## 3 Proposed Td-MPPT Method
Similarly to the rapidly changing irradiance scenario, it can be assumed that the perturbation was made in the correct direction if \(dp_{1}>dp_{2}\) under steady irradiance, i.e., MPP is beyond \(v_{k}\) and MPPT should keep perturbating in the same direction to reach it. On the contrary, if \(dp_{1}<dp_{2}\), then the proposed MPPT can assume that \(v_{k}\) is closer to MPP than \(v_{x}\) so the next perturbation needs to be performed in the opposite direction. Td-MPPT would then reduce steady-state oscillations by alternating between two operating points around the MPP instead of three. The flowchart of the overall Td-MPPT is shown in Figure 5. It becomes evident that \(T_{x}\) and \(T_{MPPT}\) need to be determined to tune the Td-MPPT algorithm properly. To simplify the required computations, \(T_{MPPT}\) should be selected as a multiple of \(T_{x}\). Traditional MPPT algorithms might present erratic behavior caused by transient states if they perturb the operating point too quickly. A safe approach would be to use a \(T_{MPPT}\) that is considerably larger than the natural oscillation period of the system. For a DC-DC boost converter (4) with capacitance (C) and inductance (\(L\)), \(T_{MPPT}\) can be selected to satisfy the following. \[T_{MPPT}\gg 2\pi\sqrt{LC}. \tag{5}\]  
However, if the perturbation time is too slow, the MPPT cannot keep up with fast irradiation changes. The authors in [32; 33; 34] have carried out an analysis on the dynamic behavior of PV systems and their converters to optimize \(T_{MPPT}\) and improve the dynamic response of traditional MPPT systems. In addition to optimizing \(T_{MPPT}\) for the proposed algorithm, an analysis of the system's transient response would allow choosing the op  
Figure 2.0 PV system voltage and power transient response after perturbation around the MPP under steady irradiance. imum time to observe the maximum overshoot caused by the step perturbation. The research in [31] showed that the transient behavior of PV systems changes depending on the operating region of the system, i.e., constant current region (CCR), constant power region (CPR), and constant voltage region (CVR). When analyzed as a small signal equivalent circuit, the simulated converter (Figure 6) contains two independent inputs: the control input (\(\hat{d}\)) and the load current (\(\hat{i}\)). The variables with hats represent the small-signal changes around steady-state values. Hence, for an ideal boost converter, the voltage output (\(\hat{\sigma}\)) can be expressed as a superposition of these to find the following: the small control signal to array voltage transfer function (\(G_{\text{\tiny{D}}}\)) and the load current to array voltage output transfer function (\(G_{\text{\tiny{V}}}\)) as follows:  
Figure 2.1  
where the following is the case. Figure 2.2  
Figure 2.3  
Small duty-cycle changes can be expressed as follows. Figure 2.4  
For this application (Figure 6), the steady-state value of the duty cycle (\(D\)) is determined by the relationship between the voltage reference defined by the MPPT algorithm (\(V_{\text{\tiny{ref}}}\)), and the output voltage of the power supply (\(V_{f}\)) as follows  
Figure 2.5  
Transfer function \(G_{\text{\tiny{V}}}\) is relevant for applications such as grid-tied systems since it models interfacing converter effects on dynamic behavior [33; 34]. However, for an application with no variation in the load current, transfer function \(G_{\text{\tiny{V}}}\) is equal to 0. For an ideal boost converter, \(R_{in}=R(1-D)^{2}\); thus, \(G_{v}\) assumes the following:  
Figure 2.6  
where \(\mu=-V_{O}\), \(\omega_{n}\approx\ 1/\sqrt{L\ C}\) and \(\xi\approx\sqrt{L\ C}/(2\ R_{in})\). It must be noted that \(R_{in}\) denotes the ratio between the PV panel voltage and current, and it depends on each panel's characteristics, the operating point (CCR, CPR, or CVR), and external conditions such as temperature and irradiance [31].