes the clear sky beam irradiance normal to the solar beam.

\[B_{0c}=G_{0} \cdot \exp\left(-0.8662 \cdot T_{LK} \cdot m \cdot \delta_{R}\right)\]
Parameters

  • G0 (float) – ext rad normal to solar beam, \(G_0\) [W/m2]

  • Tl2 (float) – airmass 2 Linke atmospheric turbidity factor, \(T_{LK}\) [-]

  • m (float) – relative optical airmass, \(m\) [-]

  • rotm (float) – Rayleigh optical thickness at airmass m, \(\delta_{R}\) [-]

  • h0 (float) – solar elevation angle, \(h_0\) [degrees]

Returns

B0c – beam irradiance normal to the solar beam, \(B_{0c}\) [W/m2]

Return type

float

beam_irradiance_horizontal_clear(B0c, h0)

Computes the clear sky beam irradiance on a horizontal surface.

\[B_{hc} = B_{0c} \cdot \sin\left(h_{0}\right)\]
Parameters

  • B0c (float) – beam irradiance normal to the solar beam, \(B_{0c}\) [W/m2]

  • h0 (float) – solar elevation angle, \(h_0\) [degrees]

Returns

Bhc – beam irradiance at a horizontal surface, \(B_{hc}\) [W/m2]

Return type

float

linke_turbidity(wv_i, aod550_i, p_air_i, p_air_0_i)

Computes the air mass 2 Linke atmospheric turbidity factor.

\[\begin{split}p_{rel} &= \frac{p}{p_{0}} \\ T_{LK} &= 3.91 \cdot \tau_{550} \cdot e^{0.689p_{rel}}+0.376 \cdot \ln\left(TCWV\right)+\left(2+0.54 \cdot p_{rel}-0.34 \cdot p_{rel}^{2}\right)\end{split}\]
Parameters

  • wv_i (float) – total column atmospheric water vapor, \(TCWV\) [kg m-2]

  • aod550_i (float) – Aerosol optical depth at 550nm, \(aod550\) [-]

  • p_air_i (float) – actual instantaneous air pressure, \(p\) [hPa]

  • p_air_0_i (float) – air pressure at sea level, \(p_0\) [-]

Returns

Tl2 – Airmass 2 Linke atmospheric turbidity factor, \(T_{LK}\) [-]

Return type

float

Examples

diffuse_irradiance_horizontal_clear(G0, Tl2, h0)

Computes the clear sky beam irradiance on a horizontal surface.

\[D_{hc}=G_{0} \cdot Tn\left(T_{LK}\right) \cdot F_{d}\left(h_{0}\right)\]

For the estimation of the transmission function \(Tn\left(T_{LK}\right)\) the following function is used:

\[Tn\left(T_{LK}\right)=-0.015843+0.030543 \cdot T_{LK}+0.0003797 \cdot T_{LK}^{2}\]

The solar altitude function \(F_{d}\left(h_{0}\right)\) is evaluated using the expression:

\[F_{d}\left(h_{0}\right)=A_{1}+A_{2} \cdot \sin (h_{0})+A_{3} \cdot sin^{2}(h_{0})\]

with:

\[\begin{split}A_{1}^{\prime} &= 0.26463-0.061581 \cdot T_{LK}+0.0031408 \cdot T_{LK}^{2} \\ A_{1} &= \frac{0.0022}{Tn\left(T_{LK}\right)} \: \text{if} \: A_{1}^{\prime} \cdot Tn\left(T_{LK}\right)<0.0022 \\ A_{1} &= A_{1}^{\prime} \: \text{if} \: A_{1}^{\prime} \cdot Tn\left(T_{LK}\right)\geq0.0022 \\ A_{2} &= 2.04020+0.018945 \cdot T_{LK}-0.011161 \cdot T_{LK}^{2} \\ A_{3} &= -1.3025+0.039231 \cdot T_{LK}+0.0085079 \cdot T_{LK}^{2}\end{split}\]
Parameters

  • G0 (float) – ext rad normal to solar beam, \(G_0\) [W/m2]

  • Tl2 (float) – Airmass 2 Linke atmospheric turbidity factor, \(T_{LK}\) [-]

  • h0 (float) – solar elevation angle, \(h_0\) [degrees]

Returns

Dhc – Diffuse irradiance at a horizontal surface, \(D_{hc}\) [W/m2]

Return type

float

Examples

ra_clear_horizontal(Bhc, Dhc)

Computes the clear sky beam irradiance on a horizontal surface.

\[G_{hc}=B_{hc}+D_{hc}\]
Parameters

  • Bhc (float) – beam irradiance at a horizontal surface, \(B_{hc}\) [W/m2]

  • Dhc (float) – Diffuse irradiance at a horizontal surface, \(D_{hc}\) [W/m2]

Returns

ra_hor_clear_i – Total clear-sky irradiance on a horizontal surface, \(G_{hc}\) [W/m2]

Return type

float

Examples

1

Marcel Šúri and Jaroslav Hofierka. A new gis-based solar radiation model and its application to photovoltaic assessments. Transactions in GIS, 8(2):175–190, 2004. URL: https://doi.org/10.1111/j.1467-9671.2004.00174.x.

2

JW Gruter. Radiation nomenclature. brussels, cec. Second Solar Energy Programme, Project F, Solar Radiation Data, 1984.

3

Fritz Kasten and Andrew T Young. Revised optical air mass tables and approximation formula. Applied optics, 28(22):4735–4738, 1989. URL: https://doi.org/10.1364/AO.28.004735.

4

F Kasten. The linke turbidity factor based on improved values of the integral rayleigh optical thickness. Solar energy, 56(3):239–244, 1996. URL: https://doi.org/10.1016/0038-092X(95)00114-7.