## 3 Modeling of Non-Inverted DC-DC Buck-Boost Topology of Converter  
DC-DC converters are normally used to change the current and voltage level provided by photo voltaic modules that is required by electrical loads [34; 35; 36]. The non-inverted DC-DC buck-boost topology of converter [37; 38] increase/decrease the output voltage of the PV system in order to get the maximum power point voltage \(V_{mpp}\). To operate the system at \(V_{mpp}\), the converter is controlled periodically by changing its duty cycle with the help of controller proposed in the existing literature. The duty cycle of DC-DC converter is defined as \(u=t_{on}/T\), where \(T=t_{on}+t_{off}\) shows the total converter time period. Figure 5 depicts converter topology. In Figure 5, \(V_{pp}\) represents the input voltage coming from PV system [39]. The input capacitor \(C_{1}\) is used for limiting the ripples in input voltage of converters. On the other hand, output capacitor \(C_{2}\) limits the ripples in the converter's output voltage. \(D_{1}\)&\(D_{2}\) are diodes and \(S_{1}\)&\(S_{2}\) are Insulated Gate Bipolar Transistors (IGBT) switches. \(R\) and \(L\) represent the resistor and the inductor of converter, respectively [40]. Before moving into further developments of the converter, the following assumptions are made. * Diodes and switches are considered ideal, i.e., losses are negligible. * Converter operation is considered in continuous conduction mode (CCM). * CCM have two switching intervals in the. The assumption for the first interval is that two switches are turned on, diodes are operating in reverse biased, and the inductor is charging from PV voltage. Figure 4: Characteristic curves of P-V and I-V under varying irradiance. Figure 5: Non-inverted buck boost converter topology. Complete discussion of both switching interval is as follows. Applying [PERSON]'s laws for the first switching interval, the state space equations can be derived as  
\[\left[ \begin{array}{c}\frac{dV_{pe}}{dt}\\ \frac{dt}{dt}\\ \frac{dV_{C2}}{dt}\end{array} \right]=\left[ \begin{array}{ccc}0&-\frac{1}{C_ {1}}&0\\ \frac{1}{L}&0&0\\ 0&0&-\frac{1}{R_{L}C_{2}}\end{array} \right]\left[ \begin{array}{c}V_{pp}\\ i_{L}\\ V_{C_{2}}\end{array} \right]+\left[ \begin{array}{c}\frac{i_{pe}}{C_{1}}\\ 0\\ 0\end{array} \right] \tag{9}\]  
Similarly, apply [PERSON]'s laws to find second switching interval using he state space equations, where the load is taken current from the inductor, the diodes are operated in forward biased and both the switches are turned off. The state space equation for this interval is as follows  
\[\left[ \begin{array}{c}\frac{dV_{pe}}{dt}\\ \frac{dt_{L}}{dt}\\ \frac{dV_{C2}}{dt}\end{array} \right]=\left[ \begin{array}{ccc}0&0&0\\ 0&0&-\frac{1}{L}\\ 0&\frac{1}{C_{2}}&-\frac{1}{R_{L}C_{2}}\end{array} \right]\left[ \begin{array}[ ]{c}V_{pv}\\ i_{L}\\ V_{C_{2}}\end{array} \right]+\left[ \begin{array}{c}\frac{i_{pe}}{C_{1}}\\ 0\\ 0\end{array} \right] \tag{10}\]  
The mathematical expression for both switching intervals of non-inverted buck boost topology converter having a resistive load can be presented using the principles of capacitor charge balance and inductor volt-second balance as follows  
\[\left[ \begin{array}{c}\frac{dV_{pe}}{dt}\\ \frac{dt_{L}}{dt}\\ \frac{dV_{C2}}{dt}\end{array} \right]=\left[ \begin{array}{ccc}0&-\frac{\mu} {C_{1}}&0\\ \frac{\mu}{L}&0&\frac{\mu}{L}-\frac{1}{L}\\ 0&\frac{1}{C_{2}}-\frac{\mu}{C_{2}}&-\frac{1}{R_{L}C_{2}}\end{array} \right] \left[ \begin{array}{c}V_{pv}\\ i_{L}\\ V_{C_{2}}\end{array} \right]+\left[ \begin{array}{c}\frac{i_{pe}}{C_{1}}\\ 0\\ 0\end{array}