## 3 The Proposed MPPT Algorithm  
The power \(P_{Grid}\) injected by the interlaced converter in the electric grid can be linked easily to the power absorbed from the PV array plus LC filter \(P_{DC}\) by neglecting the Joule or switching the losses of  
Figure 1: Topology of the grid connected photovoltaic generation system via an interleaved inverter. the converters and by imposing that the currents are uniformly distributed between the channels of the interleaved converter:  
\[P_{DC}=P_{Grid}+\frac{1}{2}L_{R}\sum_{k=1}^{3}i_{s,k}\frac{\mathrm{d}i_{s,k}}{ \mathrm{d}t}=\sum_{k=1}^{3}v_{s,k}i_{s,k}+\frac{1}{2}L_{R}\sum_{k=1}^{3}i_{s,k} \frac{\mathrm{d}i_{s,k}}{\mathrm{d}t}\cong\sum_{k=1}^{3}v_{s,k}i_{s,k}, \tag{1}\]  
where the magnetizing power of the circulating inductors has been neglected with respect to the grid power, because, by choosing proper values for the inductance and by taking into account that the current gradients are strongly limited by the considered application, it results in \(L_{R}\mathrm{d}i_{s,k}/\mathrm{d}t<<v_{s,k}\). From (1), it is clear that the \(P_{DC}\) can be imposed by properly controlling the line currents. Therefore, the system, composed of the controlled interleaved converter plus the electric grid, can be replaced by a controlled current generator for which the current value is linked to \(P_{Grid}\):  
\[i_{c}=\frac{P_{Grid}}{v_{c}}, \tag{2}\]  
Indeed, in order to discuss the proposed MPPT algorithm in detail, in Figure 2a, the topology of the system is proposed in a simplified manner. The inverter is substituted by an ideal current generator and the PV array is represented by an ideal voltage generator connected in series with an internal resistance. The values of \(E\) and \(R\) depend on the actual PV operating point following the characteristic of Figure 2b. In other words, the real PV characteristic is approximated by a linearization process identifying three different linear laws. The quantities \(v_{mpp}\) and \(i_{mpp}\) refer to the PV maximum power point, while \(v_{\infty}\) and \(i_{sc}\) are the PV open circuit voltage and the short circuit current respectively. The MPPT algorithm is based on the capacitance voltage measurements and, in particular, on its time derivative \(\mathrm{d}v_{c}/\mathrm{d}t\). Starting from zero, the power requested by the inverter increases linearly with time with a prefixed slope. The power slope can be fixed such that the system reaches its rated value in a prefixed time (\(t_{StartUp}\)):  
\[p_{slope}=\frac{P_{MPP}}{t_{StartUp}}, \tag{3}\]  
As it is seen further, the LC filter is sized such that the inductor voltage drop and the capacitance current absorption are negligible with respect to their rated values and, as a consequence, the PV current and voltage time derivatives only depend on the inverter power slope via the PV characteristic. This ensures that the time derivative of the capacitance voltage is limited to the PV characteristics (see Figure 2b). During the time when the requested power linearly increases, the PV operating point of Figure 2b moves from left toward right. Before the MPP is reached, \(\mathrm{d}v_{c}/\mathrm{d}t\) is negative and its absolute value is limited by the zone 2 of the PV characteristic. When the power requested by the inverter gets higher than the MPP, the PV power does not increase any further. Indeed, the inductor keeps the PV current stable to a value very close to the MPP and the PV operating point moves slightly at the right side of MPP. The inverter requested power keeps increasing and the extra power absorption is taken from  
Figure 2: Simplified grid connected PV system: **(a)** linear circuit; **(b)** linearized PV characteristic. the capacitor, whose voltage time derivative keeps decreasing as the difference between the inverter current and the PV current keeps increasing. This phenomenon testifies that the MPP has been reached and is being overcome.