# Indoor PV Modeling Based on the One-Diode Model
## 2 Methodology
### Parameter Extraction Technique  
The model is composed of five parameters: \(I_{ph}\), \(I_{o}\), \(a\), \(R_{s}\), and \(R_{p}\). These parameters can be retrieved using the parameter extraction technique described in [12]. To begin, the output current of the model is evaluated using Equation (3) under various operating conditions, including open-circuit, short-circuit, and maximum-power conditions. Under the open-circuit condition, \(V=V_{\alpha c}\) and \(I=0\). Hence, the output current can be written as:  
\[0=I_{ph}-I_{o}\left[\exp(\frac{qV_{oc}}{N_{s}akT})-1\right]-\frac{V_{oc}}{R_{p}} \tag{5}\]  
Under the short-circuit condition, \(V=0\) and \(I=I_{sc}\). The output current is:  
\[I_{sc}=I_{ph}-I_{o}\left[\exp(\frac{qR_{s}I_{sc}}{N_{s}akT})-1\right]-\frac{R _{s}I_{sc}}{R_{p}} \tag{6}\]  
Under the maximum-power condition, \(V=V_{m}\) and \(I=I_{m}\). The output current is:  
\[I_{m}=I_{ph}-I_{o}\left[\exp(\frac{q(V_{m}+I_{m}R_{s})}{N_{s}akT})-1\right]- \frac{V_{m}+I_{m}R_{s}}{R_{p}} \tag{7}\]  
where \(V_{oc}\) represents the open-circuit voltage, \(I_{sc}\) represents the short-circuit current, \(V_{m}\) represents the maximum-power voltage, and \(I_{m}\) represents the maximum-power current. By rearranging and simplifying the above equations, \(I_{ph}\) and \(I_{o}\) can be written as:  
\[I_{ph}=I_{o}\left[\exp(\frac{qV_{oc}}{N_{s}akT})-1\right]+\frac{V_{oc}}{R_{p}} \tag{8}\]  
\[I_{o}=\left(I_{sc}-\frac{V_{oc}-I_{sc}R_{s}}{R_{p}}\right)\exp(-\frac{qV_{oc} }{aN_{s}kT}) \tag{9}\]  
The authors of [12] included the number of parallel-connected PV cells (\(N_{p}\)) in Equation (3) to obtain a more precise expression for the parameters \(R_{s}\) and \(R_{p}\). The inclusion of \(N_{p}\) effectively scales down \(V\) to represent the voltage of a single solar cell. The accuracy improvement comes from the knowledge that the voltage of a single solar cell is within a limited range. As a result, the new equation for the model's output current is as follows:  
\[I=N_{p}I_{ph}-N_{p}I_{o}\left[\exp(\frac{q(VN_{p}+IR_{s}N_{s})}{N_{p}N_{s}akT })-1\right]-\frac{VN_{p}+IR_{s}N_{s}}{N_{s}R_{p}} \tag{10}\]  
To obtain the parameter \(R_{s}\), Equation (10) is assessed under short-circuit, open-circuit, and maximum-power conditions. The expression \((VN_{p}+IR_{s}N_{s})/(N_{p}N_{s}R_{p})\) can be omitted due to the low level of current flowing through the parallel resistor, yielding the following:  
\[R_{s}=\frac{akTN_{p}}{qI_{m}}\ln[\exp(\frac{qV_{oc}}{akTN_{s}})-\frac{I_{m}}{ I_{sc}}\left\{\exp(\frac{qV_{oc}}{akTN_{s}})\right\}]-\frac{V_{m}N_{p}}{I_{m}N_{s}} \tag{11}\]Furthermore, the formula for \(R_{p}\) can be obtained from the derivative of the current in relation to the voltage at the maximum-power point:  
\[\left(\frac{dI}{dV}\right)\Big{|}_{MPP}=-\frac{I_{m}}{V_{m}} \tag{12}\]  
Similarly, the short-circuit and open-circuit conditions in Equation (10) are substituted into Equation (12) to derive the parameter \(R_{p}\), as shown in Equation (13). The authors of [12] recommended increasing the value of parameter \(a\) from 0 until \(R_{p}\) achieves its minimal positive value. After determining the values of the five parameters, the \(R_{s}\) value can be increased iteratively to match the maximum power point, as proposed in [12]. This adjustment of \(R_{s}\) is performed to increase the model's accuracy. It is noted that the \(R_{s}\) value affects the fill factor and maximum-power point. The higher the \(R_{s}\) value, the lower the fill factor and maximum-power point. Based on the observation that the I-V curve of the PV panel under low indoor illuminance levels has a lower fill factor and maximum-power point compared to under outdoor conditions, the higher \(R_{s}\) value that matches the maximum-power point is able to produce a more accurate I-V curve. By applying this technique, the model's accuracy can be improved.