The main factor affecting the power output from a PV system is the absorbed solar radiation, S, on the PV surface. As was seen in Chapter 3, S depends on the incident radiation, air mass, and incident angle. As in the case of thermal collectors, when radiation data on the plane of the PV are unknown, it is necessary to estimate the absorbed solar radiation using the horizontal data and information on incidence angle. As in thermal collectors, the absorbed solar radiation includes the beam, diffuse, and ground-reflected components. In the case of PVs, however, a spectral effect is also included. Therefore, by assuming that the diffuse and ground-reflected radiation is isotropic, S can be obtained from (Duffie and Beckman, 2006):

2 It

where M = air mass modifier.

The air mass modifier, M, accounts for the absorption of radiation by species in the atmosphere, which cause the spectral content of the available solar radiation to change, thus altering the spectral distribution of the incident radiation and the generated electricity. An empirical relation that accounts for the changes in the spectral distribution resulting from changes in the air mass, m, from the reference air mass of 1.5 (at sea level) is given by the following empirical relation developed by King et al. (2004):

Constant a, values in Eq. (9.26) depend on the PV material, although for small zenith angles, less than about 70°, the differences are small (DeSoto et al., 2006). Table 9.4 gives the values of the a,- constants for various PV panels tested at the National Institute of Standards and Technology (NIST) (Fanney et al., 2002).

Cell type |
Silicon thin film |
Monocrystalline |
Polycrystalline |
Three-junction amorphous |

0.938110 |
0.935823 |
0.918093 |
1.10044085 | |

04 |
0.062191 |
0.054289 |
0.086257 |
-0.06142323 |

02 |
-0.015021 |
-0.008677 |
-0.024459 |
-0.00442732 |

03 |
0.001217 |
0.000527 |
0.002816 |
0.000631504 |

o4 |
-0.000034 |
-0.000011 |
-0.000126 |
-1.9184 X 10-5 |

As was seen in Chapter 2, Section 2.3.6, the air mass, m, is the ratio of the mass of air that the beam radiation has to traverse at any given time and location to the mass of air that the beam radiation would traverse if the sun were directly overhead. This can be given from Eq. (2.81) or from the following relation developed by King et al. (1998):

As the incidence angle increases, the amount of radiation reflected from the PV cover increases. Significant effects of inclination occur at incidence angles greater than 65°. The effect of reflection and absorption as a function of incidence angle is expressed in terms of the incidence angle modifier, Ke, defined as the ratio of the radiation absorbed by the cell at incidence angle e divided by the radiation absorbed by the cell at normal incidence. Therefore, in equation form, the incidence angle modifier at angle e is obtained by

It should be noted that the incidence angle depends on the PV panel slope, location, and time of the day. As in thermal collectors, separate incidence angle modifiers are required for the beam, diffuse, and ground-reflected radiation. For the diffuse and ground-reflected radiation, the effective incidence angle given by Eqs. (3.4) can be used. Although these equations were obtained for thermal collectors, they were found to give reasonable results for PV systems as well.

So, using the concept of incidence angle modifier and noting that

Eq. (9.25) can be written as 5 = (Ta)„M\0bRbK, b + GdK9, d [i+f^ ] + GpgK9, g ]}

It should be noted that, because the glazing is bonded to the cell surface, the incidence angle modifier of a PV panel differs slightly from that of a flat-plate collector and is obtained by combining the various equations presented in Chapter 2, Section 2.3.3:

(TO)e = )] ] i _ 1 sin2(9r - 9) + tan2(9r - 9) [ (9.30)

Cell type |
Silicon thin film |
Monocrystalline |
Polycrystalline |
Three-junction amorphous |

bo |
0.998980 |
1.000341 |
0.998515 |
1.001845 |

bi |
-0.006098 |
-0.005557 |
-0.012122 |
-0.005648 |

b2 |
8.117 X 10-4 |
6.553 X 10-4 |
1.440 X 10-3 |
7.250 X 10-4 |

b'i |
-3.376 X 10-5 |
-2.703 X 10-5 |
-5.576 X 10-5 |
-2.916 X 10-5 |

b4 |
5.647 X 10-7 |
4.641 X 10-7 |
8.779 X 10-7 |
4.696 X 10-7 |

b5 |
-3.371 X 10-9 |
-2.806 X 10-9 |
-4.919 X 10-9 |
-2.739 X 10-9 |

where 9 and 9r are the incidence angle and refraction angles (same as angles 0! and 02 in Section 2.3.3). A typical value of the extinction coefficient, K, for PV systems is 4 (for water white glass), glazing thickness is 2 mm, and the refractive index for glass is 1.526.

