## 3 Proposed Td-MPPT Method  
The idea behind the proposed method is that MPP can be tracked better if there is more information about the PV-curve. Traditional MPPT algorithms perturb every update interval (\(T_{MPPT}\)) and either wait for the system to reach a steady-state or filter natural oscillations before registering power changes. Nevertheless, algorithms can enrich the information they rely on by sampling transitory states. However, implementation complexity increases if too many points are registered. By selecting key measurement points, enough data to provide an effective MPP tracking can be gathered while preserving a relatively simple algorithm. The method presented in this investigation features an additional measurement stage during the transient state to the conventional P&O (\(p_{x}\)). The added measurement is registered during the overshoot produced after the perturbation made by MPPT, as observed in (Figure 3). This improvement prevents the drift and oscillation characteristic of the P&O algorithm while maintaining its relative simplicity. Figure 2: Classical drifting behavior of P&O around MPP under increasing irradiance, as reported in the literature. Figure 3 shows the proposed algorithm influence on power and voltage under the steadily increasing irradiance scenario previously shown in Figure 2. By registering three different operation points, the proposed algorithm calculates two differentials of power for each perturbation period as follows. \[dp_{1}=\frac{p_{x}\ -p_{k-1}}{T_{x}} \tag{2}\]  
\[dp_{2}=\frac{p_{k}\ -p_{x}}{T_{MPPT}-T_{x}}\]  
During the first perturbation period, \(dp_{1\,k-1}\) and \(dp_{2\,k-1}\) are calculated. It can be observed that \(dp_{1\,k-1}\) is drastically reduced when the perturbation introduced by the algorithm drives the operating point away from the MPP even under rapidly changing irradiance scenarios. In contrast, \(dp_{2\,k-1}\) depends much more on the effects of irradiance changes. For this case, it can be stated that \(dp_{2\,k-1}>dp_{1\,k-1}\) when the perturbation moved the system away from the MPP. Then, Td-MPPT executes the next perturbation in the opposite direction. In contrast, the perturbation performed at \(p_{k-1}\) brings the power closer to MPP and the following is the case:  
\[dp_{1}\ >dp_{2} \tag{3}\]  
when the perturbation is performed in the appropriate direction, assuming that the change rate in irradiance remains constant during a sample period. To reduce computational burden, \(T_{MPPT}\) can be selected as an integer multiple of the overshoot measuring time \(T_{x}\); thus, condition (3) can be expressed as follows. \[(p_{x}\ -p_{k-1})\cdot(\alpha-1)>p_{k}\ -p_{x}. \tag{4}\]  
\[\alpha=\frac{T_{MPPT}}{T_{X}}\]  
Figure 4 shows the classical, usually underdamped, transient response of a direct control MPPT [1; 2; 3; 4] oscillating around the MPP during steady irradiance conditions. The system operating at a point on the right of the MPP (\(v_{k-1}\)) is perturbed in the correct direction by the MPPT to a point on the left of it (\(v_{k}\)). The classical P&O method uses only  
Figure 3: Perturbation performed by the Td-MPPT algorithm around MPP under increasing irradiance. \(\Delta p\) to decide whether the perturbation was made in the correct direction or not. For this example, since \(\Delta p>0\), P&O would assume that the next perturbation should be made in the same direction. However, doing so would drive the operating point further away from MPP since \(v_{k}\) is already to the left of MPP. This phenomenon is why P&O is known to oscillate between three operating points around MPP. It can be noted that the overshoot can provide additional information about the PV-curve shape since it occurs in the same direction as \(\delta_{\rm k}\). By applying this principle, Td-MPPT can decide whether perturbing in the same direction again would bring the system closer to MPP or not.