## 3 Theoretical Framework
### PV System  
PV systems have three main parts: PV cell, MPPT, and DC/AC converter. Figure 4 shows the PV model used in the present study. The P&O and incremental conductance (INC) algorithms are the control solutions used for MPPT in most hybrid systems [59; 60]. Compared to adaptive controllers, these controllers are weaker in flexibility and slow in obtaining the solar cell's MPP but are more simple, applicable, and require less computation. A Z-source converter was used in the current study to connect the solar farms to the grid.  
Figure 3: Studied 9 MW wind farm compromising a DFIG and an ANFIS controller.  
Using the [PERSON]'s current law, the PV cell's output current can be expressed as follows:  
\[I_{Cell}\!=\!I_{\text{Photon}}\!-\!I_{\text{Diodel}}\!-\!I_{\text{Diode2}}\!-\! \left(\frac{V_{Cell}\!+\!I_{Cell}R_{SE}}{R_{\text{Parallel}}}\right), \tag{18}\]  
where \(I_{\text{Photon}}\) denotes the source of the circuit, which can be modeled as follows:  
\[I_{\text{Photon}}\!=\![I_{\text{Photon\_STC}}\!+\!K_{S}(T_{\text{C}}\!-\!T_{ STC})]\times\frac{G}{G_{STC}}, \tag{19}\]  
with \(I_{\text{Photon\_STC}}\) denoting the cell currently under standard test conditions (STCs), \(T_{\text{C}}\) and \(T_{\text{STC}}\) denoting ambient and STC temperatures in Celsius, \(G\) and \(G_{STC}\) denoting the surface and STC irradiance of the cell, and \(K_{S}\) denoting the coefficient of the cell short circuit current, normally provided by its manufacturer [61]. In this study, both the diode reverse saturation currents \((Is_{i})\) were assumed equal, therefore:  
\[I_{Diodei}\!=\!\!I_{\text{Si}}\!\left(e^{\frac{V_{Cell}+R_{SE}I_{Cell}}{R_{ \text{Thermal}}}}-1\right), \tag{20}\]  
where \(\eta\) denotes ideality factor, a number between 1 and 2 which typically increases as the current decreases and \(V_{\text{Thermal}}\) is equal to 26 mv. Figure 5 shows the used Z-source converter [62].  
Due to the main circuit state, the Z-source converter operates in two shoot-through and non-shoot-through states. In the shoot-through state, the switches of a phase leg are connected, and in a non-shoot-through state, the upper switch of a phase leg and the lower switch of another phase leg are on. If the capacitors and inductors of the Z-source  
Figure 4: Equivalent circuit of a PV system.  
Figure 5: Block diagram of the studied PV and Z-source converter system.  
impedance network are assumed to be similar, (\(C_{1}\)=\(C_{2}\)=\(C\) and \(L_{1}\)= \(L_{2}\)=\(L\) ), the maximum output of the Z-source output can be calculated as follows [63]:  
\[\vartheta_{i}\)=\(V_{C}\)\(-\)\(v_{L}\)= 2\(V_{C}\)\(-\)\(V_{\rm dc}\)=\(\frac{T_{sw}}{T_{1}\)\(-\)\(T_{0}\)\(V_{\rm dc}\)=\(BV_{\rm dc}\),  
where \(B\) denotes the boost factor, which can be obtained as follows:  
\[B\)=\(\frac{T_{sw}}{T_{1}\)\(-\)\(T_{0}\)\(=\)\(\frac{1}{1-2(T_{0}/T_{sw})}\)\(=\)\(\frac{1}{1-2D}\)\(\geq\)\(1\),  
where \(T_{0}\) and \(T_{1}\) denote the times of total shoot-through and total non-shoot-through states, respectively, the switching cycle, \(T_{sw}\), is equal to \(T_{0}\)\(+\)\(T_{1}\), and \(D\) is defined as the shoot-through duty ratio, which is equal to \(T_{0}\)\(/\)\(T_{sw}\).  
An ANFIS controller and a PWM system were used to control the Z-source converter. The frequency of the modulating signals was assumed to be equal to 60 Hz. Other details of the used Z-source converter can be found in [63]. Two ANFIS controllers in the wind turbine and the PV Z-source converter improve the dynamic response, reduce the power and current fluctuations, increase reliability, and improve the grid voltage profile. Regarding the robustness of solar systems with an ANFIS and Z-source converter approach, Refs. [46; 64] confirmed the interesting results. Simultaneously, the current work uses a motion detection algorithm to improve the performance of the incremental conductance (INC) MPPT controller.