## 3 Mathematical Modeling and Design Methodology
### Modeling of a PV Array Output Power Generation
#### 3.1.3 TD Model  
The TD model, shown in Figure 6, of a PV panel takes into account both the diffusion and the recombination processes by including diodes \(D_{1}\) and \(D_{2}\)[35]. The first diode (\(D_{1}\)) simulates the diffusion process of the minority carriers into the depletion layer, while diode (\(D_{2}\)) represents the recombination in the space charge region of the P-N junction [36]. The TD model is a modified and improved version of the SD circuit. The SD circuit did not take into account the effect of recombination as shown in Figure 5. Many RES studies use the SD model for a PV system design [2; 4; 21; 29; 30]. However, the TD model is more realistic and accurate, because it takes into account both the diffusion effects and recombination losses. This results in a more precise and realistic mathematical description of a solar panel and leads to a correct size for the PV array and also the size, ecological impact, and reliability of the entire system [14; 15; 16]. After adding a second diode (\(D_{2}\)) in parallel with the first one (\(D_{1}\)), Equation (2) becomes as shown in Equation (3) [37; 38]. The values of \(I_{D1}\), \(I_{D2}\), and \(I_{Sh}\) are computed as in [12; 16; 37]. Thus, the output PV current for the TD model is updated to be as in Equation (3):  
\[I=I_{Ph}-I_{o_{1}}\left(e^{\frac{V+IR_{h}}{a_{1}\,V_{T}}}-1\right)-I_{o_{2}} \left(e^{\frac{V+IR_{h}}{a_{2}\,V_{T}}}-1\right)-\frac{V+IR_{s}}{R_{sh}} \tag{3}\]  
The optically generated current of the PV module is linearly dependent on the solar irradiance and is also influenced by temperature, according to the \(I_{Ph}\) Equation found in [6; 14; 38]. The hourly measured values of the solar irradiance and ambient temperature are depicted in Figure 2.  
The diode (\(D_{1}\) or \(D_{2}\)) saturation currents \(I_{01\ or\ 2}\) depend on the temperature and can be expressed as shown in [16; 37]. Some research studies provide another approach to get the value of the saturation current of \(D_{1}\) or \(D_{2}\) in the TD model. In fact, the diode saturation current is very small and to simplify the model, in these studies, both of the reverse saturation currents \(I_{o_{1}}\) and \(I_{o_{2}}\) were set equal as in [16; 37; 38]. The diode ideality factors \(\alpha_{1}\) and \(\alpha_{2}\) represent the diffusion effects and recombination losses. The recombination has an impact on both the short circuit current and open circuit voltage. The recombination region should be minimized in order to ensure that the p-n junction is convenient to collect the light generated current, and to minimize the losses in the current due to recombination. So, the recombination is a negative effect, and thus, it is a loss. Regarding the diffusion effect, the carrier must be generated within a diffusion length of the junction, so that it will be able to diffuse to the junction before recombining. When a carrier exists with a concentration in the semiconductor, it will move from the higher to the lower concentration area in the diffusion process. The impact of diffusion on the solar cell, will increase the carrier concentrations in a device with time. Additionally, a low diffusion length means that minority carriers disappear from the junction edge quickly due to recombination, thus allowing more carriers to cross and increasing the forward bias current. Consequently, to minimize recombination and achieve a high voltage, a high diffusion length is required. So, the diffusion and  
Figure 6: Equivalent circuit of a PV panel using the TD model. recombination effects are interrelated. However, the aforementioned models (i.e., SD and ISD), do not take into account the recombination losses as the TD model does, which is a physical phenomenon. In accordance with [PERSON]'s diffusion theory, \(\alpha_{1}\) is unity [14; 37; 39]. However, the value of \(\alpha_{2}\) is flexible. It was found that if (\(1.2\leq\alpha_{2}\leq 2\)), the best match between their proposed model and the practical I-V curve is obtained.