# Solar Power System Assessments Using ANN and Hybrid Boost Converter Based MPPT Algorithm
## 3 Analytical Approach
### Analytical Modeling of a Traditional Boost Converter  
The electrical MPPT can be achieved through a DC to DC converter inserted between the photovoltaic module and the load.The DC to DC converter ensures the matching of load resistance with the varying source resistance. The converter is basically used to transfer maximum energy from source to load.  
The framework of traditional boost converter is represented in Figure 4 which transfers maximum energy by adjusting the PV module output voltage to the reference voltage. The traditional boost converter has two controllable variables: Voltage \(V_{pu}\) and the induc  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name of Parameter** & **Description** \\ \hline \(i_{ideal}\) & Diode Current \\ \hline \(i_{ac}\) & Short Circuit Current \\ \hline \(i_{o}\) & Reserve Saturation Current \\ \hline \(i_{ac}^{M}\) & Short circuit current under standard condition \\ \hline \(V_{ac}^{m}\) & open voltage under standard condition \\ \hline \(V_{T}\) & Thermal voltage in the semiconductor \\ \hline \(P_{max}\) & Power under standard condition \\ \hline \(N_{s}\) & Cell in series in module \\ \hline \(N_{p}\) & parallel branches in module \\ \hline \(N_{n}\) & module in series in an array \\ \hline \(N_{B}\) & parallel branches in array \\ \hline \(T_{a}\) & Ambient temperature \\ \hline \(T_{c}\) & Cell temperature \\ \hline \(G_{a}\) & Irradiance \\ \hline \(K\) & Boltzmann Constant \\ \hline FF & Fill Factor \\ \hline \(D\) & Duty Cycle \\ \hline \(T_{ON}\) & On time of switch \\ \hline \(V_{pu}^{*}\) & Reference Voltage \\ \hline \(i_{\overline{i}}\) & Reference inductor current \\ \hline \(m^{*}\) & Optimal duty cycle \\ \hline \(L_{1,2}\) & Inductor 1 and 2 \\ \hline \(C_{IN}\) & Capacitance \\ \hline \(I_{LAW}\) & Inductor Average Current \\ \hline \(V_{OUT}\) & Output Voltage \\ \hline \end{tabular}
\end{table}
Table 2: List of symbols used for analytical model.  
Figure 3: Equivalent circuit of photovoltaic cell.  
tor current \(i_{L}\). The mathematical model that describe the voltage and inductor current relation is  
\[\begin{cases}V_{m}\\ i_{L}\end{cases}=(1-D)\begin{cases}V_{m}\\ i_{L}\end{cases} \tag{19}\]  
where \(D\) is the duty cycle of the boost converter, which is expressed as  
\[D=\frac{T_{on}}{T_{total}}=T_{on}f_{s} \tag{20}\]  
where \(f_{s}\) is the frequency of switching of converter's switch and \(T_{total}\) is the total time and \(T_{ON}\) is the on time of the switch.  
\[\frac{dI_{L}}{T}=\frac{1}{L_{pu}}(V_{m}-V_{pu})-\frac{P_{pu}}{L_{pu}} \tag{21}\]  
where \(V_{pu}\) is the reference voltage generated by the ANN based MPPT.  
\[\frac{dV_{pu}}{dt}=\frac{(I_{L}-I_{pu})}{Pu} \tag{22}\]  
By using Equation (21) we obtain the reference inductor current  
\[i_{L}^{*}=(V_{pu}^{*}-V_{pu})+I_{pu} \tag{23}\]  
The optimal switching voltage can be expressed by using the Equations (22) and (23)  
\[V_{m}^{*}=PI(V_{pu}^{*}-V_{pu})+Vpu+\frac{R_{pu}}{L_{pu}}I_{L} \tag{24}\]  
while the boost converter command is obtained by the conversion of Equation (19)  
\[D^{*}=1-\frac{V_{m}^{*}}{V_{DC}} \tag{25}\]  
The DC link capacitor at the output play a key role to ensure energy balance between the PV module and the power injected into the system. The DC link capacitor charge and discharge that oscillate between two levels depending on actual weather condition and power injection.