## 3 Incremental Conductance with Integral Compensator (IC-INC)
### IC-INC Method and Algorithm  
The IC-INC approach is based on the incremental conductance Equation (6), written as (9).  
\[V_{PV}\frac{dI_{PV}}{dV_{PV}}+I_{PV}=0, \tag{9}\]  
This equation means that in steady state, the conductance \(I_{PV}/V_{PV}\) equals its incremental value \(dI_{PV}/dV_{PV}\). However, during transients or during the convergence to steady state, the algebraic sum of the incremental conductance value \(dI_{PV}/dV_{PV}\) with the conductance \(I_{PV}/V_{PV}\) will not be zero; this will lead to a tracking error \(e_{MPPT}\), rewritten in (10) as the error, or deviation, of the negative feedback system.  
\[V_{PV}\frac{dI_{PV}}{dV_{PV}}-(-I_{PV})=e_{MPPT}, \tag{10}\]  
This tracking error value \(e_{MPPT}\) should be enforced to zero within a finite amount of time using a suitable closed-loop controller. Upon dividing all the components of (10) by \(V_{PV}\), (11) is obtained.  
\[\frac{dI_{PV}}{dV_{PV}}-\left(-\frac{I_{PV}}{V_{PV}}\right)=\frac{e_{MPPT}}{V _{PV}}, \tag{11}\]  
Assuming the boost converter is driven by a current controller (e.g., the hysteresis controller), the inductor current \(I_{L}\) dynamics will track the current reference \(I_{L}\approx I_{L_{PV}}\approx I_{PV}\) with a small delay. This relatively small delay can be approximated using a first-order low-pass filter dynamics with a dominant pole at \(-1/Tc\). Then, the term \(-I_{PV}/V_{PV}\) in (11) can be written as follows.  
\[-\frac{I_{PV}}{V_{PV}}\approx-\frac{I_{L_{ref}}}{V_{PV}}\frac{1}{1+sTc}, \tag{12}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(V_{\alpha}\) (PV open circuit voltage) & 36.84 V \\ \(I_{sc}\) (PV short circuit current) & 8.32 A \\ \(V_{MPP}\) (PV MPP voltage) & 30.72 V \\ \(I_{MPP}\) (PV MPP current) & 7.83 A \\ \(P_{MPP}\) (PV MPP power) & 240.54 W \\ \(L\) (inductor) & 300 \(\mu\)H \\ \(C_{in}\) (input capacitor) & 150 \(\mu\)F \\ \(C_{out}\) (output capacitor) & 150 \(\mu\)F \\ \(f_{\rm{sw}}\) (switching frequency) & 50 kHz \\ \(V_{mg}\) (microgrid voltage) & 48 V \\ \(R_{mg}\) (microgrid resistance) & 50 m\(\Omega\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of implemented topology.  
If we consider the capacitor at the output of the PV \(C_{in}\) is parallel with an equivalent PV resistor, given by \(R_{PV}=V_{MPP}/I_{MPP}\), \(T_{c}\) can be estimated as follows.  
\[T_{c}=C_{in}R_{PV}=C_{in}\frac{V_{MPP}}{I_{MPP}}\, \tag{13}\]  
To ensure MPPT tracking (zero steady-state error), an integral controller \(K_{i}/s\) can then be devised using (10) and (12) (Figure 9). The integral gain \(K_{i}\) can be computed using the closed-loop transfer function represented in Equation (14).  
\[\frac{-\frac{I_{PV}}{V_{PV}}}{\frac{dI_{PV}}{dV_{PV}}}\ \approx\ \frac{-\frac{K_{i}}{T_{c}}}{s^{2}+\frac{s}{T_{c}}-\frac{K_{i}}{T_{c}}}, \tag{14}\]  
Compared with the canonical form of a second-order system \(s^{2}+2\xi\omega_{n}+\omega_{n}^{2}\), (15) is obtained. To ensure stability, \(K_{i}\) should be negative with a damping factor close to unity \(\xi\approx 1\).  
\[\omega_{n}^{2}=-\frac{K_{i}}{T_{c}};K_{i}=-\frac{1}{2\ \xi^{2}T_{c}}, \tag{15}\]