# Dependence of PV Module Temperature on Incident Time-Dependent Solar Spectrum
## 4 Proposed Spectrum Model-Combined Energy Balance Equation and Solar Spectrum Method  
The new approach of the present study to calculate the module operating temperature is based on the energy balance Equation (3) and the time-dependent solar spectrum. The irradiance \(P_{rad}\) (see Equation (5)) is determined by the solar spectrum, which varies with time during the day. The irradiance\(P_{rad}\) is calculated by:  
\[P_{rad}=\int P(\lambda)d\lambda \tag{7}\]  
and \(P_{rad}\tau\alpha\) (see Equation (5)) is the part of the irradiance absorbed by the PV cell given by:  
\[P_{rad}\tau\alpha=\int P(\lambda)\cdot(1-R_{cell}(\lambda))d\lambda \tag{8}\]  
The spectral reflectance coefficient of the PV cell is \(R_{cell}(\lambda)\) and, therefore, \((1-R_{cell}(\lambda))\) is the transmittance factor of the absorbed solar irradiance in the PV cell. The solar spectrum depends on time, location, and the atmospheric condition. The global solar radiation, neglecting the ground reflected radiation, consisting of the direct beam and the diffuse components. Therefore, the solar spectrum is divided into two spectrum models, one for the direct beam radiation and the other for the diffuse radiation. The range of wavelengths of the solar incident radiation on the PV modules corresponds to the relevant range of the spectral response of the PV cells. The effect of varying spectral irradiance on the performance of solar cells is addressed in [19; 20; 21]. The impact of solar spectral irradiance on the yield of different technologies was studied for four locations in [20]. The study in [22] states that the spectrum has little effect for low gap materials between seasons, for clear skies, but not for large band gap materials. Figure 1 presents the external quantum efficiency of four module technologies, for which the present study calculates the operating temperature. Figure 1 was derived from articles [19; 20; 21] while using interpolations to match the sampling points to specx points needed for the spectral model SPECTRAL(2), (see below). Several solar spectrum models were developed with time for clear skies, among them: LOWTRAN(2) [23], SPECTRAL(2) [24], SMARTS(2) [25], and REST(2) [26]. Some models present high spectral resolutions, however they require complex calculations. In the present article, the SPECTRAL(2) is used, because it conveniently fits in the computation program and is of sufficient spectral resolution. In addition, SPECTRAL(2) is listed in ([[http://rredc.nrel.gov/solar/pubs/spectral/model](http://rredc.nrel.gov/solar/pubs/spectral/model)]([http://rredc.nrel.gov/solar/pubs/spectral/model](http://rredc.nrel.gov/solar/pubs/spectral/model))), including parameters that are needed for the solar spectrum calculation (see also [27]), mentioning the advantage of SPECTRAL(2) on SMARTS(2)). SPECTRAL(2) was implemented by a single function and interfaced as described in Table 1. The parameters 1, 2, 3, 8 and 9 are easily obtainable, whereas the atmospheric data for the parameters 4-7 may be obtained for locations worldwide ([[http://aeronet.gsfc.nasa.gov](http://aeronet.gsfc.nasa.gov)]([http://aeronet.gsfc.nasa.gov](http://aeronet.gsfc.nasa.gov))). The output of spectral model is:  
specx--sampling points along the spectrum axis, dense for shorter waves and sparse for longer waves. specdir--direct beam solar irradiance (normal to solar rays). specdir--diffuse irradiance on horizontal plane. Two locations in Israel were chosen to demonstrate the solar spectrum for a given day and an hour--Sde Boker, 30.67\({}^{\prime}\)N (the desert) on February 1 at 16:00, and Eilat 29.55\({}^{\circ}\)N on August 4 at 12:00 noon. Table 2 contains the parameters for producing the solar spectrum at Sde Boker and Eilat. Figure 2 shows the results. A remark on the irradiances is in order. The direct beam irradiance is