# Online Identification of Photovoltaic Source Parameters by Using a Genetic Algorithm
## 2 The Optimized SDM Parameter Identification Method Based on Genetic Algorithm
### GA Boundary Constraint Definition  
In general, the genes assigned to the individuals of a GA population are constrained in ranges depending on the problem to be solved. As explained in [12] and related references, for each SDM parameter, the bounds have been selected by considering the physical constraints, applied to the equivalent electrical circuit of the photovoltaic source shown in Figure 1, as well as typical values found in the literature where experimental tests have been performed to put into evidence the variations of those parameters for the different PV technologies and environmental conditions.  
The obtained bounds are shown in Table 1. It is worth noting that, if no further information is available, the GA research space is only confined by such bounds.  
Figure 2: Test points’ selection (TP) for calculating the fitness function  
The nonlinear equation representing the I-V photovoltaic curve is strongly sensitive to the parameters' variation; considering that the number of parameter combinations is high, it is very difficult to find the optimal solution when the research space is very large and the multimodality of the objective function increases the probability to be trapped in a local optimum. However, since the operating conditions and the PV technology significantly affect the SDM parameters, this information can be properly exploited to reduce the research domain with respect to that expressed by the physical bounds, thus improving the robustness of the GA to converge towards the global optimum and consequently to enhance the performance of the on-line SDM parameters' identification method. This concept is qualitatively explained with the help of Figure 3 for a bi-dimensional case (i.e., for two generic parameters \(P1\) and \(P2\)).  
The approach used in this paper adopts a guess solution not only to initialize the genes of one individual of the population, but also to preliminarily detect the region of the physical research domain where it is highly probable to find the best solution. Then, the GA evolves by searching the optimal solution only in this region. In particular, every time the SDM parameter identification method is activated, the guess solution \(\mathbf{P_{guess}}=[I_{phg},I_{sg},\eta_{g},R_{sg},R_{hg}]\) is preliminarily calculated, and the new boundary conditions of the GA research space are computed as shown in Table 2.  
The real bounds for the parameters having large variations (\(I_{s},R_{s},R_{h}\)) are at least one order of magnitude higher and lower with respect to the guess solution values. In this way, a wide enough range is provided to account for any error in the guess solution, which is computed using approximated explicit equations, as discussed in the next section.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \(I_{ph}\) **(A)** & \(I_{s}\) **(A)** & \(\eta\) & \(R_{s}\) **(O)** & \(R_{h}\) **(O)** \\ \hline Lower Bound & 0 & \(10^{-10}\) & 0.5 & 0.01 & 10 \\ Upper Bound & 1.5 \(I_{sc,max}\) & \(10^{-2}\) & 5 & 10 & \(100,000\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Single-diode model (SDM) parameters physical bounds.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \(I_{ph}\) **(A)** & \(I_{s}\) **(A)** & \(\eta\) & \(R_{s}\) **(O)** & \(R_{h}\) **(O)** \\ \hline Lower Bound & 0.9 \(\cdot\)\(I_{phg}\) & 0.01 \(\cdot\)\(I_{sg}\) & max (0.5, \(\eta_{g}-1\)) & max (0.01, \(0.1\cdot R_{sg}\)) & max (10, \(0.1\cdot R_{hg}\)) \\ Upper Bound & 1.1 \(\cdot\)\(I_{phg}\) & 100 \(\cdot\)\(I_{sg}\) & min (5, \(\eta_{g}+1\)) & min (10, \(10\cdot R_{sg}\)) & min (105, \(100\cdot R_{hg}\)) \\ \hline \hline \end{tabular}
\end{table}
Table 2: SDM parameters’ real boundary conditions.  
Figure 3: The difference of physical boundary conditions (continuous box) and real boundary conditions (dashed boxes).