A simpler way to obtain the incidence angle modifier is given by King et al. (1998), who suggested the following equation:

Table 9.5 gives the values of the b constants for various PV panels tested at NIST (Fanney et al., 2002).

Therefore, Eq. (9.31) can be used directly for the specific type of cell to give the incidence angle modifier according to the incidence angle. Again, for the diffuse and ground-reflected radiation, the effective incidence angle given by Eq. (3.4) can be used.

A south-facing PV panel is installed at 30° in a location which is at 35°N latitude. If, on June 11 at noon, the beam radiation is 715 W/m2 and the diffuse radiation is 295 W/m2, both on a horizontal surface, estimate the absorbed solar radiation on the PV panel. The thickness of the glass cover on PV is 2 mm, the extinction coefficient K is 4 m-1, and ground reflectance is 0.2.

From Table 2.1, on June 11, 6 = 23.09°. First, the effective incidence angles need to be calculated. For the beam radiation, the incidence angle is required, estimated from Eq. (2.20):

cos (0) = sin(L - |3)sin(6) + cos(L - |3)cos(6)cos(h)

= sin(35 - 30)sin(23.09) + cos(35 - 30)cos(23.09)cos(0) = 0.951 or 0 = 18.1°

For the diffuse and ground-reflected components, Eq. (3.4) can be used:

0e,D = 59.68 - 0.1388(3 + 0.001497(2 = 59.68 - 0.1388(30) + 0.001497(30)2 = 56.7°

9e,g = 90 - 0.5788(3 + 0.002693(2 = 90 - 0.5788(30) + 0.002693(30)2 = 75.1°

Next, we need to estimate the three incidence angle modifiers. At an incidence angle of 18.1°, the refraction angle from Eq. (2.44) is sin (9r) = sin(9)/1.526 = sin(18.1)/1.526 = 0.204 or 9r = 11.75°

= e-[0.008/cos(11.75)] l1 - 1 [sin2(11.75 - 18.1) + tan2(11.75 - 18.1)

0.9487

At normal incidence, as shown in Chapter 2, Section 2.3.3, Eq. (2.49), the term in the square bracket of Eq. (9.30) is replaced with 1 - [(n - 1)/ (n + 1)]2. Therefore,

0.9490

(toQb

0.9487 0.9490

0.9997

For the diffuse radiation, sin (9r) = sin(9)/1.526 = sin(56.7)/1.526 = 0.5477 or 9r = 33.21°

0.9111

\1 _ 1 sin2(33.21 - 56.7) + tan2(33.21 - 56.7) I 2 sin2 (33.21 + 56.7) tan2(33.21 + 56.7)

0.9111 0.9490

Using Eq. (9.31) for monocrystalline cells gives Ke,D = 0.9679; and for poly-crystalline cells, Ke,D = 0.9674. Both values are close to the value just obtained, so even if the exact type of PV cell is not known, acceptable values can be obtained from Eq. (9.31) using either type of cell.

For the ground-reflected radiation, sin (9r) = sin(9)/1.526 = sin(75.1)/1.526 = 0.6333 or 9r = 39.29°

0.7325

2 sin2 (9r + 0e,G) tan2 (0r + 0e,G) f1 _ 1 sin2 (39.29 _ 75.1) + tan2(39.29 _ 75.1)

0.7325 0.9490

Using Eq. (9.31) for monocrystalline cells gives Ke,G = 0.7752, and for polycrystalline cells, Ke,G = 0.7665. Both values, again, are close to the value obtained previously.

For the estimation of the air mass, the zenith angle is required, obtained from Eq. (2.12):

= sin(35)sin(23.09) + cos(35)cos(23.09)cos(0) = 0.9785 or $ = 11.91°

The air mass is obtained from Eq. (9.27):

It should be noted that the same result is obtained using Eq. (2.81):

= 0.935823 + 0.054289 X (1.022) - 0.008677 X (1.022)2

+ 0.000527 X (1.022)3 - 0.000011 X (1.022)4 = 0.9828

R = cos(9) = cos(18.1) = 0 971 B cos($) cos(11.91) .

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