# An Evolutionary-Based MPPT Algorithm for Photovoltaic Systems under Dynamic Partial Shading
## 3 Test Case Modeling  
### Test Case PV System Model  
The test case PV system taken into account for simulating the MPPT algorithms is a simple off grid PV system that consists of a PV module, a boost DC/DC converter, a load resistor, and a MPPT controller. Figure 4 shows the diagram of the simulated off grid PV system. This model represents the MPPT test bench available at the SolarTechLab of the Department of Energy at Politecnico di Milano [24], which was used for a preliminary experimental test campaign carried out on 27 February and 3 March 2018. The MPPT algorithm controls the PV generator, regardless of the kind and the complexity of the PV system. Since the goal of this work is to investigate, by means of simulations, the capability of the MPPT controller to find the GMPP under partial and dynamic shading conditions, this simple off-grid PV system represents a general case suitable for this kind of research. The PV module taken into account in this work is a monocrystalline S19.285 produced by Aleo Solar, whose ratings are reported in Table 1. It is made of 60 cells connected in series and three bypass diodes parallel connected to groups of 20 cells, which divide the PV module in three equal electrical sections. An active non-linear and time-varying element models the PV module. The PV module I-V curve is calculated combining the I-V curves of each cell and the I-V curves of each bypass diode in order to comply with the series and parallel electrical connection constraints. The I-V curve of each cell was calculated by using the five parameter mathematical model [26], whose equivalent circuit is reported in Figure 5. The five parameters that characterize this model are the light generated current (\(I_{PV}\)), the leakage or reverse saturation current (\(I_{0}\)), the diode quality factor (\(n\)), the series resistance (\(R_{s}\)) and the shunt resistance (\(R_{sh}\)). Referring to the equivalent electric circuit in Figure 5, the I-V curve of a PV cell can be expressed based on [PERSON]'s current law, [PERSON]'s law, and the Shockley diode equation:  
\[I(G,T_{C})=I_{PV}(G,T_{C})-I_{0}(T_{C})\cdot\left(e^{\frac{V+R_{s}\cdot I(G,T_ {C})}{n\cdot V_{t}}}-1\right)-\frac{V+R_{S}\cdot I(G,T_{C})}{R_{SH}(G)} \tag{6}\]  
The light generated current as a function of irradiance, \(G\), and cell temperature, \(T_{C}\), can be expressed as:  
\[I_{PV}(G,T_{C})=I_{PV,ref}\cdot\frac{G}{G_{ref}}\cdot\frac{S_{unshaded}}{S_{ cell}}\cdot\left(1+\alpha_{I_{SC}}\cdot\left(T_{C}-T_{ref}\right)\right) \tag{7}\]  
where the subscript _ref_ stands for reference conditions, \(G\) is the irradiance, \(T_{C}\) is the cell temperature and \(\alpha_{\text{ISC}}\) is the temperature coefficient for short-circuit current. In most cases, reference values are measured at standard test conditions (STC), that is with \(G_{ref}\) equal to 1000 W/m\({}^{2}\), cell temperature equal to 25 \({}^{\circ}\)C and Air Mass equal to 1.5. In case of partial shading on a cell, its light generated current is directly proportional to the ratio of the unshaded area of the cell, \(S_{unshaded}\), and its total surface, \(S_{cell}\)  
Figure 4: Diagram of the simulated off grid photovoltaic (PV) system. The unshaded area of each cell is calculated according to the shape of the shading object and its speed and position. The diode reverse saturation current can be expressed as:  
\[I_{0}(T_{\mathrm{C}})=I_{0,ref}\cdot\left(\frac{T_{C}}{T_{ref}}\right)^{3}\cdot e ^{\frac{E_{g}(T_{BFE})}{k+T_{ref}}}-\frac{E_{g}(T_{C})}{k+\cdot C} \tag{8}\]  
where \(k\) is the Boltzmann constant (8.6173324\(\cdot\)10\({}^{-5}\) eV\(\cdot\)K\({}^{-1}\)) and \(E_{g}\) is the bandgap energy of the silicon, that is temperature dependent and it is given, in eV, as:  
\[E_{g}(T)=1.17-4.73\cdot 10^{-4}\frac{T^{2}}{T+636} \tag{9}\]  
The shunt resistance changes with absorbed solar radiation, and different equations can be found in literature. In this work, inversely proportional dependence of shunt resistance with irradiance is taken into account.