## 2 Issues in the Conventional P and O Algorithm  
The objective of the P and O algorithm is to shift the operation point (V, P) toward the MPP by continuously perturbing the PV voltage or PV current with a fixed step size. To illustrate the nonlinear characteristics of solar cells, Figures 1 and 2 present current versus voltage (I-V), and power versus voltage (P-V) curves of a commercial solar panel, SW-260-mono [24] operating at different irradiation conditions. Figure 1: Ideal I-V curve. Figure 2: P-V curves of SW-260-mono operating. In Figure 3, P\({}_{1}\) through P\({}_{3}\) are three consecutive locations of the PV operation point, and shifting of the operation point is conducted by the MPPT algorithm. S\({}_{1}\) through S\({}_{6}\) are six possibilities that can be observed in the tracking progress of the traditional P and O. For example, the case that S\({}_{1}\) consists of several steps: (1) the PV voltage increases; (2) the operation point moves from P\({}_{1}\) to P\({}_{2}\); (3) an increase in the PV power is observed; (4) the PV voltage keeps increasing because the direction of the previous perturbation results in an increase in PV power; (5) the operation point moves from P\({}_{2}\) to P\({}_{3}\). As shown in the case S\({}_{3}\), a decrease in PV voltage can lead the PV power to increase so that the PV voltage keeps decreasing. Sometimes, the direction of perturbation iteratively changes, especially whenever the operation point is located within the region near the real MPP such as S\({}_{4}\)-S\({}_{6}\). For instance, in S\({}_{4}\) in Figure 3, the decrease in the PV power causes that the operation point moves from P\({}_{1}\) to P\({}_{2}\), and a decrease in PV power can be observed. To increase the PV power, then the direction of the following perturbation (P\({}_{2}\) to P\({}_{3}\)) should be inversed. For the transitions from S\({}_{4}\) to S\({}_{6}\) and the transitions from S\({}_{6}\) to S\({}_{4}\), the PV power will periodically increase and decrease. And this is the reason why power oscillations can be observed in a PV system. Power points around MPP can generate less power, compared to the exact MPP. Plus, the direction of perturbation for the operation point will be inversed as long as decreases in power are observed. Hence, it could be predicted that the operation point will move among three reachable power points which are near MPP, as shown in Figure 4. In Figure 4, the traces of the shifting of the operation point can be concluded as two cases: P\({}_{1}\)\(\rightarrow\)P\({}_{2}\)\(\rightarrow\)P\({}_{3}\) and P\({}_{3}\)\(\rightarrow\)P\({}_{2}\)\(\rightarrow\)P\({}_{1}\). It is worthwhile to note that in a real PV system, P\({}_{2}\) may not always exactly overlay MPP due to many imperfect factors such as resolutions of the switching signal and steady-state error of the voltage regulation module. In the steady state of a traditional P and O controlled PV system, the above two cases about the shifting of PV operation point will endlessly circulate so that the steady state power oscillation cannot be avoided. Fortunately, the magnitude of such steady state power oscillation can be suppressed if the step size of the perturbation of the operation point is ideally small, and is close to zero. The conventional P and O and INC algorithms using fixed step-size (\(\Delta\)V) could waste time and energy in the low power region in which the operation point is far from the real MPP. In the simulation results shown in Figure 5, the real maximum power point voltage (\(V_{mpp}\)) and maximum power point power (\(P_{mpp}\)) are 27.03 V and 98.23 W, respectively. Different fixed step-sizes for PV voltage perturbation result in different tracking performance in terms of the tracking time and steady-state power oscillation.