## 2 System Design and Configuration
### Design and Control of the SEPIC  
The SEPIC is a versatile DC-DC converter that can step up, step down, or maintain the input voltage level depending on the operating conditions. The converter efficiently handles a broad range of input voltages, making it ideal for MPPT in PV systems. The design of the SEPIC is carried out considering operation in continuous conduction mode (CCM). As depicted in Figure 1, to achieve a non-inverted SEPIC configuration, the positions of the output inductor and the diode in the Cuk converter are reversed. With the non-inverted output, the output voltage maintains the same polarity as the input, simplifying integration with downstream components. The design also includes inductors on both the input and output sides, which minimize current ripple and ensure smooth power delivery. The design of passive components for the converter are described in the following. The design of these components is carried out based on the optimal power point. The duty cycle (\(D\)) for \(V_{dc}\) = 48 V and \(V_{mpp}\) = 58 V is determined at \(D\) = 0.45 following Equation (4). The input inductor current is \(i_{L1}\) = \(i_{mpp}\) = 7.24 A. The current through the DC bus, \(i_{dc}\), is defined as per Equation (5) considering the peak power capability \(P_{dc}\) = 1.25 kW, achieving \(i_{dc}\) = 26.04 A.  
\[D=\frac{V_{dc}}{V_{dc}+V_{mpp}} \tag{4}\]  
\[i_{dc}=\frac{P_{dc}}{V_{dc}} \tag{5}\]  
Figure 2: Mechanism for the switching pulses to the VSI. In the proposed design, the ripple current is shared equally between the two inductors, effectively halving the required inductance (\(L_{1}=L_{2}\)). The inductors are sized according to Equation (6), calculated at \(L_{1}=L_{2}=0.90\) mH, ensuring CCM while minimizing current ripple. For optimal cost and size, the switching frequency \(f_{sw}\) is set to 20 kHz, with a maximum current ripple of 10% (\(\Delta i_{L1}=0.1\cdot i_{L1}\)). \[L_{1}=L_{2}=\frac{1}{2}.\frac{D\cdot V_{mpp}}{f_{sw}\Delta i_{L1}} \tag{6}\]  
The flying or coupling capacitor \(C_{1}\) facilitates energy transfer and protects against short-circuits in the load. The capacitor reduces the voltage ripple. Its value is estimated at \(C_{1}=97.64\)\(\mu\)F following Equation (7), where \(V_{C1}=V_{dc}=48\) V and the maximum allowed voltage ripple is set at 10% (\(\Delta V_{C1}=0.1\cdot V_{C1}\)). A ceramic capacitor with a low equivalent series resistance (ESR) value is preferred for this application. \[C_{1}=\frac{D\cdot i_{dc}}{f_{sw}\cdot\Delta V_{C1}} \tag{7}\]  
Finally, the diode prevents reverse current flow, and a metal-oxide-semiconductor field-effect transistor (MOSFET) is used as the switching device controlled by a PWM signal, as developed hereafter. The SEPIC operates by regulating the duty cycle \(D\) of its switching element to control the input voltage (\(V_{pv}\)) seen by the PV panel. The input voltage of the SEPIC directly determines the operating point of the PV panel. The relationship between \(V_{pv}\) and the duty cycle is given by Equation (8):  
\[V_{pv}=V_{dc}.\frac{1-D}{D} \tag{8}\]  
By adjusting \(D\), the converter changes \(V_{pv}\), which shifts the operating point of the PV panel along its voltage-current characteristics. The MPPT controller determines the optimal \(D\) to achieve \(V_{mpp}\), thus achieving maximum solar power extraction, i.e., the MPPT algorithm monitors \(V_{pv}\) and \(i_{pv}\) to calculate the extracted power, and consequently adjusts \(D\) to move the operating point of the PV panel toward the maximum power point. The duty cycle is incremented and decremented iteratively to achieve maximum power. The optimal operating point is reached when the derivative of power with respect to voltage is zero, and the duty cycle is then stabilized. If \(V_{pv}<V_{dc}\), the converter works on step-up operation (higher \(D\)), while if \(V_{pv}>V_{dc}\), the converter works on step-down operation (lower \(D\)).